# Valentino Tosatti

McGill University

Organized by the CRM in collaboration with the Institut des sciences mathématiques (ISM), the Colloque des sciences mathématiques du Québec offers a forum for mathematicians of great reputation, who are invited to give lectures of current and general interest, and accessible to the entire Quebec mathematical community. The tradition is that these lectures are as qualitative and non-technical as possible in order to be accessible to general graduate students in mathematics and statistics.

Zoom link

December 3, 2021 from 15:30 to 16:30 (Montreal/EST time) On location

Colloquium presented by **Valentino Tosatti (McGill University)**

K3 surfaces are a class of compact complex manifolds that enjoys many special properties and play an important role in several areas of mathematics. In this colloquium I will discuss a new interplay between complex geometry and analysis on K3 surfaces equipped with their Calabi-Yau metrics, and dynamics of holomorphic diffeomorphisms of these surfaces, that Simion Filip and I have been investigating recently.

**Address**

December 10, 2021 from 14:00 to 15:00 (Montreal/EST time) On location

Colloquium presented by **Samit Dasgupta, Duke**

**Address**

December 17, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Yen-Tsung Huang**

**Address**

January 14, 2022 from 11:00 to 12:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Eva Miranda (Polytechnic University of Catalonia, Spain)**

What physical systems can be non-computational? (Roger Penrose, 1989). Is hydrodynamics capable of calculations? (Cris Moore, 1991). Can a mechanical system (including the trajectory of a fluid) simulate a universal Turing machine? (Terence Tao, 2017).

The movement of an incompressible fluid without viscosity is governed by Euler equations. Its viscid analogue is given by the Navier-Stokes equations whose regularity is one of the open problems in the list of problems for the Millenium by

the Clay Foundation. The trajectories of a fluid are complex. Can we measure its levels of complexity (computational, logical and dynamical)?

In this talk, we will address these questions. In particular, we will show how to construct a 3-dimensional Euler flow which is Turing complete. Undecidability of fluid paths is then a consequence of the classical undecidability of the halting

problem proved by Alan Turing back in 1936. This is another manifestation of complexity in hydrodynamics which is very different from the theory of chaos.

Our solution of Euler equations corresponds to a stationary solution or Beltrami field. To address this problem, we will use a mirror [5] reflecting Beltrami fields as Reeb vector fields of a contact

structure. Thus, our solutions import techniques from geometry to solve a problem in fluid dynamics. But how general are Euler flows? Can we represent any dynamics as an Euler flow? We will address this universality problem using the Beltrami/Reeb mirror again and Gromov's h-principle. We will also consider the non-stationary case. These universality features illustrate the complexity of Euler flows. However, this construction is not "physical" in the sense that the associated metric is not the euclidean metric. We will announce an euclidean construction and its implications to complexity and undecidability.

These constructions [1,2,3,4] are motivated by Tao's approach to the problem of Navier-Stokes [7,8,9] which we will also explain.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb

embeddings. https://arxiv.org/abs/1911.01963.

[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in

dimension 3. Proc. Natl. Acad. Sci. 118 (2021) e2026818118.

[3] R. Cardona, E. Miranda, D. Peralta-Salas. Turing universality of the incompressible Euler equations

and a conjecture of Moore. Int. Math. Res. Notices, , 2021;, rnab233,

https://doi.org/10.1093/imrn/rnab233

[4] R. Cardona, E. Miranda, D. Peralta-Salas. Computability and Beltrami fields in Euclidean space.

https://arxiv.org/abs/2111.03559

[5] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture.

Nonlinearity 13 (2000) 441–458.

[6] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity

4 (1991) 199–230.

[7] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.

[8] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete

Cont. Dyn. Sys. A 38 (2018) 1553–1565.

[9] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.

**Address**

January 28, 2022 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Gilles Stupfler (ENSAI)**

**Address**

Change display

Start time | Title | Speaker |
---|---|---|

2021-12-03 15:30 | K3 surfaces: geometry and dynamics | Valentino Tosatti (McGill University) |

2021-12-10 14:00 | TBA | Samit Dasgupta, Duke |

2021-12-17 10:00 | TBA | Yen-Tsung Huang |

2022-01-14 11:00 | Looking at hydrodynamics through a contact mirror: From Euler to Turing and beyond | Eva Miranda (Polytechnic University of Catalonia, Spain) |

2022-01-28 15:30 | TBA | Gilles Stupfler (ENSAI) |

December 3, 2021 from 15:30 to 16:30 (Montreal/EST time) On location

Colloquium presented by **Valentino Tosatti (McGill University)**

**Address**

December 10, 2021 from 14:00 to 15:00 (Montreal/EST time) On location

Colloquium presented by **Samit Dasgupta, Duke**

**Address**

December 17, 2021 from 10:00 to 11:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Yen-Tsung Huang**

**Address**

January 14, 2022 from 11:00 to 12:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Eva Miranda (Polytechnic University of Catalonia, Spain)**

What physical systems can be non-computational? (Roger Penrose, 1989). Is hydrodynamics capable of calculations? (Cris Moore, 1991). Can a mechanical system (including the trajectory of a fluid) simulate a universal Turing machine? (Terence Tao, 2017).

