# Colloque des sciences mathématiques du Québec

Organisé par le CRM en collaboration avec l’Institut des sciences mathématiques (ISM), le Colloque des sciences mathématiques du Québec offre une tribune à des mathématiciens de grande réputation, qui sont invités à prononcer des conférences d’intérêt actuel et général, et accessibles à l’ensemble de la communauté mathématique québécoise. La tradition veut que ces conférences soient aussi qualitatives et non-techniques que possible afin d’être accessibles aux étudiants aux cycles supérieurs en mathématiques et en statistique.

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# Horaire des colloques

Pour avoir accès au lien de la réunion Zoom, veuillez vous inscrire au colloque des sciences mathématiques du Québec. Une seule inscription vous donne accès à toutes les activités.

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# Colloques à venir Changer d'affichage

Heure de début Titre Conférencier
2021-01-22 15:00 À venir / TBA Robert Haslhofer (Université de Toronto, Canada)
2021-01-29 15:30 À venir / TBA Jonathan Wakefield (University of Washington, USA)
2021-02-05 15:00 À venir / TBA Egor Shelukhin (Université de Montréal, Canada)
2021-02-12 15:30 À venir / TBA Mikael Kuusela (Carnegie Mellon University, USA)
2021-02-26 15:00 À venir / TBA Jean-Pierre Demailly (Université Grenoble Alpes, France)
2021-03-12 15:30 À venir / TBA Jay Breidt (Colorado State University, USA)

22 janvier 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

### À venir / TBA

Lauréat 2020 du prix de mathématiques André-Aisenstadt

29 janvier 2021 de 15 h 00 à 14 h 00 (heure de Montréal/Miami)

### À venir / TBA

5 février 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

### À venir / TBA

Lauréat 2020 du prix de mathématiques André-Aisenstadt

12 février 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

### À venir / TBA

26 février 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

### À venir / TBA

12 mars 2021 de 15 h 00 à 16 h 00 (heure de Montréal/Miami)

# Colloques passés Changer d'affichage

Heure de début Titre Conférencier
2020-11-27 15:00 Moduli of unstable objects in algebraic geometry Frances Kirwan (Université d'Oxford)
2020-11-20 15:00 Hodge Theory of p-adic varieties Wieslawa Niziol (CNRS, Université Sorbonne)
2020-11-13 15:30 Approximate Cross-Validation for Large Data and High Dimensions Tamara Broderick (Massachusetts Institute of Technology, USA)
2020-10-16 15:00 Trigonometric functions and modular symbols Nicolas Bergeron (École normale supérieure (Paris), France)
2020-10-09 15:00 Hodge Theory and Moduli Phillip Griffiths (Institute for Advanced Study, Princeton, USA)
2020-10-02 15:30 Data Science, Classification, Clustering and Three-Way Data Paul McNicholas (Université McMaster, Canada)
2020-09-11 16:00 Machine Learning for Causual Inference Stefan Wager (Stanford University, USA)
2020-06-19 16:00 Quantitative approaches to understanding the immune response to SARS-CoV-2 infection Morgan Craig (Université de Montréal)
2020-04-17 16:00 Observable events and typical trajectories in finite and infinite dimensional dynamical systems Lai-Sang Young (New York University Courant)
2019-05-16 16:00 Introduction to birational classification theory in dimension three and higher Jungkay A. Chen (National Taiwan University)
2019-05-10 16:00 Quantum Jacobi forms and applications Amanda Folsom (Amherst College)
2019-05-03 16:00 The stochastic heat equation and KPZ in dimensions three and higher Lenya Ryzhik (Stanford University)
2019-04-26 16:00 Distinguishing finitely presented groups by their finite quotients Alan W. Reid (Rice University)
2019-04-12 16:00 Linking in torus bundles and Hecke L functions Nicolas Bergeron (École normale supérieure (Paris), France)
2019-03-29 16:00 Principal Bundles in Diophantine Geometry Minhyong Kim (University of Oxford)
2019-03-22 16:00 Flexibility in contact and symplectic geometry Emmy Murphy (Northwestern University)
2019-03-19 14:30 Special Colloquium : A constructive solution to Tarski’s circle squaring problem Andrew Marks (UCLA)
2019-03-15 16:00 Persistent homology as an invariant, rather than as an approximation Shmuel Weinberger (University of Chicago)
2018-11-02 16:00 The complexity of detecting cliques and cycles in random graphs
2018-09-28 16:00 A delay differential equation with a solution whose shortened segments are dense Hans-Otto Walther (Universität Giessen)
2018-09-21 16:00 Algebraic structures for topological summaries of data Ezra Miller (Duke University)
2018-05-04 16:00 Klein­-Gordon­-Maxwell­-Proca systems in the Riemannian setting Emmanuel Hebey (Université de Cergy-­Pontoise)
2018-04-13 16:00 Local-­global principles in number theory Eva Bayer (École Polytechnique Fédérale de Lausanne)
2018-02-23 16:00 Cluster theory of the coherent Satake category Sabin Cautis (University of British Columbia)
2018-02-16 16:00 Quantum n-­body problem: generalized Euler coordinates (from J-­L Lagrange to Figure Eight by Moore and Ter-­Martirosyan, then and today) Alexandre Turbiner (UNAM)
2018-02-16 15:30 The Law of Large Populations: The return of the long-­ignored N and how it can affect our 2020 vision Xiao-Li Meng (Harvard University)
2018-02-09 16:00 Persistence modules in symplectic topology Egor Shelukhin (Université de Montréal, Canada)
2018-01-12 16:00 What is quantum chaos Semyon Dyatlov (UC Berkeley / MIT)
2017-12-08 16:00 Primes with missing digits James Maynard (University of Oxford)
2017-11-24 15:30 150 years (and more) of data analysis in Canada David R. Bellhouse (Western University, London, Ontario)
2017-11-24 15:30 Complex analysis and 2D statistical physics Stanislav Smirnov (University of Geneva and Skolkovo Institute of Science and Technology)
2017-11-17 16:00 Recent progress on De Giorgi Conjecture Jun-Cheng Wei (UBC)
2017-10-27 16:00 Beneath the Surface: Geometry Processing at the Intrinsic/Extrinsic Interface Justin Solomon (Massachusetts Institute of Technology)
2017-10-13 16:00 Supercritical Wave Equations Avi Soffer (Rutgers University)
2017-09-29 16:00 The first field John H. Conway (Princeton University)
2017-09-15 16:00 Isometric embedding and quasi­-local type inequality Siyuan Lu (Rutgers University, Lauréat 2017 du Prix Carl Herz / 2017 Carl Herz Prize Winner)
2017-05-05 16:00 From the geometry of numbers to Arakelov geometry Gerard Freixas (Institut de Mathématiques de Jussieu)
2017-04-21 16:00 Introduction to the Energy Identity for Yang-­Mills Aaron Naber (Northwestern University)
2017-03-31 16:00 PDEs on non­-smooth domains Tatiana Toro (University of Washington)
2017-03-17 15:30 Inference in Dynamical Systems Sayan Mukherjee (Duke University)
2017-03-10 16:00 Probabilistic aspects of minimum spanning trees Louigi Addario-Berry (Université McGill)
2017-02-24 16:00 Spreading phenomena in integrodifference equations with overcompensatory growth function Frithjof Lutscher (Université d'Ottawa)
2017-02-10 16:00 Knot concordance Mark Powell (UQAM)
2017-01-20 16:00 The Birch­-Swinnerton Dyer Conjecture and counting elliptic curves of ranks 0 and 1 Christopher Skinner (Princeton University)
2016-12-02 16:00 Partial differential equations of mixed elliptic-­hyperbolic type in mechanics and geometry
2016-12-01 15:30 High­-dimensional changepoint estimation via sparse projection Richard Samworth (University of Cambridge)
2016-11-26 16:00 Around the Möbius function Maksym Radziwill (McGill University)
2016-11-04 16:00 The nonlinear stability of Minkowski space for self­-gravitating massive fields Philippe G. LeFloch (Université Pierre et Marie Curie, Paris 6)
2016-10-28 15:30 Efficient tests of covariate effects in two­-phase failure time studies Jerry Lawless (University of Waterloo)
2016-10-21 16:00 Integrable probability and the KPZ universality class Ivan Corwin (Columbia University)
2016-10-14 16:00 Rigorously verified computing for infinite dimensional nonlinear dynamics: a functional analytic approach Jean-Philippe Lessard (McGill University)
2016-09-30 16:00 Notions of simplicity in low­-dimensions Liam Watson (Université de Sherbrooke)
2016-09-16 16:00 Statistical Inference for fractional diffusion processes B.L.S. Prakasa Rao (CR Rao Advanced Institute, Hyderabad, India)
2016-09-16 16:00 Cubature, approximation, and isotropy in the hypercube Nick Trefethen (University of Oxford)
2015-04-09 16:00 Modular generating series and arithmetic geometry Stephen S. Kudla (University of Toronto)
2015-04-02 16:00 Uniqueness of blowups and Lojasiewicz inequalities William Minicozzi (Massachusetts Institute of Technology)
2015-03-26 16:00 Left-orderings of groups and the topology of 3-manifolds Steve Boyer (UQAM)
2015-03-19 16:00 Integrable probability Alexei Borodin (Massachusetts Institute of Technology)
2015-03-12 16:00 The upper half-planes Pierre Colmez (CNRS & Paris VI Jussieu)
2015-03-05 16:00 Periods Sophie Morel (Princeton University)
2015-02-26 16:00 Categorification in representation theory Alistair Savage (University of Ottawa)
2015-02-19 16:00 Irrationality proofs, moduli spaces and dinner parties Francis Brown (IHES, Bures-sur-Yvette)
2015-02-12 16:00 Influence des côtes sur les courants marins Laure Saint-Raymond (École normale supérieure, Paris)
2015-02-05 16:00 Cobordism and Lagrangian topology Octav Cornea (Université de Montréal)
2015-01-29 16:00 Spectres et pseudospectres Thomas Ransford (Université Laval)
2015-01-22 16:00 On the usefulness of mathematics for insurance risk theory - and vice versa Hansjoerg Albrecher (HEC, Lausanne)
2015-01-15 16:00 Functional data analysis and related topics Fang Yao (University of Toronto - Lauréat du Prix CRM-SSC)
2014-12-04 16:00 Algebraic combinatorics and finite reflection groups
2014-11-20 16:00 High-dimensional phenomena in mathematical statistics and convex analysis Martin Wainwright (University of California, Berkeley)
2014-11-13 16:00 Recent advances in the arithmetic of elliptic curves Kartik Prasanna (University of Michigan, Ann Arbor)
2014-11-06 16:00 The Cubical Route to Understanding Groups Dani Wise (McGill University)
2014-10-30 16:00 A Pedestrian Approach to Group Representations Georgia Benkart (University of Wisconsin-Madison)
2014-10-09 16:00 Applications of additive combinatorics to homogeneous dynamics Alex Kontorovich (Rutgers University)
2014-05-02 16:00 Eigenvarieties Eric Urban (Columbia University)
2014-04-11 16:00 Flat surfaces and determinants of Laplaciancs Alexey Kobotov (Concordia University)
2014-04-04 16:00 Interaction between internal and surface waves in a two layers fluid Catherine Sulem (University of Toronto)
2014-03-21 16:00 Small gaps between primes James Maynard (University of Oxford)
2014-03-14 16:00 Pretentious multiplicative functions Dimitris Koukoulopoulos (Université de Montréal)
2014-02-14 16:00 Tores plats en 3D Vincent Borrelli (Université Claude Bernard-Lyon 1)
2014-02-07 16:00 Degenerate diffusions arising in population genetics Charles Epstein
2014-01-17 16:00 Nondegenerate curves and pentagram maps Boris Khesin (University of Toronto)
2013-12-13 16:00 Combinatorics and geometry of KP solitons and application to tsunami Yuji Kodama (Ohio State University)
2013-11-29 16:00 Higher Pentagram Maps via Cluster Mutations and Networks on Surfaces Michael Gekhtman (Université of Notre-Dame)
2013-11-22 16:00 Exact formulas in random growth Jeremy Quastel (University of Toronto)
2013-11-15 16:00 Singular (arithmetic) Riemann Roch Revisited Henri Gillet (University of Illinois, Chicago)
2013-10-25 16:00 Un survol élémentaire de la topologie symplectique sans homologie de Floer et sans théorie de jauge. François Lalonde (Université de Montréal)
2013-10-18 16:00 The Sato-Tate conjecture Ram Murty (Queen's University)
2013-09-20 16:00 Quasiperiodic Schrödinger operators Svetlana Jitomirskaya (UC Irvine / Chaire Aisenstadt 2018)
2013-04-12 16:00 Quantum correlations and Tsirelson's problem Narutaka Ozawa (RIMS, Kyoto University)
2013-04-05 16:00 Integral structures in p-adic representations Ehud de Shalit (Hebrew University of Jerusalem)
2013-03-28 16:00 Moser averaging Victor Guillemin (Massachusetts Institute of Technology)
2013-03-01 16:00 Mathematical Models for River Ecosystems Frithjof Lutscher (Université d'Ottawa)
2013-02-15 16:00 Eigenproblems, numerical approximation and proof Nilima Nigam (Simon Fraser University, Canada)
2013-02-08 16:00 Pentagram Map, Twenty Years After Sergei Tabachnikov (Pennsylvania State University)
2013-02-01 16:00 Proof of a 35 Year Old Conjecture for the Entropy of SU(2) Coherent States, and its Generalization Elliott Lieb (Princeton University)
2013-01-25 16:00 Global rigidity in contact topology Sheila Margherita Sandon (CNRS, Nantes and CRM)
2012-12-07 16:00 Igusa integrals Yuri Tschinkel (New York University and Simons Foundation)
2012-11-16 16:00 On the Doi Model for the suspension of rod-like molecules & related equations Konstantina Trivisa (University of Maryland)
2012-11-02 16:00 Dissipative motion from a Hamiltonian point of view Jürg Fröhlich (ETH Zurich)
2012-10-12 16:00 Symmetry and Reflection Positivity Rupert Frank (Princeton University and Caltech)
2012-09-21 16:00 Geometry of complex surface singularities Walter Neumann (Barnard College, Columbia University)
2012-09-14 16:00 A glimpse at the differential topology and geometry of optimal transportation Robert McCann (University of Toronto)
2012-02-03 16:00 Equivalence relations, random graphs and stochastic homogenization Vadim Kaimanovich
2012-01-27 16:00 Rational billiards and the SL(2,R) action on moduli space Alex Eskin
2012-01-20 16:00 Rational curves and rational points Jason Starr
2012-01-13 16:00 Probabilité et Physique Statistique des Systèmes Désordonnés Louis-Pierre Arguin
2011-12-16 16:00 Disordered Bosons: A Complex Geometric Viewpoint Alan Huckleberry
2011-12-09 16:00 Balanced Splitting Methods / Infinite Matrices Gilbert Strang
2011-11-25 16:00 Groups with good pedigrees, or superrigidity revisited Alex Furman
2011-11-18 16:00 Tricks in Spectral Theory Michael Levitin
2011-11-11 16:00 Domains with non-compact automorphism groups Bun Wong
2011-11-04 16:00 Teichmuller spaces of Riemann surfaces with holes and algebras of geodesic functions Leonid Chekhov
2011-10-21 16:00 Divisors on graphs Sergey Norin
2011-09-30 16:00 Variation with p of the number of solutions mod p of a system of polynomial equations Jean-Pierre Serre
2011-09-23 16:00 On Langlands functoriality Jayce Getz
2011-09-16 16:00 Symplectic topology in the large - from Morse to Floer and beyond
2011-09-15 16:00 Number Theory and Dynamical Systems: A Survey
2011-09-09 16:00 Non-trivial convex bodies with maximal sections of constant volume Fedor Nazarov
2011-06-10 16:00 Symplectic homogenization Claude Viterbo
2011-05-06 16:00 Embedding questions in Symplectic Geometry Dusa McDuff
2011-04-15 16:00 Rubik's Cube in Twenty Moves or Less Morley Davidson
2011-04-08 16:00 Conférence du Prix CRM-Fields-PIMS 2011 Mark Lewis
2011-04-01 16:00 Number Theory and Dynamical Systems: A Survey Joseph Silverman
2011-03-25 16:00 Function theory on symplectic manifolds Leonid Polterovich
2011-03-18 16:00 Geometry of measures Tatiana Toro (University of Washington)
2011-03-11 16:00 Variational Methods in Materials and Imaging Irene Fonseca
2011-03-04 16:00 Some random thoughts about Cauchy's functional equation Dan Stroock
2011-02-18 16:00 Representation theory of semisimple groups: classical, quantum, geometric, categorical Joel Kamnitzer
2011-02-11 16:00 The 5-electron case of Thompson's problem Richard Schwartz
2011-02-04 16:00 Mahler measure as special values of $L$-functions Matilde Lalin
2011-01-28 16:00 Homotopy Theory and Spaces of Representations Alejandro Adem
2011-01-14 16:00 Revisiting fracture mechanics - The variational standpoint Gilles Francfort
2010-11-26 16:00 Semi-algebraic optimization theory
2010-11-19 16:00 Ramanujan reaches his hand from his grave and snatches your theorems from you Bruce Bernt
2010-10-29 16:00 The Thermodynamic Limit of Coulomb Quantum Systems Mathieu Lewin
2010-10-22 16:00 Stochastic homogenization and related problems Claude LeBris
2010-10-15 16:00 Grand Challenges in Complexity Theory Alexander Razborov
2010-09-24 16:00 Pointwise estimates and nonlinear stability of waves Bjorn Sandstede
2010-09-17 16:00 Régulation d'évolutions «viables» dans un environnement en avenir incertain Jean-Pierre Aubin
2010-08-10 16:00 The average rank of elliptic curves Manjul Bhargava
2010-04-16 16:00 Surface Evolution under Curvature Flows - Existence and Optimal Regularity Panagiota Daskalopoulos
2010-04-09 16:00 Magnetic monopoles and projective geometry Nigel Hitchin
2010-03-19 16:00 Word maps over simple groups Michael Larsen
2010-03-12 16:00 Recent progress on the arithmetic of noncongruence modular forms Winnie Li
2010-03-05 16:00 Random Schrodinger Operators and Random Matrices Balint Virag
2010-02-26 16:00 Iwasawa Theory John Croates
2010-02-19 16:00 Large scale behaviour of the continuum random polymer and KPZ Jeremy Quastel (University of Toronto)
2010-02-12 16:00 Orbitopes Frank Sottile
2010-02-05 16:00 Optimal multidimensional pricing facing informational asymmetry Robert McCann (University of Toronto)
2010-01-29 16:00 The Euler-Kronecker constant of a number field Kumar Murty
2010-01-15 16:00 The orbifold vertex: counting curves on orbifolds by counting piles of colored boxes Jim Bryan
2010-01-10 16:00 Some features and challenges of the Navier-Stokes-alpha-beta equation Eliot Fried
2010-01-08 16:00 Diophantine equations: what numbers reveal about shape and structure Henri Darmon (McGill)
2009-12-18 16:00 La nouvelle géométrie algébrique réelle François Lalonde (Université de Montréal)
2009-12-04 16:00 Galois modules in arithmetic and geometry Erez Boas
2009-11-27 16:00 Canonical metrics on Kähler manifolds Shing-Tung Yau
2009-11-20 16:00 New Invariants on Algebraic Cycles James Lewis
2009-11-06 16:00 Kakeya-Nikodym averages and Lp norms of eigenfunctions Christopher Sogge
2009-10-30 16:00 p-adic variation in the theory of automorphic forms Glenn Stevens
2009-10-09 16:00 What is a Galois Representation? Ravi Ramakrishna
2009-09-25 16:00 Structure of attractors for (a,b)-continued fraction transformations Svetlana Katok
2009-04-24 16:00 Ricci Flow, Monge-Ampere Equation and Algebraic Spaces Gang Tian (Princeton University)
2009-04-17 16:00 Arithmetic Laplacian Alexandru Buium (University of New Mexico)
2009-04-03 16:00 Problème de Riemann-Hilbert sur la sphère et combinatoire des systèmes de racines Olivier Schiffmann (CNRS ENS Ulm)
2009-03-27 16:00 Undecidability in Number Theory Bjorn Poonen (Massachusetts Institute of Technology)
2009-03-20 16:00 Lecture by André-Aisenstadt 2009 Prize Recipient Valentin Bloomer (University of Toronto)
2009-03-13 16:00 Infinitesimal Hilbert 16th Problem Sergei Yakovenko (Weizmann Institute, Rehovot, Israel, and Fields Institute, Toronto, Canada)
2009-02-20 16:00 Branching Random Walk and Searching in Trees Louigi Addario-Berry (Université McGill)
2009-02-13 16:00 Mathematics in the Light of Metaphor and Ambiguity. William Byers (Concordia University)
2009-02-06 16:00 Nonlinear High Dimensional PDE's in High Intensity Laser-matter Interactions-New Mathematics for a New Science. André D. Bandrauk (Université de Sherbrooke)
2009-01-30 16:00 Around Tarski's Problems Alexei Miasnikov (McGill University)
2009-01-23 16:00 Statistics for the Zeroes and Traces of Zeta Functions over Finite Fields Chantal David (Concordia University)
2008-12-19 16:00 Spectral-Galerkin Methods for High-Dimensional PDEs Jie Shen (Purdue University)
2008-12-12 16:00 Solvable Schroedinger Equations and Representation theory Alexander Turbiner (CRM and National University of Mexico, Mexico)
2008-12-05 16:00 Fundamental Interactions and Classical or Quantum Geometries Robert Coquereaux (CPT, Luminy-Marseille)
2008-11-28 16:00 Shape optimization for low eigenvalues of the Laplace operator Alexandre Girouard (Université Laval)
2008-11-21 16:00 Turbulence from Statistical Theory to Wigner Measure Claude Bardos (Université Paris-Diderot (Paris 7))
2008-11-14 16:00 Overcrowding and Undercrowding of Random Zeros on Complex Manifolds Bernard Shiffman (Johns Hopkins University)
2008-11-07 16:00 Combinatorial Hopf Algebras Jean-Louis Loday (CNRS, Strasbourg)
2008-10-31 16:00 Dilute Quantum Gases Robert Seiringer (Princeton University)
2008-10-24 16:00 Nonequilibrium Statistical Mechanics and Smooth Dynamical Systems David P. Ruelle (Institut des Hautes Études Scientifiques)
2008-10-17 16:00 Random Graphs: New models and the Internet Svante Janson (Uppsala University)
2008-10-10 16:00 Visual Chaos: Dispersing, Defocusing, Absolute Focusing and Astigmatism Leonid Bunimovich (Georgia Institute of Technology)
2008-10-03 16:00 Some Calculus of Variations Problems in Quantum Mechanics Elliott Lieb (Princeton University)
2008-09-26 16:00 PDE aspects of the Navier-Stokes Equations Vladimir Sverak (University of Minnesota)
2008-09-19 16:00 Some classes of random Hermitian matrices: F(Tr(V(M)) Instead of Tr(V(M)) Kenneth McLaughlin (The University of Arizona)
2008-09-12 16:00 The Algebra and Geometry of Random Surfaces Andrei Okounkov (Princeton University)
2008-09-05 16:00 Stability and Compactification of the Moduli of Abelian Varieties Iku Nakamura (Hokkaido University)
2008-04-25 16:00 New Ricci flow invariant curvature conditions and applications Burkhard Wilking (Mathematisches Institut der Uni Munster)
2008-04-18 16:00 Flexibility of singular Einstein metrics Rafe Mazzeo (Stanford University)
2008-04-11 16:00 Nodal lines of eigenfunctions, geodesics and complex analysis Steve Zelditch (John Hopkins University)
2008-04-04 16:00 Unsolved mysteries of solutions to PDEs near the boundary Vladimir Maz'ya (Ohio State University, University of Liverpool, and Linkšping University)
2008-03-14 16:00 Greedy algorithms and complexity for nonnegative matrix factorization Stephen Vavasis (University of Waterloo)
2008-03-07 16:00 Tsunami asymptotics Michael Berry (Bristol University)
2008-02-29 16:00 What is a tau function? John Harnad (Concordia University and Centre de recherches mathématiques)
2008-02-22 16:00 Combinatorics as geometry Fernando Rodriguez Villegas (University of Texas at Austin)
2008-02-15 16:00 Differential Groups and Differential Relations Michael F. Singer (North Carolina State University)
2008-02-08 16:00 Arithmetic partial differential equations Alexandru Buium (University of New Mexico)
2008-02-01 16:00 Quantization and chiralization Victor Kac (Massachusetts Institute of Technology)
2008-01-18 16:00 From Elie Cartan to Gerard Debreu: some applications of exterior differential calculus to economic theory Ivar Ekeland (PIMS, University of British Columbia)
2008-01-11 16:00 L'invariance de Thomae de 3F2 par le groupe symétrique S5 et les produits de matrices (2,2) aléatoires Gérard Letac (Université Paul Sabetier)
2008-01-04 16:00 Attractors and invariant measures in low-dimensional dynamical systems Michael Jakobson (University of Maryland)
2007-12-14 16:00 Smale's 17th Problem Michael Shub (University of Toronto)
2007-11-30 16:00 Exact Solution of the Six-Vertex Model of Statistical Physics Pavel Bleher (Indiana University-Purdue University Indianapolis)
2007-11-23 16:00 Nilsequences in Additive Combinatorics Ben Green (Cambridge University)
2007-11-16 16:00 Unexpected Properties of Dense Packings of Spheres Charles Radin (University of Texas at Austin)
2007-11-16 16:00 Symmetries of the field of algebraic numbers John Tate (University of Texas at Austin)
2007-11-02 16:00 The geometry of numbers, old and new Akshay Venkatesh (Institute for Advanced Study)
2007-10-26 16:00 Pseudo-Riemannian geodesics and billiards Boris Khesin (University of Toronto)
2007-10-12 16:00 A Lattice Boltzmann Model for Single- and Multi-Phase Fluid Flows Tim Phillips (Cardiff University, UK)
2007-10-05 16:00 Geometric and numerical rigidity for Lagrangian submanifolds Octav Cornea (Université de Montréal)
2007-09-14 16:00 Geometry and Dynamics of Surface Group Representations William Goldman (University of Maryland)
2007-09-07 16:00 Poincaré inequality and the structure of complete manifolds Peter Li (University of California, Irive)
2007-05-04 16:00 New analytic techniques in algebraic geometry Jean-Pierre Demailly (Université Grenoble Alpes, France)
2007-04-27 16:00 Counting rational points and rational curves: from Waring's problem to quantum cohomology Yuri Manin (Max-Planck-Institut fuer Mathematik)
2007-04-20 16:00 Equisingularity, Multiplicity, and Dependence Steven Kleiman (MIT)
2007-04-13 16:00 Representation densities and arithmetic geometry Stephen S. Kudla (University of Toronto)
2007-03-30 16:00 Polynomial progressions in primes Tamar Ziegler (Michigan University)
2007-03-23 16:00 Extreme heating of the sun's atmosphere and the topology of magnetic field lines Ed Stredulinsky (University of Wisconsin-Richland)
2007-03-16 16:00 Mathematical issues and opportunities in self assembly Michael Brenner (Harvard University)
2007-03-09 16:00 What do we know about four dimensional manifolds Tomasz Mrowka (MIT)
2007-03-02 16:00 Water waves over a varying bottom Catherine Sulem (University of Toronto)
2007-02-23 16:00 Integrable Combinatorics Philippe Di Francesco (Service de Physique Théorique, CEA Saclay (France))
2007-02-09 16:00 Second Hamiltonian Paths and Nash Equilibria Jack Edmonds (University of George Washington)
2007-02-02 16:00 Une courte histoire du modèle d'Ising Yvan Saint-Aubin (Université de Montréal)
2007-01-26 16:00 Turn Table, Tippy Tops, Tapped Turtles Tadashi Tokieda (Cambridge)
2007-01-19 16:00 Eigenfunctions: limits, nodal sets and critical points Dmitry Jakobson (McGill)
2007-01-12 16:00 Les marches aléatoires et les algorithmes MCMC Jeffrey S. Rosenthal (University of Toronto)
2006-12-01 16:00 The Sato-Tate conjecture Richard Taylor (Harvard University)
2006-11-24 16:00 The integral geometry of random sets Jonathan Taylor (Université de Montréal)
2006-11-17 16:00 The Trouble with Molecular Dynamics Paul Tupper (McGill University)
2006-11-10 16:00 hp-ADAPTIVE FINITE ELEMENTS a Quest for Exponential Convergence Leszek F. Demkowicz (The University of Texas at Austin)
2006-11-03 16:00 Why is nonequilibrium statistical mechanics so hard to understand? David P. Ruelle (Institut des Hautes Études Scientifiques)
2006-10-20 16:00 RIGIDITY OF ORBIT STRUCTURE FOR ACTIONS OF HIGHER RANK ABELIAN GROUPS KAM-THEORY AND ALGEBRAIC K-THEORY Anatole Katok (Penn State University)
2006-10-13 16:00 Randomness, and its effects on Spectra Jean-Pierre Bourguignon (Inst. des Hautes Etudes Scientifiques (IHES))
2006-10-06 16:00 Randomness, and its effects on Spectra Michael Aizenman (Princeton University)
2006-09-29 16:00 Understanding the curvature/comprendre la courbure Joseph Kohn (Princeton University)
2006-09-22 16:00 Quantum Mechanics, the Stability of Matter, and Quantum Electrodynamics Elliott Lieb (Princeton University)
2006-09-08 16:00 Applications of an asymptotic expansion for the one-point function of random matrix theory: Loop equations, partition function, large deviation principles Kenneth McLaughlin (The University of Arizona)