The movement of an incompressible fluid without viscosity is governed by Euler equations. Its viscid analogue is given by the Navier-Stokes equations whose regularity is one of the open problems in the list of problems for the Millenium by

the Clay Foundation. The trajectories of a fluid are complex. Can we measure its levels of complexity (computational, logical and dynamical)?

In this talk, we will address these questions. In particular, we will show how to construct a 3-dimensional Euler flow which is Turing complete. Undecidability of fluid paths is then a consequence of the classical undecidability of the halting

problem proved by Alan Turing back in 1936. This is another manifestation of complexity in hydrodynamics which is very different from the theory of chaos.

Our solution of Euler equations corresponds to a stationary solution or Beltrami field. To address this problem, we will use a mirror [5] reflecting Beltrami fields as Reeb vector fields of a contact

structure. Thus, our solutions import techniques from geometry to solve a problem in fluid dynamics. But how general are Euler flows? Can we represent any dynamics as an Euler flow? We will address this universality problem using the Beltrami/Reeb mirror again and Gromov's h-principle. We will also consider the non-stationary case. These universality features illustrate the complexity of Euler flows. However, this construction is not "physical" in the sense that the associated metric is not the euclidean metric. We will announce an euclidean construction and its implications to complexity and undecidability.

These constructions [1,2,3,4] are motivated by Tao's approach to the problem of Navier-Stokes [7,8,9] which we will also explain.

[1] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Universality of Euler flows and flexibility of Reeb

embeddings. https://arxiv.org/abs/1911.01963.

[2] R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. Constructing Turing complete Euler flows in

dimension 3. Proc. Natl. Acad. Sci. 118 (2021) e2026818118.

[3] R. Cardona, E. Miranda, D. Peralta-Salas. Turing universality of the incompressible Euler equations

and a conjecture of Moore. Int. Math. Res. Notices, , 2021;, rnab233,

https://doi.org/10.1093/imrn/rnab233

[4] R. Cardona, E. Miranda, D. Peralta-Salas. Computability and Beltrami fields in Euclidean space.

https://arxiv.org/abs/2111.03559

[5] J. Etnyre, R. Ghrist. Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture.

Nonlinearity 13 (2000) 441–458.

[6] C. Moore. Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity

4 (1991) 199–230.

[7] T. Tao. On the universality of potential well dynamics. Dyn. PDE 14 (2017) 219–238.

[8] T. Tao. On the universality of the incompressible Euler equation on compact manifolds. Discrete

Cont. Dyn. Sys. A 38 (2018) 1553–1565.

[9] T. Tao. Searching for singularities in the Navier-Stokes equations. Nature Rev. Phys. 1 (2019) 418–419.

**Address**

January 28, 2022 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Gilles Stupfler (ENSAI)**

**Address**

Change display

Start time | Title | Speaker |
---|---|---|

2021-11-26 15:30 | Adventures with Partial Identifications in Studies of Marked Individuals | Simon Bonner (University of Western Ontario) |

2021-11-19 15:30 | Exploring string vacua through geometric transitions | Tristan Collins (MIT) |

2021-11-12 15:30 | Estimating the mean of a random vector | Gabor Lugosi (ICREA-UPF and BSE) |

2021-11-05 15:30 | Mathematics has a history and a geography | Louise Poirier (Université de Montréal) |

2021-10-29 15:30 | Opinionated practices for teaching reproducibility: motivation, guided instruction and practice | Tiffany Timbers (University of British Columbia) |

2021-10-15 15:30 | Entropy along the Mandelbrot set | Giulio Tiozzo (University of Toronto) |

2021-10-08 11:00 | (Nirenberg Lecture) Recent progress on the Kannan-Lovasz-Simonovits (KLS) conjecture and Bourgain's slicing problem II | Yuansi Chen (Duke University) |

2021-10-01 11:00 | (Nirenberg Lecture) Convexity and High-Dimensional Phenomena | Bo'az Klartag (Weizmann Institute of Science) |

2021-09-24 15:00 | Deep down, everyone wants to be causal | Jennifer Hill (NYU Steinhardt) |

2021-04-30 15:00 | Knots, polynomials and signatures | Eva Bayer-Fluckiger (École Polytechnique Fédérale de Lausanne) |

2021-04-23 15:00 | Generalized gradients, conservative fields, tame potentials, and deep learning | Adrian Lewis (Cornell University) |

2021-04-16 15:00 | Reflected Brownian motion in a wedge: from probability theory to Galois theory of difference equations | Kilian Raschel (Université de Tours) |

2021-04-09 15:00 | Insect Flight from Newton’s law to Neurons | Jane Wang (Cornell University) |

2021-03-19 15:00 | ABCD asymptotic expansion for lattice Boltzmann schemes and application to compressible Navier Stokes equations | François Dubois (Le CNAM - Paris, Member of IRL-CRM CNRS) |