27 novembre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Moduli of unstable objects in algebraic geometry

Moduli spaces arise naturally in classification problems in geometry. The study of the moduli spaces of nonsingular complex projective curves (or equivalently of compact Riemann surfaces) goes back to Riemann himself in the nineteenth century. The construction of the moduli spaces of stable curves of fixed genus is one of the classical applications of Mumford's geometric invariant theory (GIT), developed in the 1960s; many other moduli spaces of 'stable' objects can be constructed using GIT and in other ways. A projective curve is stable if it has only very mild singularities (nodes) and its automorphism group is finite; similarly in other contexts stable objects are usually better behaved than unstable ones.

The aim of this talk is to explain how recent methods from a version of GIT for non-reductive group actions can help us to classify singular curves in such a way that we can construct moduli spaces of unstable curves (of fixed type). More generally our aim is to use suitable 'stability conditions' to stratify other moduli stacks into locally closed strata with coarse moduli spaces. The talk is based on joint work with Gergely Berczi, Vicky Hoskins and Joshua Jackson.

20 novembre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Hodge Theory of p-adic varieties

Conférence de la Chaire Aisenstadt

Semestre thématique : Théorie des nombres - Cohomologie en arithmétique

p-adic Hodge Theory is one of the most powerful tools in modern Arithmetic Geometry. In this talk, I will review p-adic Hodge Theory of algebraic varieties, present current developments in p-adic Hodge Theory of analytic varieties, and discuss some of its applications to problems in Number Theory.

Dans la cadre du Semestre thématique, Wieslawa Niziol donnera une série de quatre conférences dont celle-ci (Hodge Theory of p-adic  varieties). Les trois suivantes auront lieu de 9 h 30 à 10 h 30, le lundi 30 novembre, le mardi 1er décembre et le mercredi 2 décembre 2020. Pour informations et inscription: http://www.crm.umontreal.ca/2020/Niziol20/

13 novembre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Approximate Cross-Validation for Large Data and High Dimensions

The error or variability of statistical and machine learning algorithms is often assessed by repeatedly re-fitting a model with different weighted versions of the observed data. The ubiquitous tools of cross-validation (CV) and the bootstrap are examples of this technique. These methods are powerful in large part due to their model agnosticism but can be slow to run on modern, large data sets due to the need to repeatedly re-fit the model. We use a linear approximation to the dependence of the fitting procedure on the weights, producing results that can be faster than repeated re-fitting by orders of magnitude. This linear approximation is sometimes known as the "infinitesimal jackknife" (IJ) in the statistics literature, where it has mostly been used as a theoretical tool to prove asymptotic results. We provide explicit finite-sample error bounds for the infinitesimal jackknife in terms of a small number of simple, verifiable assumptions. Without further modification, though, we note that the IJ deteriorates in accuracy in high dimensions and incurs a running time roughly cubic in dimension. We additionally show, then, how dimensionality reduction can be used to successfully run the IJ in high dimensions when data is sparse or low rank. Simulated and real-data experiments support our theory.

16 octobre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Trigonometric functions and modular symbols

Conférence de la Chaire Aisenstadt

Semestre thématique : Théorie des nombres - Cohomologie en arithmétique

In his fantastic book “Elliptic functions according to Eisenstein and Kronecker”, Weil writes:

“As Eisenstein shows, his method for constructing elliptic functions applies beautifully to the simpler case of the trigonometric functions. Moreover, this case provides […] the simplest proofs for a series of results, originally discovered by Euler.”

The results Weil alludes to are relations between product of trigonometric functions. I will first explain how these relations are quite surprisingly governed by relations between modular symbols (whose elementary theory I will sketch). I will then show how this story fits into a wider picture that relates the topological world of group homology of some linear groups to the algebraic world of trigonometric and elliptic functions. To conclude I will briefly describe a number theoretical application.

This is based on a work-in-progress with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

Trigonometric functions and modular symbols

9 octobre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Hodge Theory and Moduli

The theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space with a canonical completion  has been proved by Kollar/Shepard-Barron/Alexeev. Aside from the classical case of algebraic curves, very little is known about the structure of , especially it’s boundary. The period mapping from Hodge theory provides a tool for studying these issues.

In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.

Hodge Theory and Moduli

2 octobre 2020 de 15 h 00 à 16 h 00 (heure de Montréal/Miami) Réunion Zoom

### Data Science, Classification, Clustering and Three-Way Data

Data science is discussed along with some historical perspective. Selected problems in classification are considered, either via specific datasets or general problem types. In each case, the problem is introduced before one or more potential solutions are discussed and applied. The problems discussed include data with outliers, longitudinal data, and three-way data. The proposed approaches are generally mixture model-based.

Responsables :
Erica E. M. Moodie (erica.moodie@mcgill.ca)
Yogendra P. Chaubey (yogen.chaubey@concordia.ca)

11 septembre 2020 de 16 h 00 à 17 h 00 (heure de Montréal/Miami) Réunion Zoom

### Machine Learning for Causual Inference

Given advances in machine learning over the past decades, it is now possible to accurately solve difficult non-parametric prediction problems in a way that is routine and reproducible. In this talk, I’ll discuss how machine learning tools can be rigorously integrated into observational study analyses, and how they interact with classical statistical ideas around randomization, semiparametric modeling, double robustness, etc. I’ll also survey some recent advances in methods for treatment heterogeneity. When deployed carefully, machine learning enables us to develop causal estimators that reflect an observational study design more closely than basic linear regression based methods.

Machine Learning for Causual Inference

19 juin 2020 de 16 h 00 à 17 h 00 (heure de Montréal/Miami) Réunion Zoom

### Quantitative approaches to understanding the immune response to SARS-CoV-2 infection

La COVID-19 est généralement caractérisée par une série de symptômes respiratoires qui, dans les cas graves, évoluent vers un syndrome de détresse respiratoire aiguë (SDRA). Ces symptômes sont aussi fréquemment accompagnés d'une série d'indications inflammatoires, en particulier des réponses inflammatoires hyper-réactives et dérégulées sous forme de tempêtes de cytokines et d'une immunopathologie sévère. Il reste beaucoup à découvrir sur les mécanismes qui conduisent à des résultats disparates dans la COVID-19. Ici, des approches quantitatives, en particulier des modèles mathématiques mécanistes, peuvent être utilisées pour améliorer notre compréhension de la réponse immunitaire à l'infection par le SRAS-CoV-2.

En nous appuyant sur nos travaux antérieurs de modélisation de la production de sous-ensembles de cellules immunitaires innées et de la dynamique virale du VIH et des virus oncolytiques, nous développons un cadre quantitatif pour interroger les questions ouvertes sur la réaction immunitaire innée et adaptative dans COVID-19. Dans cet exposé, je présenterai nos récents travaux de modélisation de la dynamique virale du SRAS-CoV-2 et de la réponse immunitaire qui en découle, tant au niveau tissulaire que systémique. Une partie de ces travaux est réalisée dans le cadre d'une coalition internationale et multidisciplinaire qui travaille à la mise en place d'un simulateur tissulaire complet (physicell.org/covid19 [1]), dont je parlerai également plus en détail.

Quantitative approaches to understanding the immune response to SARS-CoV-2 infection

17 avril 2020 de 16 h 00 à 17 h 00 (heure de Montréal/Miami) Réunion Zoom

### Observable events and typical trajectories in finite and infinite dimensional dynamical systems

Les termes "événements observables" et "trajectoires typiques" dans le titre devraient vraiment être entre guillemets, car ce qui est typique et/ou observable est une question d'interprétation. Pour les systèmes dynamiques sur des espaces à dimensions finies, on assimile souvent les événements observables à des ensembles de mesures de Lebesgue positives, et les distributions invariantes qui reflètent les comportements en grand temps des ensembles de mesures de Lebesgue positives des conditions initiales (telles que la mesure de Liouville pour les systèmes hamiltoniens) sont considérées comme particulièrement importantes. Je commencerai par introduire ces concepts pour les systèmes dynamiques généraux - y compris ceux avec des attracteurs - en décrivant une image dynamique simple que l'on pourrait espérer vraie. Cette image ne tient pas toujours, malheureusement, mais une petite quantité de bruit aléatoire la fera apparaître. Dans la deuxième partie de mon exposé, je considérerai les systèmes dimensionnels infinis tels que les semi-flux issus des PDE évolutifs dissipatifs. Je discuterai de la mesure dans laquelle les idées ci-dessus peuvent être généralisées à des dimensions infinies, et je proposerai une notion de "solutions typiques".

Observable events and typical trajectories in finite and infinite dimensional dynamical systems

16 mai 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Introduction to birational classification theory in dimension three and higher

One of the main themes of algebraic geometry is to classify algebraic varieties and to study various geometric properties of each of the interesting classes. Classical theories of curves and surfaces give a beautiful framework of classification theory. Recent developments provide more details in the case of dimension three. We are going to introduce the three-dimensional story and share some expectations for even higher dimensions.

UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, salle PK-5115

10 mai 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Quantum Jacobi forms and applications

Quantum modular forms were defined in 2010 by Zagier; they are somewhat analogous to ordinary modular forms, but they are defined on the rational numbers as opposed to the upper half complex plane, and have modified transformation properties. In 2016, Bringmann and the author defined the notion of a quantum Jacobi form, naturally marrying the concept of a quantum modular form with that of a Jacobi form (the theory of which was developed by Eichler and Zagier in the 1980s). We will discuss these intertwined topics, emphasizing recent developments and applications. In particular, we will discuss applications to combinatorics, topology (torus knots), and representation theory (VOAs).

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

3 mai 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The stochastic heat equation and KPZ in dimensions three and higher

The stochastic heat equation and the KPZ equation appear as the macroscopic limits for a large class of probabilistic models, and the study of KPZ, in particular, led to many fascinating developments in probability over the last decade or so, from the regularity structures to integrable probability. We will discuss a small group of recent results on these equations in simple settings, of the PDE flavour, that fall in line with what one may call naive expectations by an applied mathematician.

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

26 avril 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Distinguishing finitely presented groups by their finite quotients

If G is a finitely generated group, let C(G) denote the set of finite quotients of G. This talk will survey work on the question of to what extent C(G) determines G up to isomorphism, culminating in a discussion of examples of Fuchsian and Kleinian groups that are determined by C(G) (amongst finitely generated residually finite groups).

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

12 avril 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Linking in torus bundles and Hecke L functions

Torus bundles over the circle are among the simplest and cutest examples of 3- dimensional manifolds. After presenting some of these examples, using in particular animations realized by Jos Leys, I will consider periodic orbits in these fiber bundles over the circle. We will see that their linking numbers --- that are rational numbers by definition --- can be computed as certain special values of Hecke L-functions. Properly generalized this viewpoint makes it possible to give new topological proof of now classical rationality or integrality theorems of Klingen-Siegel and Deligne-Ribet. It also leads to interesting new "arithmetic lifts" that I will briefly explain. All this is extracted from an on going joint work with Pierre Charollois, Luis Garcia and Akshay Venkatesh.

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

29 mars 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Principal Bundles in Diophantine Geometry

Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments.

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

22 mars 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Flexibility in contact and symplectic geometry

We discuss a number of h-principle phenomena which were recently discovered in the field of contact and symplectic geometry. In generality, an h-principle is a method for constructing global solutions to underdetermined PDEs on manifolds by systematically localizing boundary conditions. In symplectic and contact geometry, these strategies typically are well suited for general constructions and partial classifications. Some of the results we discuss are the characterization of smooth manifolds admitting contact structures, high dimensional overtwistedness, the symplectic classification of flexibile Stein manifolds, and the construction of exotic Lagrangians in C^n.

UQAM, Pavillon Président-Kennedy, 201, ave du Président-Kennedy, salle PK-5115

19 mars 2019 de 14 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Special Colloquium : A constructive solution to Tarski’s circle squaring problem

In 1925, Tarski posed the problem of whether a disc in R^2 can be partitioned into finitely many pieces which can be rearranged by isometries to form a square of the same area. Unlike the Banach-Tarski paradox in R^3, it can be shown that two Lebesgue measurable sets in R^2 cannot be equidecomposed by isometries unless they have the same measure. Hence, the disk and square must necessarily be of the same area. In 1990, Laczkovich showed that Tarski’s circle squaring problem has a positive answer using the axiom of choice. We give a completely constructive solution to the problem and describe an explicit (Borel) way to equidecompose a circle and a square. This answers a question of Wagon. Our proof has three main ingredients. The first is work of Laczkovich in Diophantine approximation. The second is recent progress in a research program in descriptive set theory to understand how the complexity of a countable group is related to the complexity of the equivalence relations generated by its Borel actions. The third ingredient is ideas coming from the study of flows in networks. This is joint work with Spencer Unger.

McGill University, Burnside Hall , 805 O., rue Sherbrooke, salle 1104

15 mars 2019 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Persistent homology as an invariant, rather than as an approximation

Persistent homology is a very simple idea that was initially introduced as a way of understanding the underlying structure of an object from, perhaps noisy, samples of the object, and has been used as a tool in biology, material sciences, mapping and elsewhere. I will try to explain some of this, but perhaps also some more mathematical applications within geometric group theory. Then I'd like to pivot and study the part that traditionally has been thrown away, and show that this piece is relevant to approximation theory (a la Chebyshev), closed geodesics (a la Gromov), and to problems of quantitative topology (joint work with Ferry, Chambers, Dotter, and Manin).

McGill University, Burnside Hall , 805, rue Sherbrooke O., salle/Room1104

2 novembre 2018 de 16 h 00 à 17 h 00 (heure de Montréal/Miami)

### The complexity of detecting cliques and cycles in random graphs

A strong form of the P ≠ NP conjecture holds that no algorithm faster than n^{O(k)} solves the k-clique problem with high probability when the input is an Erdös–Rényi random graph with an appropriate edge density. Toward this conjecture, I will describe a line of work lower-bounding the average-case complexity of k-clique (and other subgraph isomorphism problems) in weak models of computation: namely, restricted classes of booleancircuits and formulas. Along the way I will discuss some of the history and current frontiers in Circuit Complexity. Joint work with Ken-ichi Kawarabayashi, Yuan Li and Alexander Razborov.

CRM, Université de Montréal, Pavillon André-Aisenstadt, salle 1355

28 septembre 2018 de 16 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### A delay differential equation with a solution whose shortened segments are dense

Simple-looking autonomous delay differential equations  with a real function and single time lag  can generate complicated (chaotic) solution behaviour, depending on the shape of . The same could be shown for equations with a variable, state-dependent delay , even for the linear case  with . Here the argument  of the {\it delay functional}  is the history of the solution  between  and t defined as the function  given by . So the delay alone may be responsible for complicated solution behaviour. In both cases the complicated behaviour which could be established occurs in a thin dust-like invariant subset of the infinite-dimensional Banach space or manifold of functions  on which the delay equation defines a nice semiflow. The lecture presents a result which grew out of an attempt to obtain complicated motion on a larger set with non-empty interior, as certain numerical experiments seem to suggest. For some  we construct a delay functional an infinite-dimensional subset of the space , so that the equation  has a solution whose {\it short segments} , , are dense in the space . This implies a new kind of complicated behaviour of the flowline . Reference: H. O. Walther, {\em A delay differential equation with a solution whose shortened segments are dense}.\\ J. Dynamics Dif. Eqs., to appear.

McGill University, Burnside Hall, salle 1104, 805 rue Sherbrooke O

21 septembre 2018 de 16 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### Algebraic structures for topological summaries of data

This talk introduces an algebraic framework to encode, compute, and analyze topological summaries of data. The main motivating problem, from evolutionary biology, involves statistics on a dataset comprising images of fruit fly wing veins, which amount to embedded planar graphs with varying combinatorics. Additional motivation comes from statistics more generally, the goal being to summarize unknown probability distributions from samples. The algebraic structures for topological summaries take their cue from graded polynomial rings and their modules, but the theory is complicated by the passage from integer exponent vectors to real exponent vectors. The key to making the structures practical for data science applications is a finiteness condition that encodes topological tameness -- which occurs in all modules arising from data -- robustly, in equivalent combinatorial and homological algebraic ways. Out of the tameness condition surprisingly falls much of ordinary commutative algebra, including syzygy theorems and primary decomposition.

UQAM, Pavillon Président-Kennedy, 201, av. du Président-Kennedy, salle PK-5115

4 mai 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Klein­-Gordon­-Maxwell­-Proca systems in the Riemannian setting

We intend to give a general talk about Klein­-Gordon-­Maxwell-­Proca systems which we aim to be accessible to a broad audience. We will insist on the Proca contribution and then discuss the kind of results one can prove in the electromagneto static case of the equations.

UQAM, pavillon Président­-Kennedy, 201, av. du Président-­Kennedy, salle PK­5115

13 avril 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Local-­global principles in number theory

One of the classical tools of number theory is the so­called local­global principle, or Hasse principle, going back to Hasse's work in the 1920's. His first results concern quadratic forms, and norms of number fields. Over the years, many positive and negative results were proved, and there is now a huge number of results in this topic. This talk will present some old and new results, in particular in the continuation of Hasse's cyclic norm theorem. These have been obtained jointly with Parimala and Tingyu Lee.

UQAM, pavillon Président-­Kennedy, 201, av. du Président-­Kennedy, salle PK­5115

23 février 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Cluster theory of the coherent Satake category

The affine Grassmannian, though a somewhat esoteric looking object at first sight, is a fundamental algebro­geometric construction lying at the heart of a series of ideas connecting number theory (and the Langlands program) to geometric representation theory, low dimensional topology and mathematical physics. Historically it is popular to study the category of constructible perverse sheaves on the affine Grassmannian. This leads to the *constructible* Satake category and the celebrated (geometric) Satake equivalence. More recently it has become apparent that it makes sense to also study the category of perverse *coherent* sheaves (the coherent Satake category). Motivated by certain ideas in mathematical physics this category is conjecturally governed by a cluster algebra structure. We will illustrate the geometry of the affine Grassmannian in an elementary way, discuss what we mean by a cluster algebra structure and then describe a solution to this conjecture in the case of general linear groups.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

16 février 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Quantum n-­body problem: generalized Euler coordinates (from J-­L Lagrange to Figure Eight by Moore and Ter-­Martirosyan, then and today)

The potential of the n-­body problem, both classical and quantum, depends only on the relative (mutual) distances between bodies. By generalized Euler coordinates we mean relative distances and angles. Their advantage over Jacobi coordinates is emphasized. The NEW IDEA is to study trajectories in both classical, and eigenstates in quantum systems which depends on relative distances ALONE. We show how this study is equivalent to the study of (i) the motion of a particle (quantum or classical) in curved space of dimension n(n-­1)/2 or the study of (ii) the Euler-Arnold (quantum or classical) ­ - sl(n(n-­1)/2, R) algebra top. The curved space of (i) has a number of remarkable properties. In the 3­body case the {\it de­Quantization} of quantum Hamiltonian leads to a classical Hamiltonian which solves a ~250­-years old problem posed by Lagrange on 3­-body planar motion.