2021-03-12 15:30 | Nonparametric Tests for Informative Selection in Complex Surveys | Jay Breidt (Colorado State University, USA) |

2021-02-26 15:00 | Analytic solutions to algebraic equations, and a conjecture of Kobayashi | Jean-Pierre Demailly (Université Grenoble Alpes, France) |

2021-02-19 15:30 | Local smoothing for the wave equation | Larry Guth (MIT) |

2021-02-12 15:30 | Spatio-temporal methods for estimating subsurface ocean thermal response to tropical cyclones | Mikael Kuusela (Carnegie Mellon University, USA) |

2021-02-05 15:00 | Symmetry, barcodes, and Hamiltonian dynamics | Egor Shelukhin (Université de Montréal, Canada) |

2021-01-29 15:30 | Small Area Estimation in Low- and Middle-Income Countries | |

2021-01-22 15:00 | Mean curvature flow through neck-singularities | Robert Haslhofer (University of Toronto, Canada) |

2020-11-27 15:00 | Moduli of unstable objects in algebraic geometry | Frances Kirwan (University of Oxford) |

2020-11-20 15:00 | Hodge Theory of p-adic varieties | Wieslawa Niziol (CNRS, Sorbonne University) |

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 (McMaster University, 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-Fluckiger (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 (M) |

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 (Centre de recherches mathématiques) |

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 | |

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 | The role of boundary layers in the global ocean circulation | Laure Saint-Raymond (École normale supérieure, Paris) |

2015-02-05 16:00 | Cobordism and Lagrangian topology | |

2015-01-29 16:00 | Spectra and pseudospectra | 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 | |

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. | |

2013-10-18 16:00 | The Sato-Tate conjecture | Ram Murty (Queen's University) |

2013-09-20 16:00 | Quasiperiodic Schrödinger operators | |

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 | Probability and Statistical Physics of Disordered | 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 | Lecture by the 2011 CRM-Fields-PIMS Prize Recipient | 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 | |

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 | |

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 | |

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 | |

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) |

November 26, 2021 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Simon Bonner (University of Western Ontario)**

Monitoring marked individuals is a common strategy in studies of wild animals (referred to as mark-recapture or capture-recapture experiments) and hard to track human populations (referred to as multi-list methods or multiple-systems estimation). A standard assumption of these techniques is that individuals can be identified uniquely and without error, but this can be violated in many ways. In some cases, it may not be possible to identify individuals uniquely because of the study design or the choice of marks. Other times, errors may occur so that individuals are incorrectly identified. I will discuss work with my collaborators over the past 10 years developing methods to account for problems that arise when are only individuals are only partially identified. I will present theoretical aspects of this research, including an introduction to the latent multinomial model and algebraic statistics, and also describe applications to studies of species ranging from the golden mantella (an endangered frog endemic to Madagascar measuring only 20 mm) to the whale shark (the largest known species of sh measuring up to 19 m).

**Address**

November 19, 2021 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Tristan Collins (MIT)**

2021 André Aisenstadt Prize in Mathematics Recipient

A fundamental problem in string theory is the multitude of distinct geometries which give rise to consistent solutions of the vacuum equations of motion.

One possible resolution of this "vacuum degeneracy" problem is the "fantasy" that the moduli space of string vacua is connected through the process of "geometric transitions".

I will discuss some geometric problems associated to this fantasy and their applications.

**Address**

November 12, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Gabor Lugosi (ICREA-UPF and BSE)**

One of the most basic problems in statistics is the estimation of the mean of a random vector, based on independent observations. This problem has received renewed attention in the last few years, both from statistical and computational points of view. In this talk, we review some recent results on the statistical performance of mean estimators that allow heavy tails and adversarial contamination in the data. In particular, we are interested in estimators that have a near-optimal error in all directions in which the variance of the one dimensional marginal of the random vector is not too small. The material of this talk is based on a series of joint papers with Shahar Mendelson.

November 5, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Louise Poirier (Université de Montréal)**

The presentation is divided in two parts. First, the main results of our study on the “mathematical portrait” of Québec students carried out as part of the En avant math! Project (CRM-CIRANO joint project supported by the Ministry of Finance) will be presented. This report is based on the one hand on the results of the international TIMS and PISA tests for Québec elementary and secondary school students and on the other hand, on the situation of mathematics in Québec universities based on data from the Bureau de coopération interuniversitaire (BCI) in terms of evolution of student enrollments and portrait of students ( gender and status). BCI data shows that the typical student enrolled in math at Québec universities is a Canadian citizen, white and male. And the total number of registrants decreases each year (except, perhaps for the PhD students). Where are the girls? Where are the students from recent immigration? And yet on the PISA and TIMMS tests, immigrant students perform better in Canada than Canadian students (the opposite is true for most of the OECD countries).

Then, in the light of the results of the student portrait, we will discuss the social issues for more inclusive mathematics. Collaborative research with the Inuit communities of Nunavik will illustrate our point.