CRM, Université de Montréal, Pavillon André­-Aisenstadt, salle 6254

16 février 2018 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### The Law of Large Populations: The return of the long-­ignored N and how it can affect our 2020 vision

For over a century now, we statisticians have successfully convinced ourselves and almost everyone else, that in statistical inference the size of the population N can be ignored, especially when it is large. Instead, we focused on the size of the sample, n, the key driving force for both the Law of Large Numbers and the Central Limit Theorem. We were thus taught that the statistical error (standard error) goes down with n typically at the rate of 1/√n. However, all these rely on the presumption that our data have perfect quality, in the sense of being equivalent to a probabilistic sample. A largely overlooked statistical identity, a potential counterpart to the Euler identity in mathematics, reveals a Law of Large Populations (LLP), a law that we should be all afraid of. That is, once we lose control over data quality, the systematic error (bias) in the usual estimators, relative to the benchmarking standard error from simple random sampling, goes up with N at the rate of √N. The coefficient in front of √N can be viewed as a data defect index, which is the simple Pearson correlation between the reporting/recording indicator and the value reported/recorded. Because of the multiplier√N, a seemingly tiny correlation, say, 0.005, can have detrimental effect on the quality of inference. Without understanding of this LLP, “big data” can do more harm than good because of the drastically inflated precision assessment hence a gross overconfidence, setting us up to be caught by surprise when the reality unfolds, as we all experienced during the 2016 US presidential election. Data from Cooperative Congressional Election Study (CCES, conducted by Stephen Ansolabehere, Douglas River and others, and analyzed by Shiro Kuriwaki), are used to estimate the data defect index for the 2016 US election, with the aim to gain a clearer vision for the 2020 election and beyond.

McGill University, OTTO MAASS 217

9 février 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Persistence modules in symplectic topology

In order to resolve Vladimir Arnol'd's famous conjecture from the 1960's, giving lower bounds on the number of fixed points of Hamiltonian diffeomorphisms of a symplectic manifold, Andreas Floer has associated in the late 1980's a homology theory to the Hamiltonian action functional on the loop space of the manifold. It was known for a long time that this homology theory can be filtered by the values of the action functional, yielding information about metric invariants in symplectic topology (Hofer's metric, for example). We discuss a recent marriage between the filtered version of Floer theory and persistent homology, a new field of mathematics that has its origins in data analysis, providing examples of new ensuing results.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

12 janvier 2018 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### What is quantum chaos

Where do eigenfunctions of the Laplacian concentrate as eigenvalues go to infinity? Do they equidistribute or do they concentrate in an uneven way? It turns out that the answer depends on the nature of the geodesic flow. I will discuss various results in the case when the flow is chaotic: the Quantum Ergodicity theorem of Shnirelman, Colin de Verdière, and Zelditch, the Quantum Unique Ergodicity conjecture of Rudnick-­Sarnak, the progress on it by Lindenstrauss and Soundararajan, and the entropy bounds of Anantharaman­-Nonnenmacher. I will conclude with a recent lower bound on the mass of eigenfunctions obtained with Jin. It relies on a new tool called "fractal uncertainty principle" developed in the works with Bourgain and Zahl.

CRM, Université de Montréal, Pavillon André-­Aisenstadt, salle 6254

8 décembre 2017 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Primes with missing digits

Many famous open questions about primes can be interpreted as questions about the digits of primes in a given base. We will talk about recent work showing there are infinitely many primes with no 7 in their decimal expansion. (And similarly with 7 replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most X^{1­c} elements less than X) which is typically very difficult. The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory

UQAM, Pavillon Président-­Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

24 novembre 2017 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### 150 years (and more) of data analysis in Canada

As Canada celebrates its 150th anniversary, it may be good to reflect on the past and future of data analysis and statistics in this country. In this talk, I will review the Victorian Statistics Movement and its effect in Canada, data analysis by a Montréal physician in the 1850s, a controversy over data analysis in the 1850s and 60s centred in Montréal, John A. MacDonald’s use of statistics, the Canadian insurance industry and the use of statistics, the beginning of mathematical statistics in Canada, the Fisherian revolution, the influence of Fisher, Neyman and Pearson, the computer revolution, and the emergence of data science.

Université McGill, Leacock Building, salle LEA 232

24 novembre 2017 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### Complex analysis and 2D statistical physics

Over the last decades, there was much progress in understanding 2D lattice models of critical phenomena. It started with several theories, developed by physicists. Most notably, Conformal Field Theory led to spectacular predictions for 2D lattice models: e.g., critical percolation cluster a.s. has Hausdorff dimension $91/48$, while the number of selfavoiding length $N$ walks on the hexagonal lattice grows like $(\sqrt{2+\sqrt{2}})^N N^{11/32}$. While the algebraic framework of CFT is rather solid, rigorous arguments relating it to lattice models were lacking. More recently, mathematical approaches were developed, allowing not only for rigorous proofs of many such results, but also for new physical intuition. We will discuss some of the applications of complex analysis to the study of 2D lattice models.

CRM, Université de Montréal, Pavillon André-­Aisenstadt, salle 6254

17 novembre 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Recent progress on De Giorgi Conjecture

Classifying solutions to nonlinear partial differential equations are fundamental research in PDEs. In this talk, I will report recent progress made in classifying some elementary PDEs, starting with the De Giorgi Conjecture (1978). I will discuss the classification of global minimizers and finite Morse index solutions, relation with minimal surfaces and Toda integrable systems, as well as recent exciting developments in fractional De Giorgi Conjecture.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

27 octobre 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Beneath the Surface: Geometry Processing at the Intrinsic/Extrinsic Interface

Algorithms for analyzing 3D surfaces find application in diverse fields from computer animation to medical imaging, manufacturing, and robotics. Reflecting a bias dating back to the early development of differential geometry, a disproportionate fraction of these algorithms focuses on discovering intrinsic shape properties, or those measurable along a surface without considering the surrounding space. This talk will summarize techniques to overcome this bias by developing a geometry processing pipeline that treats intrinsic and extrinsic geometry democratically. We describe theoretically­justified, stable algorithms that can characterize extrinsic shape from surface representations. In particular, we will show two strategies for computational extrinsic geometry. In our first approach, we will show how the discrete Laplace­Beltrami operator of a triangulated surface accompanied with the same operator for its offset determines the surface embedding up to rigid motion. In the second, we will treat a surface as the boundary of a volume rather than as a thin shell, using the Steklov (Dirichlet­to­Neumann) eigenproblem as the basis for developing volumetric spectral shape analysis algorithms without discretizing the interior.

UdeM, Pavillon André­-Aisenstadt, salle 6254

13 octobre 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Supercritical Wave Equations

I will review the problem of Global existence for dispersive equations, in particular, supercritical equations. These equations who play a fundamental role in science, have been , and remain a major challenge in the field of Partial Differential Equations. They come in various forms, derived from Geometry, General Relativity, Fluid Dynamics, Field Theory. I present a new approach to classify the asymptotic behavior of wave equations, supercritical and others, and construct global solutions with large initial data. I will then describe current extensions to Nonlinear Schroedinger Equations.

UdeM, Pavillon André-­Aisenstadt, salle 6254

29 septembre 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The first field

The “first field” is obtained by making the entries in its addition and multiplication tables be the smallest possibilities. It is really an interesting field that contains the integers, but with new addition and multiplication tables. For example, 2 x 2 = 3, 5 x 7 = 13, ... It extends to the infinite ordinals and the first infinite ordinal is the cube root of 2!

UdeM, Pavillon André-­Aisenstadt, salle 1140

15 septembre 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Isometric embedding and quasi­-local type inequality

In this talk, we will first review the classic Weyl's embedding problem and its application in quasi­local mass. We will then discuss some recent progress on Weyl's embedding problem in general Riemannian manifold. Assuming isometric embedding into Schwarzschild manifold, we will further establish a quasi­local type inequality. This talk is based on works joint with Pengfei Guan and Pengzi Miao.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

5 mai 2017 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### From the geometry of numbers to Arakelov geometry

Arakelov geometry is a modern formalism that extends in various directions the geometry of numbers founded by Minkowski in the nineteenth century. The objects of study are arithmetic varieties, namely complex varieties that can be defined by polynomial equations with integer coefficients. The theory exploits the interplay between algebraic geometry and number theory and complex analysis and differential geometry. Recently, the formalism found beautiful and important applications to the so­called Kudla programme and the Colmez conjecture. In the talk, I will first introduce elementary facts in Minkowski's geometry of numbers. This will provide a motivation for the sequel, where I will give my own view of Arakelov geometry, by focusing on toy (but non­trivial) examples of one of the central theorems in the theory, the arithmetic Riemann­Roch theorem mainly due to Bismut, Gillet and Soulé, and generalizations. I hope there will be ingredients to satisfy different tastes, for instance modular forms (arithmetic aspect), analytic torsion (analytic aspect) and Selberg zeta functions (arithmetic, analytic and dynamic aspects).

UQAM, Pavillon Président­-Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

21 avril 2017 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Introduction to the Energy Identity for Yang-­Mills

In this talk we give an introduction to the analysis of the Yang­Mills equation in higher dimensions. In particular, when studying sequences of solutions we will study the manner in which blow up can occur, and how this blow up may be understood through the classical notions of the defect measure and bubbles. The energy identity is an explicit conjectural relationship, known to be true in dimension four, relating the energy density of the defect measure at a point to the bubbles which occur at that point, and we will give a brief overview of the recent proof of this result for general stationary Yang Mills in higher dimensions. The work is joint with Daniele Valtorta.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

31 mars 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### PDEs on non­-smooth domains

In these lecture we will discuss the relationship between the boundary regularity of the solutions to elliptic second order divergence form partial differential equations and the geometry of the boundary of the domain where they are defined. While in the smooth setting tools from classical PDEs are used to address this question, in the nonsmooth setting techniques from harmonic analysis and geometric measure theory are needed to tackle the problem. The goal is to present an overview of the recent developments in this very active area of research.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

17 mars 2017 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### Inference in Dynamical Systems

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Furthermore, we show how some well­studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures. We also relate Bayesian inference to the thermodynamic formalism in tracking dynamical systems.

McGill University, Burnside Hall, 805 Sherbrooke Ouest, salle 1205

10 mars 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Probabilistic aspects of minimum spanning trees

One of the most dynamic areas of probability theory is the study of the behaviour of discrete optimization problems on random inputs. My talk will focus on the probabilistic analysis of one of the first and foundational combinatorial optimization problems: the minimum spanning tree problem. The structure of a random minimum spanning tree (MST) of a graph G turns out to be intimately linked to the behaviour of critical and near­critical percolation on G. I will describe this connection, and present some results on the structure, scaling limits, and volume growth of random MSTs. It turns out that, on high­dimensional graphs, random minimum spanning trees are expected to be threedimensional when viewed intrinsically, and six­dimensional when viewed as embedded objects. Based on joint works with Nicolas Broutin, Christina Goldschmidt, Simon Griffiths, Ross Kang, Gregory Miermont, Bruce Reed, Sanchayan Sen.

CRM, Université de Montréal, Pavillon André-­Aisenstadt, 2920 Chemin de la Tour, salle 6254

24 février 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Spreading phenomena in integrodifference equations with overcompensatory growth function

The globally observed phenomenon of the spread of invasive biological species with all its sometimes detrimental effects on native ecosystems has spurred intense mathematical research and modelling efforts into corresponding phenomena of spreading speeds and travelling waves. The standard modelling framework for such processes is based on reaction­ diffusion equations, but several aspects of an invasion can only be appropriately described by a discrete­time analogues, called integrodifference equations. The theory of spreading speeds and travelling waves in such integrodifference equations is well established for the "mono­stable" case, i.e. when the non­spatial dynamics show a globally stable positive steady state. When the positive state of the non­spatial dynamics is not stable, as is the case with the famous discrete logistic equation, it is unclear how the corresponding spatial spread profile evolves and at what speed. Previous simulations seemed to reveal a travelling profile in the form of a two­cycle, with or without spatial oscillations. The existence of a travelling wave solution has been proven, but its shape and stability remain unclear. In this talk, I will show simulations that suggest that there are several travelling profiles at different speeds. I will establish corresponding generalizations of the concept of a spreading speed and prove the existence of such speeds and travelling waves in the second­ iterate operator. I conjecture that rather than a travelling two­cycle for the next­generation operator, one observes a pair of stacked fronts for the second­iterate operator. I will relate the observations to the phenomenon of dynamic stabilization.

CRM, Université de Montréal, Pavillon André-­Aisenstadt, 2920 Chemin de la Tour, salle 6254

10 février 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Knot concordance

I will introduce the knot concordance group, give a survey of our current understanding of it and discuss some relationships with the topology of 4-­manifolds.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

20 janvier 2017 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The Birch­-Swinnerton Dyer Conjecture and counting elliptic curves of ranks 0 and 1

This colloquium talk will begin with an introduction to the Birch-­­Swinnerton­-Dyer conjecture for elliptic curves -- just curves defined by the equations y^2=x^3+Ax+B -- and then describe recent advances that allow us to prove that lots of elliptic curves have rank zero or one.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

2 décembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Partial differential equations of mixed elliptic-­hyperbolic type in mechanics and geometry

As is well­-known, two of the basic types of linear partial differential equations (PDEs) are hyperbolic PDEs and elliptic PDEs, following the classification for linear PDEs first proposed by Jacques Hadamard in the 1920s; and linear theories of PDEs of these two types have been well established, respectively. On the other hand, many nonlinear PDEs arising in mechanics, geometry, and other areas naturally are of mixed elliptic­hyperbolic type. The solution of some longstanding fundamental problems in these areas greatly requires a deep understanding of such nonlinear PDEs of mixed type. Important examples include shock reflection­-diffraction problems in fluid mechanics (the Euler equations) and isometric embedding problems in differential geometry (the Gauss-­Codazzi­Ricci equations), among many others. In this talk we will present natural connections of nonlinear PDEs of mixed elliptic­-hyperbolic type with these longstanding problems and will then discuss some recent developments in the analysis of these nonlinear PDEs through the examples with emphasis on developing and identifying mathematical approaches, ideas, and techniques for dealing with the mixed­-type problems. Further trends, perspectives, and open problems in this direction will also be addressed.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

1 décembre 2016 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### High­-dimensional changepoint estimation via sparse projection

Changepoints are a very common feature of Big Data that arrive in the form of a data stream. We study highdimensional time series in which, at certain time points, the mean structure changes in a sparse subset of the coordinates. The challenge is to borrow strength across the coordinates in order to detect smaller changes than could be observed in any individual component series. We propose a two­stage procedure called 'inspect' for estimation of the changepoints: first, we argue that a good projection direction can be obtained as the leading left singular vector of the matrix that solves a convex optimisation problem derived from the CUSUM transformation of the time series. We then apply an existing univariate changepoint detection algorithm to the projected series. Our theory provides strong guarantees on both the number of estimated changepoints and the rates of convergence of their locations, and our numerical studies validate its highly competitive empirical performance for a wide range of data generating mechanisms.

Room 1205, Burnside Hall, 805 Sherbrooke West

26 novembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Around the Möbius function

The Moebius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Moebius function. In recent joint work with K. Matomaki the speaker showed that the sum of the Moebius function exhibits cancellations in "almost all intervals'' of increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis. The result holds in fact in greater generality. Exploiting this generality one can show that between a fixed number of consecutive squares there is always an integer composed of only "small'' prime factors. This is related to the running time of Lenstra's factoring algorithm. I will also discuss some further developments : the work of Tao on correlations between consecutive values of Chowla, and his application of this result to the resolution of the Erdos discrepancy problem.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président­-Kennedy, salle PK­5115

4 novembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The nonlinear stability of Minkowski space for self­-gravitating massive fields

will review results on the global evolution of self­gravitating massive matter in the context of Einstein's theory as well as the f(R)­theory of gravity. In collaboration with Yue Ma (Xian), I have investigated the global existence problem for the Einstein equations coupled with a Klein­Gordon equation describing the evolution of a massive scalar field. Our main theorem establishes the global nonlinear stability of Minkowski spacetime upon small perturbations of the metric and the matter field. Recall that the fully geometric proof by Christodoulou and Klainerman in 1993, as well as the proof in wave gauge by Lindblad and Rodnianski in 2010, both apply to vacuum spacetimes and massless fields only. Our new technique of proof, which we refer to as the Hyperboloidal Foliation Method, does not use Minkowski's scaling field and is based on a foliation of the spacetime by asymptotically hyperboloidal spacelike hypersurfaces, on sharp estimates for wave and Klein­Gordon equations, and on an analysis of the quasi­null hyperboloidal structure (as we call it) of the Einstein equations in wave gauge.