Note: Denis Gued will excuse me for borrowing this title. These words are reflected in his interview "Let's make mathematics amiable" with L’express.

https://www.lexpress.fr/informations/rendons-les-mathematiques-aimables_640643.html

October 29, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Tiffany Timbers (University of British Columbia)**

In the data science courses at the University of British Columbia, we define data science as the study, development and practice of reproducible and auditable processes to obtain insight from data. While reproducibility is core to our definition, most data science learners enter the field with other aspects of data science in mind, for example predictive modelling, which is often one of the most interesting topic to novices. This fact, along with the highly technical nature of the industry standard reproducibility tools currently employed in data science, present out-ofthe gate challenges in teaching reproducibility in the data science classroom. Put simply, students are not as intrinsically motivated to learn this topic, and it is not an easy one for them to learn. What can a data science educator do? Over several iterations of teaching courses focused on reproducible data science tools and workflows, we have found that providing extra motivation, guided instruction and lots of practice are key to effectively teaching this challenging, yet important subject. Here we present examples of how we deeply motivate, effectively guide and provide ample practice opportunities to data science students to effectively engage them in learning about this topic.

October 15, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Giulio Tiozzo (University of Toronto)**

2021 André Aisenstadt Prize in Mathematics Recipient

The notion of topological entropy, arising from information theory, is a fundamental tool to understand the complexity of a dynamical system. When the dynamical system varies in a family, the natural question arises of how the entropy changes with the parameter.

In the last decade, W. Thurston introduced these ideas in the context of complex dynamics by defining the "core entropy" of a quadratic polynomials as the entropy of a certain forward-invariant set of the Julia set (the Hubbard tree).

As we shall see, the core entropy is a purely topological/combinatorial quantity which nonetheless captures the richness of the fractal structure of the Mandelbrot set. In particular, we will relate the variation of such a function to the geometry of the Mandelbrot set. We will also prove that the core entropy on the space of polynomials of a given degree varies continuously, answering a question of Thurston.

Finally, we will provide a new interpretation of core entropy in terms of measured laminations, and discuss its finer regularity properties such as its Holder exponent.

October 8, 2021 from 11:00 to 12:00 (Montreal/EST time)

Colloquium presented by **Yuansi Chen (Duke University)**

In recent work, Chen (2020) improved Eldan's stochastic localization proof technique, which was deployed in Lee and Vempala (2017), to prove an almost constant Cheeger isoperimetric coefficient in the KLS conjecture with dimension dependency d^o(1). Consequently, his proof also provides a substantial advance toward the resolution of Bourgain's slicing conjecture and the thin-shell conjecture. After getting conformable with Eldan's stochastic localization technique, in this talk we navigate through how to refine the technique to provide the current best bound. We will complete the self-contained proof of Chen (2020) and highlight the new ideas involved. Finally, we will discuss some extensions and provide an outlook for future research directions.

**Address**

October 1, 2021 from 11:00 to 12:00 (Montreal/EST time)

Colloquium presented by **Bo'az Klartag (Weizmann Institute of Science)**

High-dimensional problems with a geometric flavor appear in a number of branches of mathematics and mathematical physics. A priori, it seems that the immense diversity observed in high dimensions would make it impossible to formulate general, interesting theorems that apply to large classes of high-dimensional geometric objects. In this talk we will discuss situations in which high dimensionality, when viewed correctly, induces remarkable order and simplicity rather than complication. For example, Dvoretzky's theorem demonstrates that any high-dimensional convex body possesses nearly-Euclidean sections of large dimension. Another example is the central limit theorem for convex bodies, according to which any high-dimensional convex body has approximately-Gaussian marginals. There are strong motifs in high-dimensional geometry, such as the concentration of measure, which appear to compensate for the large number of different configurations. Convexity allows us to harness these motifs in order to formulate elegant and non-trivial theorems.

**Address**

September 24, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Jennifer Hill (NYU Steinhardt)**

Most researchers in the social, behavioral, and health sciences are taught to be extremely cautious in making causal claims. However, causal inference is a necessary goal in research for addressing many of the most pressing questions around policy and practice. In the past decade, causal methodologists have increasingly been using and touting the benefits of more complicated machine learning algorithms to estimate causal effects. These methods can take some of the guesswork out of analyses, decrease the opportunity for “p-hacking,” and may be better suited for more fine-tuned tasks such as identifying varying treatment effects and generalizing results from one population to another. However, should these more advanced methods change our fundamental views about how difficult it is to infer causality? In this talk I will discuss some potential advantages and disadvantages of using machine learning for causal inference and emphasize ways that we can all be more transparent in our inferences and honest about their limitations.

April 30, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Eva Bayer-Fluckiger (École Polytechnique Fédérale de Lausanne)**

After a historical introduction to knot theory, the talk will be centered around two knot invariants, the Alexander polynomial and the signature. The aim is to introduce a finite abelian group that controls their relationship, and to illustrate this by several examples. Using Seifert matrices, the geometric questions are translated into arithmetic ones.

April 23, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Adrian Lewis (Cornell University)**

To the dismay and irritation of the variational analysis community, practitioners of deep learning often implement gradient-based optimization via automatic differentiation and blithely apply the result to nonsmooth objectives. Worse, they then gleefully point out numerical convergence. In fact, as elegantly remarked by Bolte and Pauwels, automatic differentiation produces a novel generalized gradient: a conservative field with enough calculus to prove convergence of stochastic subgradient descent, as practiced in deep learning. I will sketch this interplay of analytic and algorithmic ideas, and explain how, for concrete objectives (typically semi-algebraic), this novel generalized gradient just slightly modifies Clarke's original notion.

Joint work with Tonghua Tian.

April 16, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Kilian Raschel (Université de Tours)**

We consider a reflected Brownian motion in a two-dimensional wedge. Under standard assumptions on the parameters of the model (opening of the wedge, angles of the reflections on the axes, drift), we study the algebraic and differential nature of the Laplace transform of its stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. These conditions are explicit linear dependencies among the angles involved in the definition of the model.

To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte's invariant approach, which originates in enumerative combinatorics. To establish differential transcendence, we turn the functional equation into a difference equation and apply Galoisian results on the nature of the solutions to such equations.

This is a joint work with M. Bousquet-Mélou, A. Elvey Price, S. Franceschi and C. Hardouin (https://arxiv.org/abs/2101.01562).

April 9, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Jane Wang (Cornell University)**

Why do animals move the way they do? Bacteria, insects, birds, and fish share with us the necessity to move so as to live. Although each organism follows its own evolutionary course, it also obeys a set of common laws. At the very least, the movement of animals, like that of planets, is governed by Newton’s law: All things fall. On Earth, most things fall in air or water, and their motions are thus subject to the laws of hydrodynamics. Through trial and error, animals have found ways to interact with fluid so they can float, drift, swim, sail, glide, soar, and fly. This elementary struggle to escape the fate of falling shapes the development of motors, sensors, and mind. Perhaps we can deduce parts of their neural computations by understanding what animals must do so as not to fall.

We have been seeking mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, we worked to build a theoretical framework for computing flight and for studying the control of flight. I will discuss our recent computational and experimental studies of the balancing act of dragonflies and fruit flies: how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case, the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.

March 19, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **François Dubois (Le CNAM - Paris, Member of IRL-CRM CNRS)**

We first recall some elements of history of the construction of lattice Boltzmann schemes. Then we present our "ABCD" approach, founded on the property that the numerical scheme is exact for the advection equation with the velocities of the lattice. This asymptotic analysis allows to write at several orders the conservative partial differential equations equivalent to the numerical scheme. A fit of parameters permits in favorable cases a precise approximation of compressible fluids equations.

March 12, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Jay Breidt (Colorado State University, USA)**

Informative selection, in which the distribution of response variables given that they are sampled is different from their distribution in the population, is pervasive in complex surveys. Failing to take such informativeness into account can produce severe inferential errors, including biased and inconsistent estimation of population parameters. While several parametric procedures exist to test for informative selection, these methods are limited in scope and their parametric assumptions are difficult to assess. We consider two classes of nonparametric tests of informative selection. The first class is motivated by classic nonparametric two-sample tests. We compare weighted and unweighted empirical distribution functions and obtain tests for informative selection that are analogous to Kolmogorov-Smirnov and Cramer-von Mises. For the second class of tests, we adapt a kernel-based learning method that compares distributions based on their maximum mean discrepancy. The asymptotic distributions of the test statistics are established under the null hypothesis of noninformative selection. Simulation results show that our tests have power competitive with existing parametric tests in a correctly specified parametric setting, and better than those tests under model misspecification. A recreational angling application illustrates the methodology.

This is joint work with Teng Liu, Colorado State University.

February 26, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Jean-Pierre Demailly (Université Grenoble Alpes, France)**

A projective algebraic variety is defined as the zero locus of a finite family of homogeneous polynomials. Over the field of complex numbers, the geometry of such varieties is governed to a large extent by the sign, in a suitable sense, of the Ricci curvature form. When this sign is negative, the variety is expected to exhibit certain hyperbolicity properties in the sense of Kobayashi - as well as further very deep number-theoretic properties that are mostly conjectural, in the arithmetic situation. In particular, all entire holomorphic curves drawn on it should be contained in a proper algebraic subvariety: this is a famous conjecture of Green-Griffiths and Lang. Following recent ideas of D. Brotbek, we will try to explain here a rather elementary proof of a related conjecture of Kobayashi, stating that a general algebraic hypersurface of sufficiently high degree is hyperbolic, i.e. does not contain any entire holomorphic curve.

February 19, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Larry Guth (MIT)**

The local smoothing problem asks about how much solutions to the wave equation can focus. It was formulated by Chris Sogge in the early 90s. Hong Wang, Ruixiang Zhang, and I recently proved the conjecture in two dimensions.