CRM, Pavillon André-­Aisenstadt, 2920 chemin de la tour, salle 6254

28 octobre 2016 de 15 h 00 à 17 h 00 (heure de Montréal/Miami) Sur place

### Efficient tests of covariate effects in two­-phase failure time studies

Two-­phase studies are frequently used when observations on certain variables are expensive or difficult to obtain. One such situation is when a cohort exists for which certain variables have been measured (phase 1 data); then, a subsample of individuals is selected, and additional data are collected on them (phase 2). Efficiency for tests and estimators can be increased by basing the selection of phase 2 individuals on data collected at phase 1. For example, in large cohorts, expensive genomic measurements are often collected at phase 2, with oversampling of persons with “extreme” phenotypic responses. A second example is case­cohort or nested case­control studies involving times to rare events, where phase 2 oversamples persons who have experienced the event by a certain time. In this talk I will describe two­phase studies on failure times, present efficient methods for testing covariate effects. Some extensions to more complex outcomes and areas needing further development will be discussed.

Room 1205, Burnside Hall, 805 Sherbrooke West

21 octobre 2016 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Integrable probability and the KPZ universality class

I will explain how certain integrable structures give rise to meaningful probabilistic systems and methods to analyze them. Asymptotics reveal universal phenomena, such as the Kardar­Parisi­Zhang universality class. No prior knowledge will be assumed.

CRM, Pavillon André-­Aisenstadt, 2920 chemin de la tour, salle 6254

14 octobre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Rigorously verified computing for infinite dimensional nonlinear dynamics: a functional analytic approach

Studying and proving existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for partial differential equations, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. The goal of this talk is to present rigorous numerical technique relying on functional analytic and topological tools to prove existence of steady states, time periodic solutions, traveling waves and connecting orbits for the above mentioned dynamical systems. We will spend some time identifying difficulties of the proposed approach as well as time to identify future directions of research.

CRM, Pavillon André­-Aisenstadt, 2920 chemin de la tour, salle 6254

30 septembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Notions of simplicity in low­-dimensions

Various auxiliary structures arise naturally in low­dimensions. I will discuss three of these: left­orders on the fundamental group, taut foliations on three­manifolds, and non­trivial Floer homological invariants. Perhaps surprisingly, for (closed, connected, orientable, irreducible) three­manifolds, it has been conjectured that the existence of any one of these structures implies the others. I will describe what is currently known about this conjectural relationship, as well as some of the machinery — particularly in Heegaard Floer theory — that has been developed in pursuit of the conjecture.

UQAM, Pavillon Président-­Kennedy, 201, ave du Président-­Kennedy, salle PK­5115

16 septembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Statistical Inference for fractional diffusion processes

There are some time series which exhibit long­range dependence as noticed by Hurst in his investigations of river water levels along Nile river. Long­range dependence is connected with the concept of self­similarity in that increments of a self­similar process with stationary increments exhibit long­range dependence under some conditions. Fractional Brownian motion is an example of such a process. We discuss statistical inference for stochastic processes modeled by stochastic differential equations driven by a fractional Brownian motion. These processes are termed as fractional diffusion processes. Since fractional Brownian motion is not a semimartingale, it is not possible to extend the notion of a stochastic integral with respect to a fractional Brownian motion following the ideas of Ito integration. There are other methods of extending integration with respect to a fractional Brownian motion. Suppose a complete path of a fractional diffusion process is observed over a finite time interval. We will present some results on inference problems for such processes.

Université Concordia, Library Building, 1400 de Maisonneuve O., salle LB­921.04

16 septembre 2016 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Cubature, approximation, and isotropy in the hypercube

The hypercube is the standard domain for computation in higher dimensions. We describe two respects in which the anisotropy of this domain has practical consequences. The first is a matter well known to experts (and to Chebfun users): the importance of axis­alignment in low­rank compression of multivariate functions. Rotating a function by a few degrees in two or more dimensions may change its numerical rank completely. The second is new. The standard notion of degree of a multivariate polynomial, total degree, is isotropic – invariant under rotation. The hypercube, however, is highly anisotropic. We present a theorem showing that as a consequence, the convergence rate of multivariate polynomial approximations in a hypercube is determined not by the total degree but by the {\em Euclidean degree,} defined in terms of not the 1­norm but the 2­norm of the exponent vector $\bf k$ of a monomial $x_1^{k_1}\cdots x_s^{k_s}$. The consequences, which relate to established ideas of cubature and approximation going back to James Clark Maxwell, are exponentially pronounced as the dimension of the hypercube increases. The talk will include numerical demonstrations.

UQAM, Pavillon Président­-Kennedy, 201, ave du Président-Kennedy, salle PK­5115

9 avril 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Modular generating series and arithmetic geometry

I will survey the development of the theory of theta series and describe some recent advances/work in progress on arithmetic theta series. The construction and modularity of theta series as counting functions for lattice points for positive definite quadratic forms is a beautiful piece of classical mathematics with its origins in the mid 19th century. Siegel initiated the study of the analogue for indefinite quadratic forms. Millson and I introduced a geometric variant in which the theta series give rise to modular generating series for the cohomology classes of "special" algebraic cycles on locally symmetric varieties. These results motivate the definition of analogous generating series for the classes of such special cycles in the Chow groups and for the classes in the arithmetic Chow groups of their integral extensions. The modularity of such series is a difficult problem. I will discuss various cases in which recent progress has been made and some of the difficulties involved.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6254

2 avril 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Uniqueness of blowups and Lojasiewicz inequalities

The mean curvature flow (MCF) of any closed hypersurface becomes singular in finite time. Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, by Federer-Fleming in 1959, is that they weakly resemble cones. For MCF, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a MCF looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is perhaps the most fundamental question about singularities. We will discuss the proof of this long standing open question for MCF at all generic singularities and for mean convex MCF at all singularities. This is joint work with Toby Colding.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

26 mars 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Left-orderings of groups and the topology of 3-manifolds

Many decades of work culminating in Perelman's proof of Thurston's geometrisation conjecture showed that a closed, connected, orientable, prime 3-dimensional manifold $W$ is essentially determined by its fundamental group $\pi_1(W)$. This group consists of classes of based loops in $W$ and its multiplication corresponds to their concatenation. An important problem is to describe the topological and geometric properties of $W$ in terms of $\pi_1(W)$. For instance, geometrisation implies that $W$ admits a hyperbolic structure if and only if $\pi_1(W)$ is infinite, freely indecomposable, and contains no $\mathbb Z \oplus \mathbb Z$ subgroups. In this talk I will describe recent work which has determined a surprisingly strong correlation between the existence of a left-order on $\pi_1(W)$ (a total order invariant under left multiplication) and the following two measures of largeness for $W$: a) the existence of a co-oriented taut foliation on $W$ - a special type of partition of $W$ into surfaces which fit together locally like a deck of cards. b) the condition that $W$ not be an Lspace - an analytically defined condition representing the non-triviality of its Heegaard-Floer homology. I will introduce each of these notions, describe the results which connect them, and state a number of open problems and conjectures concerning their precise relationship.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

19 mars 2015 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Integrable probability

The goal of the talk is to survey the emerging field of integrable probability, whose goal is to identify and analyze exactly solvable probabilistic models. The models and results are often easy to describe, yet difficult to find, and they carry essential information about broad universality classes of stochastic processes.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

12 mars 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The upper half-planes

The upper half-planes (complex and p-adic) are very elementary objects, but they have a surprisingly rich structure that I will explore in the talk.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

5 mars 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Periods

We will discuss periods, in particular the periods conjecture of Kontsevich and Zagier and the relationship between formal periods and Nori motives.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

26 février 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Categorification in representation theory

This will be an expository talk concerning the idea of categorification and its role in representation theory. We will begin with some very simple yet beautiful observations about how various ideas from basic algebra (monoids, groups, rings, representations etc.) can be reformulated in the language of category theory. We will then explain how this viewpoint leads to new ideas such as the "categorification" of the abovementioned algebraic objects. We will conclude with a brief synopsis of some current active areas of research involving the categorification of quantum groups. One of the goals of this idea is to produce four-dimensional topological quantum field theories. Very little background knowledge will be assumed.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

19 février 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Irrationality proofs, moduli spaces and dinner parties

After introducing an elementary criterion for a real number to be irrational, I will discuss Apery’s famous result proving the irrationality of zeta(3). Then I will give an overview of subsequent results in this field, and finally propose a simple geometric interpretation based on a classical dinner party game.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

12 février 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Influence des côtes sur les courants marins

Comprendre les mécanismes qui régissent la circulation océanique est un défi pour les géophysiciens, mais aussi pour les mathématiciens qui doivent développer de nouveaux outils d'analyse pour ces modèles complexes (qui font intervenir en particulier de très nombreuses échelles de temps et d'espace). Un mécanisme particulièrement important pour la circulation à l'échelle planétaire est le phénomène de couche limite qui explique une partie des échanges énergétiques. On montrera ici au travers d'un modèle très simplifié qu'il permet d'expliquer notamment l'intensification des courants de bord Ouest. On évoquera ensuite les difficultés mathématiques liées à la prise en compte de la géométrie. Note : l'exposé sera en anglais avec des transparents en français.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

5 février 2015 de 16 h 00 à 16 h 00 (heure de Montréal/Miami) Sur place

### Cobordism and Lagrangian topology

This talk aims to discuss how two different basic organizing principles in topology come together in the study of Lagrangian submanifolds. The first principle is cobordism and it emerged in topology in the 1950’s, mainly starting with the work of Thom. It was introduced in Lagrangian topology by Arnold in the 1970’s. The second principle is to reconstruct a subspace of a given space from a family of slices’’, each one obtained by intersecting the subspace with a member of a preferred class of special test’’ subspaces. For instance, a subspace of 3d euclidean space can be described as the union of all its intersections with horizontal planes. The key issue from this point of view is, of course, how to assemble all the slices together. The perspective that is central for my talk originates in the work of Gromov and Floer in the 1980’s: if the ambient space is a symplectic manifold M, and if the subspace to be described is a Lagrangian submanifold, then, surprisingly,the glue’' that puts the slices together in an efficient algebraic fashion is a reflection of the combinatorial properties of J-holomorphic curves in M. This point of view has been pursued actively since then by many researchers such as Hofer, Fukaya, Seidel leading to a structure called the Fukaya category. Through recent work of Paul Biran and myself, cobordism and the Fukaya category turn out to be intimately related and at the end of the talk I intend to give an idea about this relation.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

29 janvier 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Spectres et pseudospectres

Les valeurs propres sont parmi les notions les plus utiles en mathématiques: elles permettent la diagonalisation des matrices, elles décrivent l'asymptotique et la stabilité, elles donnent de la personnalité à une matrice. Cependant, lorsque la matrice en question n'est pas normale, l'analyse par des valeurs propres ne donne qu'une information très partielle, et peut même nous induire en erreur. Cet exposé se veut une introduction à la théorie des pseudospectres, un raffinement de la théorie spectrale standard qui s'est avéré utile dans des applications concernant des matrices non normales. Je vais m'intéresser surtout à la question suivante: À quel point les pseudospectres d'une matrice déterminent-ils le comportement de la matrice?

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

22 janvier 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### On the usefulness of mathematics for insurance risk theory - and vice versa

This talk is on applications of various branches of mathematics in the field of risk theory, a branch of actuarial mathematics dealing with the analysis of the surplus process of a portfolio of insurance contracts over time. At the same time such practical problems frequently trigger mathematical research questions, in some cases leading to remarkable identities and connections. Next to the close interactions with probability and statistics, examples will include the branches of real and complex analysis, algebra, symbolic computation, number theory and discrete mathematics.

McGill University, Burnside Hall, 805 rue Sherbrooke 0., Montréal, salle 920

15 janvier 2015 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Functional data analysis and related topics

Functional data analysis (FDA) has received substantial attention, with applications arising from various disciplines, such as engineering, public health, finance etc. In general, the FDA approaches focus on nonparametric underlying models that assume the data are observed from realizations of stochastic processes satisfying some regularity conditions, e.g., smoothness constraints. The estimation and inference procedures usually do not depend on merely a finite number of parameters, which contrasts with parametric models, and exploit techniques, such as smoothing methods and dimension reduction, that allow data to speak for themselves. In this talk, I will give an overview of FDA methods and related topics developed in recent years.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

4 décembre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Algebraic combinatorics and finite reflection groups

The lecture will be delivered in French, with English slides, so that anyone may enjoy it. ----- La conférence sera présentée en français, avec des transparents en anglais, pour que tous puissent suivre. Les dernières années ont vu une explosion d’activités à la frontière entre la combinatoire algébrique, la théorie de la représentation et la géométrie algébrique, avec des liens captivants avec la théorie des nœuds et la physique mathématique. En gardant un large auditoire en tête, nous esquisserons en quoi cette interaction a été très fructueuse et a soulevé de nouvelles questions intrigantes dans les divers domaines concernés. Nous essaierons de donner la saveur des résultats obtenus, des techniques utilisées, du grand nombre de questions ouvertes, et du pourquoi de leur intérêt. Ce fascinant échange entre combinatoire et algèbre fait d’une part intervenir des généralisations au contexte des rectangles des « chemins de Dyck ». Il est bien connu, depuis Euler, que ces chemins sont comptés par les nombres de Catalan, dans le cas d’un carré. De plus, les fonctions de stationnement (parking functions) sont intimement reliées à ces chemins. D’autre part, du côté algébrique, apparaissent des S_n-module bigradué de polynômes harmoniques diagonaux du groupe symétrique S_n. Il a été conjecturé qu’une énumération adéquate des fonctions de stationnement, associées à certaines familles de chemins de Dyck, fournit une formule combinatoire explicite du caractère bigradué de ces modules. Cette conjecture, connue sous le nom de conjecture « shuffle », a récemment été grandement étendue pour couvrir tous les cas rectangulaires. Interviennent dans tout ceci, des opérateurs sur les polynômes de Macdonald, l’algèbre de Hall elliptique, les algèbres de Hecke affines doubles (DAHA), le schéma de Hilbert de points dans le plan, etc.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 5340

20 novembre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### High-dimensional phenomena in mathematical statistics and convex analysis

Statistical models in which the ambient dimension is of the same order or larger than the sample size arise frequently in different areas of science and engineering. Although high-dimensional models of this type date back to the work of Kolmogorov, they have been the subject of intensive study over the past decade, and have interesting connections to many branches of mathematics (including concentration of measure, random matrix theory, convex geometry, and information theory). In this talk, we provide a broad overview of the general area, including vignettes on phase transitions in high-dimensional graph recovery, and randomized approximations of convex programs.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

13 novembre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Recent advances in the arithmetic of elliptic curves

In the past few years there have been several spectacular advances in understanding the arithmetic of elliptic curves including results about ranks on average and on the conjecture of Birch and Swinnerton-Dyer. I will give an introduction to the main problems of interest and survey some of these developments. This talk will be addressed to a general mathematical audience.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

6 novembre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The Cubical Route to Understanding Groups

Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will primarily aim to introduce nonpositively curved cube complexes, and then describe some of the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

30 octobre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### A Pedestrian Approach to Group Representations

Determining the number of walks of n steps from vertex A to vertex B on a graph often involves clever combinatorics or tedious treading. But if the graph is the representation graph of a group, representation theory can facilitate the counting and provide much insight. This talk will focus on connections between Schur-Weyl duality and walking on representation graphs. Examples of special interest are the simplylaced affine Dynkin diagrams, which are the representation graphs of the finite subgroups of the special unitary group SU(2) by the McKay correspondence. The duality between the SU(2) subgroups and certain algebras enables us to count walks and solve other combinatorial problems, and to obtain connections with the Temperley-Lieb algebras of statistical mechanics, with partitions, with Stirling numbers, and much more.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

9 octobre 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Applications of additive combinatorics to homogeneous dynamics

CRM, Université de Montréal, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1140

2 mai 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Eigenvarieties

After discussing some down-to-earth examples, I will explain what Eigenvarieties are for general reductive groups, present some important conjectures about them, and some basic number theory applications.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

11 avril 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Flat surfaces and determinants of Laplaciancs

In 2008 D. Korotkin and the author found an explicit formula for the determinant of the Friedrichs Laplacian on a compact 2d surface of genus g >1 provided with flat conical metric with trivial holonomy. This formula can be considered as a higher genus version of the classical Ray-Singer formula for the determinant of the Laplace operator on an elliptic curve provided with (smooth) flat conformal metric. We will discuss further generalizations of this result for 1) General polyhedral metrics on compact surfaces. 2) Other (i.e. non Friedrichs) self-adjoint extensions of conical Laplacians. 3) Noncompact flat surfaces with cylindrical and Euclidean ends. The talk is based on the joint works with L. Hillairet and V. Kalvin.