February 12, 2021 from 15:30 to 16:30 (Montreal/EST time)

Colloquium presented by **Mikael Kuusela (Carnegie Mellon University, USA)**

Tropical cyclones (TCs), driven by heat exchange between the air and sea, pose a substantial risk to many communities around the world. Accurate characterization of the subsurface ocean thermal response to TC passage is crucial for accurate TC intensity forecasts and for understanding the role TCs play in the global climate system, yet that characterization is complicated by the high-noise ocean environment, correlations inherent in spatio-temporal data, relative scarcity of in situ observations and the entanglement of the TC-induced signal with seasonal signals. We present a general methodological framework that addresses these difficulties, integrating existing techniques in seasonal mean field estimation, Gaussian process modeling, and nonparametric regression into a functional ANOVA model. Importantly, we improve upon past work by properly handling seasonality, providing rigorous uncertainty quantification, and treating time as a continuous variable, rather than producing estimates that are binned in time. This functional ANOVA model is estimated using in situ subsurface temperature profiles from the Argo fleet of autonomous floats through a multi-step procedure, which (1) characterizes the upper ocean seasonal shift during the TC season; (2) models the variability in the temperature observations; (3) fits a thin plate spline using the variability estimates to account for heteroskedasticity and correlation between the observations. This spline fit reveals the ocean thermal response to TC passage. Through this framework, we obtain new scientific insights into the interaction between TCs and the ocean on a global scale, including a three-dimensional characterization of the near-surface and subsurface cooling along the TC storm track and the mixing-induced subsurface warming on the track's right side. Joint work with Addison Hu, Ann Lee, Donata Giglio and Kimberly Wood.

February 5, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Egor Shelukhin (Université de Montréal, Canada)**

2020 André Aisenstadt Prize in Mathematics Recipient

In the early 60s Arnol'd has conjectured that Hamiltonian diffeomorphisms, the motions of classical mechanics, often possess more fixed points than required by classical topological considerations. In the late 80s and early 90s Floer has developed a powerful theory to approach this conjecture, considering fixed points as critical points of a certain functional. Recently, in joint work with L. Polterovich, we observed that Floer theory filtered by the values of this functional fits into the framework of persistence modules and their barcodes, originating in data sciences. I will review these developments and their applications, which arise from a natural time-symmetry of Hamiltonians. This includes new constraints on one-parameter subgroups of Hamiltonian diffeomorphisms, as well as my recent solution of the Hofer-Zehnder periodic points conjecture. The latter combines barcodes with equivariant cohomological operations in Floer theory recently introduced by Seidel to form a new method with further consequences.

January 29, 2021 from 15:30 to 16:30 (Montreal/EST time)

The under-five mortality rate (U5MR) is a key barometer of the health of a nation. Unfortunately, many people living in low- and middle-income countries are not covered by civil registration systems. This makes estimation of the U5MR, particularly at the subnational level, difficult. In this talk, I will describe models that have been developed to produce the official United Nations (UN) subnational U5MR estimates in 22 countries. Estimation is based on household surveys, which use stratified, two-stage cluster sampling. I will describe a range of area- and unit-level models and describe the rationale for the modeling we carry out. Data sparsity in time and space is a key challenge, and smoothing models are vital. I will discuss the advantages and disadvantages of discrete and continuous spatial models, in the context of estimation at the scale at which health interventions are made. Other issues that will be touched upon include: design-based versus model-based inference; adjustments for HIV epidemics; the inclusion of so-called indirect (summary birth history) data; reproducibility through software availability; benchmarking; how to deal with incomplete geographical data; and working with the UN to produce estimates.

January 22, 2021 from 15:00 to 16:00 (Montreal/EST time)

Colloquium presented by **Robert Haslhofer (University of Toronto, Canada)**

2020 André Aisenstadt Prize in Mathematics Recipient

A family of surfaces moves by mean curvature flow if the velocity at each point is given by the mean curvature vector. Mean curvature flow first arose as a model of evolving interfaces and has been extensively studied over the last 40 years.

In this talk, I will give an introduction and overview for a general mathematical audience. To gain some intuition we will first consider the one-dimensional case of evolving curves. We will then discuss Huisken’s classical result that the flow of convex surfaces always converges to a round point. On the other hand, if the initial surface is not convex we will see that the flow typically encounters singularities. Getting a hold of these singularities is crucial for most striking applications in geometry, topology and physics. Specifically, singularities can be either of neck-type or conical-type. We will discuss examples from the 90s, which show, both experimentally and theoretically, that flow through conical singularities is utterly non-unique.

In the last part of the talk, I will report on recent work with Kyeongsu Choi, Or Hershkovits and Brian White, where we proved that mean curvature flow through neck-singularities is unique. The key for this is a classification result for ancient asymptotically cylindrical flows that describes all possible blowup limits near a neck-singularity. In particular, this confirms the mean-convex neighborhood conjecture. Assuming Ilmanen’s multiplicity-one conjecture, we conclude that for embedded two-spheres mean curvature flow through singularities is well-posed.