CRM, Université de Montréal, pavillon André-Aisenstadt, 2920 chemin de la Tour, salle 6214

4 avril 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Interaction between internal and surface waves in a two layers fluid

Internal waves occur within a fluid that is density-stratified, most commonly by temperature or salinity variation. In the oceans, such disturbances in internal layers are often generated by tides. They appear as large amplitude, long wavelength nonlinear waves and can propagate over large distances. Photographs taken from orbital shuttle as well as local measurements show that their presence has a significant effect on the surface of the ocean. In some instances, the visible signature of internal waves on the surface of the ocean is a band of roughness which propagates at the same velocity as the internal wave, followed after its passage, by the mill pond effect, the complete calmness of the sea. I will show an asymptotic analysis and a derivation of an effective system of PDEs modeling the coupling between the interface and the free surface of a two layers fluid in a scaling regime chosen to capture these observations.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

21 mars 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Small gaps between primes

It is believed that there should be infinitely many pairs of primes which differ by 2; this is the famous twin prime conjecture. More generally, it is believed that for every positive integer $m$ there should be infinitely many sets of $m$ primes, with each set contained in an interval of size roughly $m\log{m}$. Although proving these conjectures seems to be beyond our current techniques, recent progress has enabled us to obtain some partial results. We will introduce a refinement of the GPY sieve method' for studying these problems. This refinement will allow us to show (amongst other things) that $\liminf_n(p_{n+m}-p_n)<\infty$ for any integer $m$, and so there are infinitely many bounded length intervals containing $m$ primes.

CRM, Université de Montréal, pavillon André-Aisenstadt, 2920 chemin de la Tour, salle 6214

14 mars 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Pretentious multiplicative functions

When trying to understand extreme phenomena in mathematics, one of the natural things to study is whether the extremizer has any special structure. Indeed, the more information one has on the extremiser, the better one should be ably to analyze the phenomenon under investigation. This approach has been proven very effective when studying the average behaviour of general multiplicative functions. These are complex-valued functions defined over the integers which respect the multiplicative structure of the integers, i.e. f(mn)=f(m)f(n) when m and n are coprime. They are of central importance to number theory as several important questions in number theory can be formulated in terms of the average behaviour of them. Perhaps the most prominent example is the Riemann Hypothesis, which is equivalent to proving that the partial sums of a certain multiplicative function exhibit square-root cancellation. During the recent years, Granville and Soundararajan pioneered a new theory whose goal is to unify and extend the theory of general multiplicative functions. The starting point is a theorem of Halasz which states that if a multiplicative function assumes values inside the unit circle then its partial sums can be large only if it “pretends to be” a very special multiplicative function, the function n^{it} with t fixed. So Halasz’s theorem gives a very elegant description of the extremizers for the problem of maximizing the partial sums of a function. This simple idea, of a one multiplicative function pretending to be another one, turns out to be very potent. Indeed, using it we now have new proofs of famous old theorems, such as the Prime Number Theorem and Linnik’s theorem, concerning the existence of primes in short arithmetic progression. More importantly, the theory of pretentious multiplicative functions has shed light to problems which were previously unattackable, most prominently concerning character sums and the Quantum Unique Ergodicity conjecture. My goal in this talk is to present this new and evolving theory, and some of my contributions to it.

CRM, Université de Montréal, pavillon André-Aisenstadt, 2920 chemin de la Tour, salle 6214

14 février 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Tores plats en 3D

Un tore plat est un quotient du plan euclidien par un réseau. Topologiquement, ce n'est rien d'autre qu'une surface en forme de bouée. Métriquement en revanche, l'image de la bouée ne convient plus car celle-ci est courbée alors que le tore est plat. A cause de cette différence de courbure, on a longtemps pensé qu'il était impossible de représenter isométriquement un tore plat comme une surface dans l'espace 3D. Cette croyance va cesser au milieu des années 50 avec les travaux de J. Nash et N. Kuiper montrant l'existence d'applications isométriques des tores plats dans l'espace euclidien 3D. En utilisant une technique inventée par M. Gromov -- l'intégration convexe -- nous avons pu récemment visualiser ces applications et comprendre en partie la géométrie paradoxale de leurs images.

CRM, Université de Montréal, pavillon André-Aisenstadt, 2920 chemin de la Tour, salle 6214

7 février 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Degenerate diffusions arising in population genetics

I will speak on recent work, joint with Rafe Mazzeo and Camelia Pop, on the analysis of solutions to a class of degenerate diffusion equations that arise as limits of Markov chain models used in population genetics and mathematical finance. These equations are naturally defined on spaces with rather singular boundaries, like simplices and orthants. In addition to basic existence, uniqueness and regularity results, I will discuss Harnack inequalities and heat kernel estimates.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

17 janvier 2014 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Nondegenerate curves and pentagram maps

A plane curve is called nondegenerate if it has no inflection points. How many classes of closed nondegenerate curves exist on a sphere? We are going to see how this geometric problem, solved in 1970, reappeared along with its generalizations in the context of the Korteweg-de Vries and Boussinesq equations. Its discrete version is related to the 2D pentagram map defined by R.Schwartz in 1992. We will also describe its generalizations, pentagram maps on polygons in any dimension and discuss their integrability properties. This is a joint work with Fedor Soloviev.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

13 décembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Combinatorics and geometry of KP solitons and application to tsunami

Let $Gr(N,M)$ be the real Grassmann manifold defined by the set of all $N$-dimensional subspaces of ${\mathbb R}^M$. Each point on $Gr(N,M)$ can be represented by an $N\times M$ matrix $A$ of rank $N$. If all the $N\times N$ minors of $A$ are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by $Gr(N,M)_{\ge 0}$. In this talk, I start to give a realization of $Gr(N,M)_{\ge 0}$ in terms of the (regular) soliton solutions of the KP (KadomtsevPetviashvili) equation which is a two-dimensional extension of the KdV equation. The KP equation describes small amplitude and long waves on a surface of shallow water. I then construct a cellular decomposition of $Gr(N,M)_{\ge 0}$ with the asymptotic form of the soliton solutions. This leads to a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group $S_M$. Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Lediagram provides a complete classification of the ''entire'' spatial patterns of the soliton solutions coming from the $Gr(N,M)_{\ge 0}$ for asymptotic values of the time. I will also present some movies of real experiments of shallow water waves which represent some of those solutions obtained in the classification problem. Finally I will discuss an application of those results to analyze the Tohoku-tsunami on March 2011. The talk is elementary, and shows interesting connections among combinatorics, geometry and integrable systems.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

29 novembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Higher Pentagram Maps via Cluster Mutations and Networks on Surfaces

The pentagram map that associates to a projective polygon a new one formed by intersections of short diagonals was introduced by R. Schwartz and was shown to be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick demonstrated that the pentagram map can be put into the framework of the theory of cluster algebras, a new and rapidly developing area with many exciting connections to diverse fields of mathematics. In this talk I will explain that one possible family of higher-dimensional generalizations of the pentagram map is a family of discrete integrable systems intrinsic to a certain class of cluster algebras that are related to weighted directed networks on a torus and a cylinder. After presenting necessary background information on Poisson geometry of cluster algebras, I will show how all ingredients necessary for integrability - Poisson brackets, integrals of motion - can be recovered from combinatorics of a network. The talk is based on a joint project with M. Shapiro, S. Tabachnikov and A. Vainshtein.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

22 novembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Exact formulas in random growth

In the past few years a number of exact solutions have been discovered for the distribution of fluctuations in discrete and continuous models in the KPZ (Kardar-Parisi-Zhang) universality class. We will review some of the history of the equations, solutions, and some of the new developments.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

15 novembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Singular (arithmetic) Riemann Roch Revisited

I shall discuss an alternative approach to proving the "classical" Baum-Fulton-MacPherson singular Riemann-Roch theorem, and how this allows one to prove an arithmetic RiemannRoch Theorem. I will also give an overview of arithmetic Remann-Roch.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

25 octobre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Un survol élémentaire de la topologie symplectique sans homologie de Floer et sans théorie de jauge.

La topologie symplectique peut être pensée comme le versant mathématique de la théorie des cordes: elles sont nées toutes les deux, indépendamment, dans les années 80, la seconde comme entreprise fantastique d'unification des physiques à grande et à petite échelle, et la première pour résoudre les problèmes dynamiques classiques sur les orbites périodiques des systèmes physiques, notamment les conjectures d'Arnold. Dans les années 80, le travail révolutionnaire de Gromov a permis de présenter la topologie symplectique comme géométrie presque Kähler (un concept qu'il a défini) en construisant une théorie qui est covariante, alors que la géométrie algébrique est contravariante. Quelques années plus tard, on a compris que les aspects dynamiques et kahlériens de la topologie symplectique sont intimement reliés: c'est ce que Lalonde-McDuff ont montré en établissant l'équivalence entre le Non Squeezing theorem et l'inégalité capacité-énergie. De nos jours, la topologie symplectique est l'un des sujets les plus actifs, et il n'y a peut-être pas d'autre discipline qui produise tant de nouveaux espaces de modules à un tel rythme ! Des résultats plus récents seront aussi présentés.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

18 octobre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The Sato-Tate conjecture

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 1360

20 septembre 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Quasiperiodic Schrödinger operators

We will give an overview and briefly present some recent developments in the spectral theory of discrete one-dimensional quasiperiodic operators, focusing on several phenomena that distinguish this class from both random and periodic models: metalinsulator transitions, Cantor spectra, statistics of eigenvalues.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

12 avril 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Quantum correlations and Tsirelson's problem

The EPR paradox tells us quantum theory is incompatible with classic realistic theory. Indeed, Bell has shown that quantum correlations of independent bipartite systems have more possibility than the classical correlations. To study what the possibilities are, Tsirelson has introduced the set of quantum correlation matrices, but depending on the interpretation of independence, there are two plausible definitions of it. Tsirelson's problem asks whether these definitions are equivalent. It turned out that this problem in quantum information theory is in fact equivalent to Connes's embedding conjecture, one of the most important open problems in theory of operator algebras. I will talk some recent progress on Tsirelson's problem.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

5 avril 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Integral structures in p-adic representations

Representation theory of p-adic Lie groups such as GL_2(F), where Fis a p-adic field, is central to many problems in number theory. The coefficients of these representations have been classically taken to be the complex numbers.Although it has been known for many years that p-adic representations of the same groups offer new exciting possibilities, their study has begun, in earnest,only recently. After a brief review of the p-adic numbers we shall explain why it is important to shift from complex representations to p-adic ones, and their relation to the “p-adicLanglands program”. We shall then concentrate on the BreuilSchneider conjecture

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

28 mars 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Moser averaging

Moser averaging is a method for detecting periodic trajectories in classical mechanical systems which are small perturbations of periodic systems. (The Kepler system: the earth rotating about the sun, is probably the most familiar example of a system of this type.) In this talk I'll describe how, in the late nineteen seventies, Weinstein and Colin de Verdiere adapted Moser's techniques to the quantum mechanical setting and describe some recent applications of their results to inverse problems.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

1 mars 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Mathematical Models for River Ecosystems

River ecosystems are characterized by unidirectional flow; individuals are at risk of being transported downstream. This movement bias gives rise to the drift paradox': How can a population persist if individuals are washed out of the system? More generally, advection introduces an asymmetry into riverine ecosystems that affects not only persistence of a single population but also spatial spread and interactions between two or more species. In this talk, I will present a number of reaction-advectiondiffusion models for populations in rivers and other advective environments. I will start with fairly simple equations and move to increasingly complex models of individual behavior and species interactions. I will explain how advection affects population-level patterns, such as persistence, spread or competitive dominance. This talk is aimed at a general audience.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

15 février 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Eigenproblems, numerical approximation and proof

In this talk, we investigate the role of numerical analysis and scientific computing in the construction of rigorous proofs of conjectures. We focus on eigenproblems, and present recent progress on three unusual, conceptually simple, eigenvalue problems. We explore how validated numerics and provable convergence and error estimates are helpful in proving theorems about the eigenvalue problems. The first of these problems concerns sharp bounds on the eigenvalue of the Laplace-Beltrami operator of closed Riemannian surfaces of genus higher than one. One may ask: for a fixed genus, and a given fixed surface area, which surface maximizes the first Laplace eigenvalue? The second of these concerns eigenvalue problems for the Laplacian, with mixed Dirichlet-Neumann data. If the Neumann and Dirichlet curves meet at an angle which is $\pi$ or larger, reflection strategies will not work. The third problem is about the famous Hot Spot conjecture: the extrema of the 2nd Neumann eigenfunction of the Laplacian in an acute triangle will be at the vertices.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, salle 5340

8 février 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Pentagram Map, Twenty Years After

Introduced by R. Schwartz about 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. The pentagram map is a discrete completely integrable system whose continuous limit is the Boussinesq equation, a completely integrable PDE of soliton type. In this talk I shall survey recent work on the pentagram map and its generalizations, emphasizing its close ties with the theory of cluster algebras, a new and rapidly developing field with numerous connections to diverse areas of mathematics.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

1 février 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Proof of a 35 Year Old Conjecture for the Entropy of SU(2) Coherent States, and its Generalization

35 years ago Wehrl defined a classical entropy of a quantum density matrix using Gaussian (Schrödinger, Bargmann, ...) coherent states. This entropy, unlike other classical approximations, has the virtue of being positive. He conjectured that the minimum entropy occurs for a density matrix that is itself a projector onto a coherent state and this was proved soon after. It was then conjectured that the same thing would occur for SU(2) coherent states (maximal weight vectors in a representation of SU(2)). This conjecture, and a generalization of it, have now been proved with J.P. Solovej. (arxiv: 1208.3632). After a review of coherent states in general, a summary of the proof will be given. Obviously, one would like to prove similar conjectures for SU(n) and other Lie groups. This is open and the audience is invited to join the fun. Another question the audience is invited to think about is the meaning of all this for group representation theory. If this conjecture is correct, it must have some general significance.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

25 janvier 2013 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Global rigidity in contact topology

Contact topology studies odd-dimensional manifolds endowed with a maximally non-integrable field of hyperplanes. It is commonly considered the odd-dimensional sister of symplectic topology, with which it shares the local flexibility property. Following the work of Eliashberg-Kim-Polterovich and of myself (partly jointly with Vincent Colin) I will discuss some global rigidity phenomena for contact manifolds, that can be seen as contact analogues (but with some specific and still quite mysterious features) of the symplectic non-squeezing theorem by Gromov, of the Arnold conjecture on fixed points of Hamiltonian symplectomorphisms and of the Hofer metric on the Hamiltonian group.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

7 décembre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Igusa integrals

Geometric Igusa integrals appear as important technical tools in the study of rational and integral points on algebraic varieties. I will describe some of these applications (joint work with A. Chambert-Loir).

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

16 novembre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### On the Doi Model for the suspension of rod-like molecules & related equations

The Doi model for the suspensions of rod-like molecules in a dilute regime describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained (cf. Doi and Edwards (1986)). The orientation distribution of the rods on the microscopic level is described by a Fokker-Planck-type equation on the sphere, while the fluid flow is given by the Navier-Stokes equations, which are now enhanced by an additional macroscopic stress reflecting the orientation of the rods on the molecular level. Prescribing arbitrarily the initial velocity and the initial orientation distribution in suitable spaces we establish the global-in-time existence of a weak solution to our model defined on a bounded domain in the three dimensional space. The proof relies on a quasi-compressible approximation of the pressure, the construction of a sequence of approximate solutions and the establishment of compactness.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

2 novembre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Dissipative motion from a Hamiltonian point of view

I will study the motion of a classical particle interacting with a dispersive wave medium. (Concretely, one may think of a heavy particle interacting with an ideal Bose gas at zero temperature, in the large-density or mean-field limit.) This is an example of a Hamiltonian system with infinitely many degrees of freedom that describes dissipative phenomena. I will show that the particle experiences a friction force with memory, which is caused by the particle's emission of Cherenkov radiation of sound waves into the medium. This friction force decelerates the particle until its speed has dropped to the minimal speed of sound in the medium (=0, for an ideal Bose gas). Various open problems that I suspect might be of interest to analysts will be described. (The results presented in this lecture have been found in joint work with Daniel Egli, Gang Zhou, Avy Soffer and Israel Michael Sigal.)