November 27, 2020 from 15:00 to 16:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Frances Kirwan (University of Oxford)**

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.

November 20, 2020 from 15:00 to 16:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Wieslawa Niziol (CNRS, Sorbonne University)**

**Chaire Aisenstadt Chair Conference**

**Thematic Semester: Number Theory - Cohomology in Arithmetic**

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.

As part of the **Thematic Semester**, Wieslawa Niziola will give a series of four lectures including this one (Hodge Theory of p-adic varieties). The following three will take place from 9:30 a.m. to 10:30 a.m. on Monday November 30, Tuesday December 1st and Wednesday December 2, 2020. Info and registration: http://www.crm.umontreal.ca/2020/Niziol20/index_e.php

November 13, 2020 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Tamara Broderick (Massachusetts Institute of Technology, USA)**

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.

October 16, 2020 from 15:00 to 16:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Nicolas Bergeron (École normale supérieure (Paris), France)**

**Chaire Aisenstadt Chair Conference**

**Thematic Semester: Number Theory - Cohomology in Arithmetic**

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.

October 9, 2020 from 15:00 to 16:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Phillip Griffiths (Institute for Advanced Study, Princeton, USA)**

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

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.

October 2, 2020 from 15:30 to 16:30 (Montreal/EST time) Zoom meeting

Colloquium presented by **Paul McNicholas (McMaster University, Canada)**

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.

Organizers:

Erica E. M. Moodie (erica.moodie@mcgill.ca)

Yogendra P. Chaubey (yogen.chaubey@concordia.ca)

September 11, 2020 from 16:00 to 17:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Stefan Wager (Stanford University, USA)**

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.

June 19, 2020 from 16:00 to 17:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Morgan Craig (Université de Montréal)**

COVID-19 is typically characterized by a range of respiratory symptoms that, in severe cases, progress to acute respiratory distress syndrome (ARDS). These symptoms are also frequently accompanied by a range of inflammatory indications, particularly hyper-reactive and dysregulated inflammatory responses in the form of cytokine storms and severe immunopathology. Much remains to be uncovered about the mechanisms that lead to disparate outcomes in COVID-19. Here, quantitative approaches, especially mechanistic mathematical models, can be leveraged to improve our understanding of the immune response to SARS-CoV-2 infection.

Building upon our prior work modelling the production of innate immune cell subsets and the viral dynamics of HIV and oncolytic viruses, we are developing a quantitative framework to interrogate open questions about the innate and adaptive immune reaction in COVID-19. In this talk, I will outline our recent work modelling SARS-CoV-2 viral dynamics and the ensuing immune response at both the tissue and systemic levels. A portion of this work is done as part of an international and multidisciplinary coalition working to establish a comprehensive tissue simulator (physicell.org/covid19 [1]), which I will also discuss in more detail.

April 17, 2020 from 16:00 to 17:00 (Montreal/EST time) Zoom meeting

Colloquium presented by **Lai-Sang Young (New York University Courant)**

The terms "observable events" and "typical trajectories" in the title should really be between quotation marks, because what is typical and/or observable is a matter of interpretation. For dynamical systems on finite dimensional spaces, one often equates observable events with positive Lebesgue measure sets, and invariant distributions that reflect the large-time behaviors of positive Lebesgue measure sets of initial conditions (such as Liouville measure for Hamiltonian systems) are considered to be especially important. I will begin by introducing these concepts for general dynamical systems -- including those with attractors -- describing a simple dynamical picture that one might hope to be true. This picture does not always hold, unfortunately, but a small amount of random noise will bring it about. In the second part of my talk I will consider infinite dimensional systems such as semi-flows arising from dissipative evolutionary PDEs. I will discuss the extent to which the ideas above can be generalized to infinite dimensions, and propose a notion of "typical solutions".

May 16, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jungkay A. Chen (National Taiwan University)**

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.

**Address**

May 10, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Amanda Folsom (Amherst College)**

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).

**Address**

May 3, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Lenya Ryzhik (Stanford University)**

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.

**Address**

April 26, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alan W. Reid (Rice University)**

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).

**Address**

April 12, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Nicolas Bergeron (École normale supérieure (Paris), France)**

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.

**Address**

March 29, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Minhyong Kim (University of Oxford)**

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.

**Address**

March 22, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Emmy Murphy (Northwestern University)**

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.

**Address**

March 19, 2019 from 14:30 to 16:30 (Montreal/EST time) On location

Colloquium presented by **Andrew Marks (UCLA)**

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.

**Address**

March 15, 2019 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Shmuel Weinberger (University of Chicago)**

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).

**Address**

November 2, 2018 from 16:00 to 17:00 (Montreal/EST time)

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.

**Address**

September 28, 2018 from 16:00 to 17:00 (Montreal/EST time) On location

Colloquium presented by **Hans-Otto Walther (Universität Giessen)**

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.