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

12 octobre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Symmetry and Reflection Positivity

There are many examples in mathematics, both pure and applied, in which problems with symmetric formulations have non-symmetric solutions. Sometimes this symmetry breaking is total, but often the symmetry breaking is only partial. One technique that can sometimes be used to constrain the symmetry breaking is reflection positivity. It is a simple and useful concept that will be explained in the talk, together with some examples. One of these concerns the minimum eigenvalues of the Laplace operator on a distorted hexagonal lattice. Another example that we will discuss is a functional inequality due to Onofri. The talk is based on joint work with E. Lieb.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

21 septembre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Geometry of complex surface singularities

A complex variety has two intrinsic metric space structures in the neighborhood of any point ("inner" and "outer" metric) which are uniquely determined from the complex structure up to bilipschitz change of the metric (changing distances by at most a constant factor). In dimension 1 the inner metric (given by minimal arc-length within the variety) carries no interesting information, and it is only very recently, starting with a 2008 paper of Birbrair and Fernandes, that it has become clear how rich metric information is in higher dimensions. Dimension 2 is now very well understood through work of Birbrair, Pichon and the speaker. The talk will give an overview of this work and some applications.

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

14 septembre 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### A glimpse at the differential topology and geometry of optimal transportation

The Monge-Kantorovich optimal transportation problem is to pair producers with consumers so as to minimize a given transportation cost. When the producers and consumers are modeled by probability densities on two given manifolds or subdomains, it is interesting to try to understand the structure of the optimal pairing as a subset of the product manifold. This subset may or may not be the graph of a map. The talk will expose the differential topology and geometry underlying many basic phenomena in optimal transportation. It surveys questions concerning Monge maps and Kantorovich measures: existence and regularity of the former, uniqueness of the latter, and estimates for the dimension of its support, as well as the associated linear programming duality. It shows the answers to these questions concern the differential geometry and topology of the chosen transportation cost. It establishes new connections --- some heuristic and others rigorous --- based on the properties of the cross-difference of this cost, and its Taylor expansion at the diagonal.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

3 février 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Equivalence relations, random graphs and stochastic homogenization

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420

27 janvier 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Rational billiards and the SL(2,R) action on moduli space

Université de Montréal, Pav. AndréAisenstadt, 2920, chemin de la Tour, SALLE 6214

20 janvier 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Rational curves and rational points

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420

13 janvier 2012 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Probabilité et Physique Statistique des Systèmes Désordonnés

Université de Montréal, Pav. AndréAisenstadt, 2920, chemin de la Tour, SALLE 6214

16 décembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Disordered Bosons: A Complex Geometric Viewpoint

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420

9 décembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Balanced Splitting Methods / Infinite Matrices

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

25 novembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Groups with good pedigrees, or superrigidity revisited

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., SALLE SH-3420

18 novembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Tricks in Spectral Theory

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

11 novembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Domains with non-compact automorphism groups

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

4 novembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Teichmuller spaces of Riemann surfaces with holes and algebras of geodesic functions

Université de Montréal, Pav. AndréAisenstadt, 2920, chemin de la Tour, SALLE 6214

21 octobre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Divisors on graphs

UQAM, 200, rue Sherbrooke O. / salle SH-3420

30 septembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Variation with p of the number of solutions mod p of a system of polynomial equations

UQAM, 200, rue Sherbrooke Ouest / SALLE SH-3420

23 septembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### On Langlands functoriality

Université de Montréal, Pav. André-Aisenstadt, 2920, chemin de la Tour, SALLE 6214

16 septembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Symplectic topology in the large - from Morse to Floer and beyond

Université de Montréal, Pav. AndréAisenstadt, 2920, chemin de la Tour, SALLE 6214

15 septembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Number Theory and Dynamical Systems: A Survey

CRM, UdeM, Pav. AndréAisenstadt, 2920, ch. de la Tour, salle 6214

9 septembre 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Non-trivial convex bodies with maximal sections of constant volume

UQAM, 200, rue Sherbrooke O. / salle SH-3420

10 juin 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Symplectic homogenization

Given a Hamiltonian on $T^n\times R^n$, we shall explain how the sequence of rescaled Hamiltonians, $(\theta,p)\to H(k\theta , p)$, converges, for a suitably defined symplectic metric, as $k$ goes to infinity. We shall then explain some applications, in particular to symplectic topology and invariant measures of dynamical systems.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

6 mai 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Embedding questions in Symplectic Geometry

There has been a lot of recent progress in understanding when one open set embeds symplectically in another. This talk will describe some of the recent results and open problems. It should be accessible to those who know little or no symplectic geometry. Le colloque sera suivi d'une réception en l'honneur de M. Peter Russell, directeur du CRM,dont le mandat se terminera le 31 mai 2011. The colloquium will be followed by a reception to honour Dr. Peter Russell, whose term as CRM director will end on May 31, 2011.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

15 avril 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Rubik's Cube in Twenty Moves or Less

In July 2010 a team of four researchers led by Tomas Rokicki of Palo Alto announced that "God's Number" for the Rubik's Cube is 20, that is, any scramble can be solved in at most 20 moves (where a 90-degree or 180-degree twist counts as one move). Stated in group theory language, the problem asked for the diameter of the Cayley graph of the Rubik's Cube group using the so-called half-turn metric. The speaker had the privilege of being part of its solution, ultimately achieved through Rokicki's adaptation of Herbert Kociemba's two-step solution algorithm together with the solution of an auxiliary set cover problem and the help of Google's computing infrastructure. In this talk we will outline the thirty-year history of the problem and discuss the primary mathematical and computational breakthroughs that led to its solution.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 1360

8 avril 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Conférence du Prix CRM-Fields-PIMS 2011

"The Mathematics Behind Biological Invasion Processes" Models for invasions track the front of an expanding wave of population density. They take the form of parabolic partial differential equations and related integral formulations. These models can be used to address questions ranging from the rate of spread of introduced invaders and diseases to the ability of vegetation to shift in response to climate change. In this talk I will focus on scientific questions that have led to new mathematics and on mathematics that have led to new biological insights. I will investigate the mathematical and empirical basis for multispecies invasions, for accelerating invasion waves, and for nonlinear stochastic interactions that can determine spread rates.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

1 avril 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Number Theory and Dynamical Systems: A Survey

Recent years have seen a flourishing new field in which one studies dynamical analogues of classical results and conjectures in algebraic number theory and arithmetic geometry. In this talk I will give a survey of fundamental problems and recent results in arithmetic dynamics. To give a flavor of the talk, I mention two examples. The first is the study of the arithmetic properties of (pre)periodic points. Preperiodic points are dynamical analogues of torsion points on abelian varieties. There are many interesting arithmetic questions that one can ask about preperiodic points, including the problem of uniform boundedness, equidistribution in various topologies, and arithmetic properties of the towers of number fields that they generate. A second topic, which is also an area of much current research, is to describe the intersection of a subvariety with a special set of points such as preperiodic points, points of small height, or orbits of non-preperiodic points.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

25 mars 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Function theory on symplectic manifolds

It has been recently observed that function spaces associated to a symplectic manifold exhibit unexpected properties and surprising structures, giving rise to new tools and intuition in symplectic topology. In the talk I shall discuss these developments as well as links to other subjects such as dynamics, group theory, Lie algebras and quantum-classical correspondence. The talk is based on a series of joint works with Michael Entov.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

18 mars 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Geometry of measures

In this talk we will discuss how the regularity properties of a measure determine the geometry of its support. This classical question in geometric measure theory is easy to state. Giving an answer yields some interesting challenges.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. / salle SH-3420

11 mars 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Variational Methods in Materials and Imaging

Several questions in applied analysis motivated by issues in computer vision, physics, materials sciences and other areas of engineering may be treated variationally leading to higher order problems and to models involving lower dimension density measures. Their study often requires state-of-the-art techniques, new ideas, and the introduction of innovative tools in partial differential equations, geometric measure theory, and the calculus of variations. In this talk it will be shown how some of these questions may be reduced to well understood first order problems, while in others the higher order plays a fundamental role. Applications to phase transitions, to the equilibrium of foams under the action of surfactants, imaging, micromagnetics, thin films, and quantum dots will be addressed.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O. / salle SH-3420

4 mars 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Some random thoughts about Cauchy's functional equation

In this lecture I will discuss a variation on Cauchy's functional equation (*) $f(x + y) = f(x) + f(y)$ for all $(x; y) \in R^2$: After reviewing the familiar fact that any measurable f which satises (*) must be linear, I will investigate what can be said when (*) is replaced by (**) $f(x + y) = f(x) + f(y)$ for Lebesgue almost every $(x; y) \in R^2$: Borrowing ideas from probability theory, I will show that any measurable solution to (**) is Lebesgue almost everywhere equal to a linear function. If time permits, I will also show how the same ideas apply in more exotic settings.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

18 février 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Representation theory of semisimple groups: classical, quantum, geometric, categorical

The representation theory of semisimple Lie groups is a classical subject going back to Weyl. In the 1980s, Drinfeld, Jimbo, Reshetikhin, and others invented a quantum version of this theory. These quantum groups were used by Reshetikhin and Turaev to construct knot and 3-manifold invariants. Also during the 1980s and 1990s, two geometric approaches to the representation theory of semisimple groups emerged. The first approach, due to Lusztig, Ginzburg, Drinfeld, and Mirkovic-Vilonen, used the geometry of the affine Grassmannian. The second, due to Lusztig, Ginzburg, and Nakajima, used the geometry of quiver varieties. These geometric approaches led to further understanding of classical and quantum representation theory and in particular to the construction of canonical bases. More recently, a new categorical representation theory of semisimple groups has been developed by Khovanov, Rouquier, and others. On one hand, this categorical representation theory leads to homological knot invariant such as Khovanov homology. On the other hand, the work of Vasserot-Varagnolo, Webster, and Cautis-Licata-Kamnitzer has provided a strong interaction between this categorical representation theory and geometric representation theory. In my talk, I will attempt to survey these all these developments.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

11 février 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The 5-electron case of Thompson's problem

Thompson's problem, which dates from 1904, asks how electrons arrange themselves on the sphere so as to minimize their total potential (Coulomb) energy. Chemists and physicists have long "known" that 5 electrons will arrange themselves into a triangular bi-pyramid so as to minimize their potential energy. Recently I worked out a rigorous, computer-aided proof of this result. In my talk, I'll explain the main ideas of the proof.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

4 février 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Mahler measure as special values of $L$-functions

The Mahler measure of an $n$-variable polynomial $P$ is the integral of $log|P|$ over the $n$-dimensional unit torus $T^n$ with respect to the Haar measure. Moreover, it is often connected to special values of $L$-functions. These conections can often be explained in terms of number theoretic statements that relate L-functions to periods (such as Beilinson's conjectures). We are going to explore examples of these relationships.

CRM, UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

28 janvier 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Homotopy Theory and Spaces of Representations

Spaces of representations play a key role in many areas of mathematics and physics. In this talk we will apply basic methods from topology to analyze them, seeking to determine the number and structure of their path components. In particular we will discuss the space of commuting n-tuples in a compact Lie group.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

14 janvier 2011 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Revisiting fracture mechanics - The variational standpoint

The basic principles of brittle fracture are viewed as non-negotiable by most mechanicians. Yet, the theory, for all its achievements, has failed in various ways and it is hard-pressed when attempting any kind of prediction of the crack path in response to a loading process. Fracture has recently been revisited in a variational light. I will describe that viewpoint and discuss its impact on crack initiation, notably in the presence of cohesive forces, as well as on crack path prediction, this time in a pure Griffith framework. I will conclude the talk with an illustration of the computational power of the variational approach through examples of 2d and 3d computations under thermal" loading; those are courtesy of Blaise Bourdin at LSU.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

26 novembre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Semi-algebraic optimization theory

Concrete optimization problems, while often nonsmooth, are not pathologically so. The class of "semi-algebraic" sets and functions - those arising from polynomial inequalities - nicely exemplifies nonsmoothness in practice. Semi-algebraic sets (and their generalizations) are common, easy to recognize, and richly structured, supporting powerful variational properties. In particular I will discuss a generic property of such sets - partial smoothness - and its relationship with a proximal algorithm for nonsmooth composite minimization, a versatile model for practical optimization.

UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

19 novembre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Ramanujan reaches his hand from his grave and snatches your theorems from you

The lecture provides a survey of results of Ramanujan for which late credit has been bestowed (if at all). These theorems have the names of others attached to them, because it was not known at the time of publication that they can be found in the unpublished work of Ramanujan.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

29 octobre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The Thermodynamic Limit of Coulomb Quantum Systems

In this talk I will review the methods for studying the limit of infinitely many quantum particles interacting through the Coulomb potential, like electrons and nuclei in ordinary matter. I will in particular present a new approach which generalizes previous results of Fefferman, Lieb and Lebowitz. This is joint work with Christian Hainzl (Birmingham, Alabama) and Jan Philip Solovej (Copenhagen, Denmark).

UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

22 octobre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Stochastic homogenization and related problems

The talk will focus on homogenization theory in the non periodic context. It will be shown how some appropriately chosen deterministic generalizations of the periodic setting, and some "weakly random" generalizations can lead to theories that are both practically relevant and computationally efficient. The material presented in the talk covers joint work with X. Blanc, PL. Lions, F. Legoll, A. Anantharaman, R. Costaouec, F. Thomines.

UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

15 octobre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Grand Challenges in Complexity Theory

Conférence s'adressant à un large auditoire / Suitable for a general audience Modern complexity theory is a vast, loosely defined area. Its methods and methodology can be successfully applied whenever one has a set of tasks, a class of admissible solutions, and numerical characteristics to measure the "quality" of a solution. This talk will focus on two specific scenarios: classical computational complexity and proof complexity. The story we will tell, however, is highly representative of the methods that have been tried in the field at large, with its mixed bag of successes, setbacks, and promising directions. I will try to reveal some of the tight, beautiful and unexpected connections existing between different branches of complexity theory. I will discuss the "grand challenges" of the field, including "P vs NP" and questions about the power of classical proof systems. This talk will assume no prior knowledge in theoretical computer science.

CRM, UdeM, Pavillon André-Aisenstadt, 2920, chemin de la tour, salle 1360

24 septembre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Pointwise estimates and nonlinear stability of waves

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

17 septembre 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Régulation d'évolutions «viables» dans un environnement en avenir incertain

Les techniques de «viabilité» on été conçues pour construire des correspondances de régulation pilotant des évolutions viable dans un environnement, toujours ou jusqu'à un instant fini ou elle atteint une cible. Motivées par les sciences du vivant, elle traduisent la confrontation de la chance et de la nécessité en façonnant des techniques mathématiques pour rendre compte de mécanismes dynamiques contingents et impulsionnels d'essence darwinienne faisant face à une incertitude «tychastique» en lieu et place des techniques d'optimisation intertemporelle, d'essence téléologique. Ces techniques mathématiques ont trouvé des applications en robotique (navigation autonome pour atteindre une cible en évitant des obstacles), en … mathématiques (fractales et attracteurs de Lorenz), en gestion de trafic, débouchant sur les solutions de viabilité d'équations d'Hamilton-Jacobi, et en finance, pour gérer dynamiquement un portefeuille d'actifs couvrant un passif dans la pire des prévisions sur l'évolution des rendements des actifs risqués.

CRM, Pavillon André Aisenstadt, Université de Montréal, salle 6214

10 août 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The average rank of elliptic curves

A {\it rational elliptic curve} may be viewed as the set of solutions to an equation of the form $y^2=x^3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The {\it rank} of a rational elliptic curve measures {\it how many} rational points are needed to generate all the rational points on the curve. There is a standard conjecture---originating in work of Goldfeld and Katz-Sarnak---that states that the {\it average} rank of all elliptic curves should be 1/2; however, it has not previously been known that the average rank is even finite! In this lecture, we describe recent work that shows that the average rank is finite (in fact, we show that the average rank is bounded by 1.05). This is joint work with Arul Shankar.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

16 avril 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Surface Evolution under Curvature Flows - Existence and Optimal Regularity

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

9 avril 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Magnetic monopoles and projective geometry

Solutions to the gauge-theoretic Bogomolny equations in Euclidean space have been much studied, but their hyperbolic analogues present a number of unanswered questions. One aspect translates into the study of rational normal curves in projective space, and we shall show how a natural symmetry of the equations corresponds to a duality theorem in projective geometry.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

19 mars 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Word maps over simple groups

Given a group G, a set of variables x_1,...,x_n, and a word in the x_i and their inverses, one can define the evaluation map or *word map* from G^n to G. There has been a flurry of recent interest in results asserting that under various hypotheses, word maps are surjective or at least have large image. I will discuss some of these results, the varied techniques (from algebraic geometry, analytic number theory, and harmonic analysis) which have been employed, and a number of questions which remain open.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

12 mars 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Recent progress on the arithmetic of noncongruence modular forms

After more than one century's effort, the arithmetic of congruence modular forms is well-understood. Contrary to this, the understanding for the arithmetic of noncongruence forms is quite primitive. A main obstacle is the lack of efficient Hecke operators. However, Atkin and Swinnerton-Dyer have come up with a conjecture which is meant to play the role of Hecke operators. Further, Scholl has attached to the space of noncongruence forms a compatible family of l-adic Galois representations. In this talk we'll survey recent progress on the arithmetic of noncongruence forms and modularity of Scholl representations.

CRM, Pavillon André Aisenstadt, UniversitÃ© de Montréal, salle 6214

5 mars 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Random Schrodinger Operators and Random Matrices

It has been conjectured that the eigenvalues of the adjacency matrix of a large box in $Z^d$, $d>=3$, perturbed by the right amount of randomness, behave like the eigenvalues of a random matrix. I will discuss this and related conjectures, explain what happens in one dimension, and present a very special provable case of long boxes. Based on joint work with E. Kritchevski and B. Valko.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

26 février 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Iwasawa Theory

CRM, Pavillon André Aisenstadt, Université de Montréal, salle 6214

19 février 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Large scale behaviour of the continuum random polymer and KPZ

About 30 years ago it was observed that a large class of one dimensional random systems have highly anomalous fluctuations. We will describe some of these models and survey recent progress in proving some of the conjectured scalings and limiting distributions.

UdeM, Pav. André-Aisenstadt, 2920, ch. de la Tour, salle 6214

12 février 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Orbitopes

An orbitope is the convex hull of an orbit of a compact group acting linearly on a vector space. Instances of these highly symmetric conex bodies have appeared in many areas of mathematics and its applications, including protein reconstruction, symplectic geometry, and calibrations in differential geometry. In this talk, I will discuss Orbitopes from the perpectives of classical convexity, algebraic geometry, and optimization with an emphasis on ten motivating problems and concrete examples. This is joint work with Raman Sanyal and Bernd Sturmfels.

CRM, Pavillon André Aisenstadt, Université de Montréal, salle 6214

5 février 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Optimal multidimensional pricing facing informational asymmetry

The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory. The solution to this problem when the space of products and of buyers can each be parameterized by a single variable (say quality X, and income Y) garnered Mirrlees (1971) and Spence (1974) their Nobel prizes in 1996 and 2001. The multidimensional version of this question is a largely open problem in the calculus of variations (see Basov's book "Multidimensional Screening".) I plan to describe recent work with A Figalli and Y-H Kim, identifying structural conditions on the value b(X,Y) of product X to buyer Y which reduce this problem to a convex program in a Banach space--- leading to uniqueness and stability results for its solution, confirming robustness of certain economic phenomena observed by Armstrong (1996) such as the desirability for the monopolist to raise prices enough to drive a positive fraction of buyers out of the market, and yielding conjectures about the robustness of other phenomena observed Rochet and Chone (1998), such as the clumping together of products marketed into subsets of various dimension. The passage to several dimensions relies on ideas from differential geometry / general relativity, optimal transportation, and nonlinear PDE.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

29 janvier 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The Euler-Kronecker constant of a number field

Ihara has defined an invariant of a number field that for the rational numbers is Euler's constant and for imaginary quadratic fields is related to Kronecker's limit formula. In this talk, we will discuss various properties of this invariant and its relation to zeros of L-functions.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

15 janvier 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### The orbifold vertex: counting curves on orbifolds by counting piles of colored boxes

Over the last few decades, sophisticated theories, often inspired by string theory, have been developed for counting curves on Calabi-Yau threefolds. For the particularly nice class of toric threefolds, these theories reduce to a beautiful combinatorial problem: how many different ways are there of piling boxes in a corner? When the curve counting is considered for toric orbifolds, the combinatorial problem transforms into counting colored boxes. We will assume no knowledge of Calabi-Yau threefolds, toric geometry, orbifolds, or string theory. Experience stacking boxes in a moving van is helpful, but not necessary.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

10 janvier 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Some features and challenges of the Navier-Stokes-alpha-beta equation

The Navier-Stokes-alpha equation regularizes the Navier-Stokes equation by including additional dispersive and dissipative terms. The former term is proportional to the divergence of the corotational time-rate of the symmetric part of the gradient of the filtered velocity. The latter term is proportional to the bi-Laplacian of the filtered velocity. Both terms involve factors of the square of alpha where, roughly, alpha represents the characteristic size of the smallest resolvable eddy. Combining dispersion and dissipation yields a model with certain attractive features. In particular, the Navier-Stokes-alpha equation possesses circulation properties analogous to those of the Navier-Stokes equation and allows for simulations with less artificial damping than those arising from more conventional subgrid-scale and Reynolds stress models. One drawback concerns boundary conditions. Except for flows in periodic domains, the additional dissipative term entering the Navier-Stokes-alpha equation necessitates additional boundary conditions. Unfortunately, the conventional method used to derive the Navier-Stokes-alpha equation does not provide such conditions. The absence of physically meaningful boundary conditions limits the applicability of the model. Using a framework for fluid-dynamical theories with gradient dependencies, we have derived a flow equation — the Navier-Stokes-alphabeta equation — that includes the Navier-Stokes-alpha equation as a special case. Aside from alpha, this equation involves an additional length scale beta. For beta=alpha, our flow equation reduces to the Navier-Stokes-alpha equation. Our formulation also yields boundary conditions at walls and free surfaces. We will consider the effects of alpha and beta on the energy spectrum and the alignment between the filtered vorticity and the eigenvalues of the filtered stretching tensor in three-dimensional homogeneous and isotropic turbulent flows in a periodic cubic domain, including the limiting cases of the Navier-Stokes-alpha and Navier-Stokes equations. We will also discuss some ongoing work and open mathematical challenges associated with the model.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

8 janvier 2010 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Diophantine equations: what numbers reveal about shape and structure

This talk will be a survey of a few of the most classical Diophantine equations, and of the insights they give into mathematical structures arising in algebra, geometry and topology. Since this lecture is primarily aimed at the students participating in the McGill winter school, no prior background in mathematics at the graduate level will be assumed (at least, consciously.)

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

18 décembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### La nouvelle géométrie algébrique réelle

Je présenterai les travaux récents en topologie symplectique réelle, et notamment les travaux de Jean-Yves Welschinger sur les invariants de Gromov-Witten en géométrie algébrique réelle. Ces résultats, maintenant connus sous le nom de Gromov-Witten-Welschinger constituent une révolution et ouvrent la voie de la géométrie algébrique réelle énumérative qui stagnait depuis son enfance il y a quatre siècles.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

4 décembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Galois modules in arithmetic and geometry

At first, Galois module theory was about describing the algebraic structure of classical arithmetical objects like rings of algebraic integers and their groups of units. The theory acquired more interest when it appeared that the Galois structure of such modules is related to the behaviour of L-functions. In the 1990s it was shown that these relations could be generalised to "higher dimensional number theory", namely to relations among analogous objects arising from algebraic varieties equipped with a group action. In this talk we shall go over the basic results of the classical theory, present the geometric set-up and give indications on some current directions of research."

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

27 novembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Canonical metrics on Kähler manifolds

We will give a brief tour of the beautiful role played by canoncial metrics in the modern developments in the study of Kähler manifolds and in algebraic geometry. We will end with some new results that illustrate the current vitality and importance of this area of research and indicate its future directions.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

20 novembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### New Invariants on Algebraic Cycles

I will explain the intertwining role of Hodge theory and algebraic cycles, beginning from the classical constructions in the 1960's to the more recent developments using arithmetical normal functions.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

6 novembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Kakeya-Nikodym averages and Lp norms of eigenfunctions

On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions ${\phi_{\lambda}}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunctiongrowth. The problem which motivates us is to determine the $(M, g )$ with maximal eigenfunction growth. In an earlier work, two of us showed that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We strengthen this result here by showing that such a manifold must have a point where the set ${\mathcal R}_x$ of recurrent directions for the geodesic flow through x satisfies $|{\mathcal R}_x|>0$. We also show that if there are no such points, $L^2$-normalized quasimodes have sup-norms that are $o(\lambda^{n-1)/2})$, and, in the other extreme, we show that if there is a point blow-down $x$ at which the first return map for the flow is the identity, then there is a sequence of quasi-modes with $L^\infty$-norms that are $\Omega(\lambda^{(n-1)/2})$.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

30 octobre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### p-adic variation in the theory of automorphic forms

Automorphic forms provide powerful analytic tools for investigating subtle arithmetic properties of elliptic curves and other algebraic varieties. Nowadays this principle has been turned on its head, allowing us to apply arithmetic tools to gain insight into the existence and nature of associated automorphic forms. Wiles' proof of the modularity conjecture (from which Fermat's Last Theorem was derived) is just one well known example. In this talk we will explore the theme of p-adic variation through an investigation of simple concrete examples, and will discuss some of the unifying aspects this theme brings to arithmetic, geometry and analysis.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

9 octobre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### What is a Galois Representation?

In introductory graduate classes we learn about the Galois correspondance for finite extensions of fields. It is often useful to study infinite Galois groups. It turns out that many of the important arithmetic questions of the last 30 years, such as the Weil Conjectures, Fermat's Last Theorem, Serre's Conjecture and the Sato-Tate Conjecture involve studying representations of these infinite Galois groups. This talk will be a leisurely tour of Galois representations and their applications to a breadth of problems.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

25 septembre 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Structure of attractors for (a,b)-continued fraction transformations

I will discuss one-dimensional maps related to a family of (a,b)-continued fractions, suggested for consideration by Don Zagier, and give a sufficient condition for validity of the Reduction theory conjecture that states that the associated natural extension maps have attractors with finite rectangular structure where every point of the plane is mapped after finitely many iterations. I will show how the structure of these attractors can be computed from the data $(a,b)$, and give a dynamical interpretation of the reduction theory" that underlines these constructions. The set of parameter pairs $(a,b)$ for which the conjecture is not valid is also well-understood; in particular, the points for which the attractors do not have finite rectangular structure is a non-empty nowhere dense subset of the boundary $b=a+1$ of the set of parameters . If time permits, I will also explain how these continued fractions can be used for coding of geodesics on the modular surface. This is a joint work with Ilie Ugarcovici.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

24 avril 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Ricci Flow, Monge-Ampere Equation and Algebraic Spaces

This is an expository talk. I will discuss recent progresses on the KŠhler-Ricci flow. I will show how complex Monge-Ampere equations can be applied to studying the KŠhler-Ricci flow and how singularity formation of the KŠhler-Ricci flow interacts with the classification of algebraic manifolds. Some open problems will be also discussed.

Salle SH-3420, UQÀM, Pavillon Sherbrooke, 200, rue Sherbrooke Ouest

17 avril 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Arithmetic Laplacian

We develop an arithmetic analogue of elliptic partial differential equations. The role of the space coordinates is played by a family of primes, and that of the space derivatives along the various primes are played by corresponding Fermat quotient operators subjected to certain commutation relations. This leads to arithmetic linear partial differential equations on algebraic groups that are analogues of certain operators in analysis constructed from Laplacians. We classify all such equations on one dimensional groups, in particular on elliptic curves, and analyze their spaces of solutions.

Salle 6214, Pavillon André Aisenstadt, 2920 ch. de la Tour, Université de Montréal

de 3 avril 2009 16 h 00 à 4 mars 2009 18 h 00 (heure de Montréal/Miami) Sur place

### Problème de Riemann-Hilbert sur la sphère et combinatoire des systèmes de racines

Soient C_1, ...C_l des classes de conjugaison dans GL(n,C). Quand peut-on choisir des éléments c_1, c_2, ... appartenant respectivement a C_1,C_2, ... tels que c_1c_2...c_l=1 ? Ce problème, qui vient de la géometrie des fibres vectoriels plats sur la sphère de Riemann privée de l points, a récemment été résolu par B. Crawley-Boevey et fait intervenir de manière surprenante la combinatoire de systèmes de racines exotiques. Nous exposerons le problème, sa solution, et nous en présenterons diverses ramifications.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

27 mars 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Undecidability in Number Theory

Hilbert's Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. But the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

20 mars 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Lecture by André-Aisenstadt 2009 Prize Recipient

13 mars 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Infinitesimal Hilbert 16th Problem

The Hilbert 16th problem (2nd part) is about the maximal possible number of isolated ovals on the phase portraits of planar polynomial vector fields (these ovals are called limit cycles). Despite the progress of analysis, geometry and algebra in the 20th century, the general question remains open as it was hundred years before. Only various local or semilocal versions of this problem seem to be amenable, every time with great efforts. In this talk I will describe the recent progress in another direction of research going back to Petrovskii and Landis. This approach deals with limit cycles born from continuous families of (nonisolated) ovals. The corresponding infinitesimal Hilbert problem was intensely studied for the last 40 years or so. I will describe the first explicit uniform global upper bound for the number of limit cycles of near-Hamiltonian polynomial vector fields. The talk (based on works by G. Binyamini, D. Novikov and the speaker) is aimed for a general audience.

UQAM, Pav. Sherbrooke, 200, rue Sherbrooke O., salle SH-3420

20 février 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Branching Random Walk and Searching in Trees

The problem is related to searching in trees. Suppose we are given a complete binary tree (a rooted tree in which the root has degree 3 and every other vertex has degree 2) with independent, identically distributed random edge weights (say copies of some random variable X). The depth d(v) of a vertex v is the number of edges on the path from v to the root. We give each vertex v the label S_v which is the sum of the edge weights on the path from v to the root. For positive integers n, we let M_n be the maximum label of any vertex at depth n, and let M^* = sup {M_n: n =0,1,...}. It is of course possible that M^* is infinity. Under suitable moment assumptions on X, it is known that there is a constant A such that M_n/n --> A almost surely and in expectation. We call the cases A>0, A=0, and A< 0 supercritical, critical, and subcritical, respectively. We derive more precise information about the expected value E(M_n) than is captured by the above "law of large numbers"-style result, and derive exponential tail bounds for M_n-E(M_n). These results are "branching random walk" analogues of results Kesten (1987) proved for branching Brownian motion. Our techniques also allow us to derive information about branching random walks in higher dimensions (with steps in R^d, d > 1). When A <= 0 it makes sense to try to find the vertex of maximum weight M* in the whole tree. One possible strategy is to only explore the subtree T_0 containing the root and consisting only of vertices of non-negative weight. With probability bounded away from zero this strategy finds the vertex of maximum weight. We derive precise information about the expected running time of this strategy. Equivalently, we derive precise information about the random variable |T_0|. In the process, we also derive rather precise information about M*. This answers two questions posed by Aldous (1997). Parts of this work are joint with Nicolas Broutin and with Bruce Reed.

UQAM-Pavillon Sherbrooke- 200 Sherbooke o., salle SH 3420

13 février 2009 de 16 h 00 à 18 h 00 (heure de Montréal/Miami) Sur place

### Mathematics in the Light of Metaphor and Ambiguity.

Mathematics is often taught and discussed as though the only thing that is going on is the logical structure. From the point of view of formal logic, ambiguity is something that must be avoided at all costs. However I shall show that a kind of metaphoric ambiguity is not only very common in mathematics but also is often the essential thing that is gong on. The conventional, formal approach to math misses what is most important, the creative essence of math, which are the mathematical ideas. Ideas are where the action is but ideas do not have to be logical. This talk will explore another way to think about mathematics and point to a totally different perspective on the philosophy of math. It will be based on my recent book, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics (Princeton University Press 2007). This is a non-technical talk that will be accessible to everyone who loves mathematics and will especially interest those who love to talk and think about mathematics..