**Address**

September 21, 2018 from 16:00 to 17:00 (Montreal/EST time) On location

Colloquium presented by **Ezra Miller (Duke University)**

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.

**Address**

May 4, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Emmanuel Hebey (Université de Cergy-Pontoise)**

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.

**Address**

April 13, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Eva Bayer-Fluckiger (École Polytechnique Fédérale de Lausanne)**

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.

**Address**

February 23, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Sabin Cautis (University of British Columbia)**

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.

**Address**

February 16, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alexandre Turbiner (UNAM)**

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.

**Address**

February 16, 2018 from 15:30 to 17:30 (Montreal/EST time) On location

Colloquium presented by **Xiao-Li Meng (Harvard University)**

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.

**Address**

February 9, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Egor Shelukhin (Université de Montréal, Canada)**

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.

**Address**

January 12, 2018 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Semyon Dyatlov (UC Berkeley / MIT)**

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.

**Address**

December 8, 2017 from 16:00 to 16:00 (Montreal/EST time) On location

Colloquium presented by **James Maynard (University of Oxford)**

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

**Address**

November 24, 2017 from 15:30 to 17:30 (Montreal/EST time) On location

Colloquium presented by **David R. Bellhouse (Western University, London, Ontario)**

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.

**Address**

November 24, 2017 from 15:30 to 17:30 (Montreal/EST time) On location

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.

**Address**

November 17, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jun-Cheng Wei (UBC)**

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.

**Address**

October 27, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Justin Solomon (M)**

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.

**Address**

October 13, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Avi Soffer (Rutgers University)**

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.

**Address**

September 29, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **John H. Conway (Princeton University)**

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!

**Address**

September 15, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

May 5, 2017 from 16:00 to 16:00 (Montreal/EST time) On location

Colloquium presented by **Gerard Freixas (Institut de Mathématiques de Jussieu)**

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).

**Address**

April 21, 2017 from 16:00 to 16:00 (Montreal/EST time) On location

Colloquium presented by **Aaron Naber (Northwestern University)**

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.

**Address**

March 31, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Tatiana Toro (University of Washington)**

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.

**Address**

March 17, 2017 from 15:30 to 17:30 (Montreal/EST time) On location

Colloquium presented by **Sayan Mukherjee (Duke University)**

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.

**Address**

March 10, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Louigi Addario-Berry (Université McGill)**

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.

**Address**

February 24, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Frithjof Lutscher (Université d'Ottawa)**

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.

**Address**

February 10, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Mark Powell (UQAM)**

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.

**Address**

January 20, 2017 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Christopher Skinner (Princeton University)**

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.

**Address**

December 2, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

December 1, 2016 from 15:30 to 17:30 (Montreal/EST time) On location

Colloquium presented by **Richard Samworth (University of Cambridge)**

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.

**Address**

November 26, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Maksym Radziwill (McGill University)**

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.

**Address**

November 4, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Philippe G. LeFloch (Université Pierre et Marie Curie, Paris 6)**

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.

**Address**

October 28, 2016 from 15:30 to 17:30 (Montreal/EST time) On location

Colloquium presented by **Jerry Lawless (University of Waterloo)**

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.

**Address**

October 21, 2016 from 16:00 to 16:00 (Montreal/EST time) On location

Colloquium presented by **Ivan Corwin (Columbia University)**

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.

**Address**

October 14, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jean-Philippe Lessard (Centre de recherches mathématiques)**

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.

**Address**

September 30, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Liam Watson (Université de Sherbrooke)**

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.

**Address**

September 16, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **B.L.S. Prakasa Rao (CR Rao Advanced Institute, Hyderabad, India)**

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.

**Address**

September 16, 2016 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Nick Trefethen (University of Oxford)**

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.

**Address**

April 9, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Stephen S. Kudla (University of Toronto)**

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.

**Address**

April 2, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **William Minicozzi (Massachusetts Institute of Technology)**

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.

**Address**

March 26, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

March 19, 2015 from 16:00 to 16:00 (Montreal/EST time) On location

Colloquium presented by **Alexei Borodin (Massachusetts Institute of Technology)**

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.

**Address**

March 12, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Pierre Colmez (CNRS & Paris VI Jussieu)**

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.

**Address**

March 5, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Sophie Morel (Princeton University)**

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

**Address**

February 26, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alistair Savage (University of Ottawa)**

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.

**Address**

February 19, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Francis Brown (IHES, Bures-sur-Yvette)**

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.

**Address**

February 12, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Laure Saint-Raymond (École normale supérieure, Paris)**

Understanding the mechanisms that govern ocean circulation is a challenge for geophysicists, but also for mathematicians who must develop new analytical tools for these complex models (which involve in particular very numerous time and space scales). A particularly important mechanism for broad-scale planetary circulation is the boundary layer phenomenon which explains part of the energy exchanges. We will show here through a very simplified model that it allows us to explain in particular the intensification of the western edge currents. We will then evoke the mathematical difficulties linked to the consideration of geometry. Note: the presentation will be in English with transparencies in French.

**Address**

February 5, 2015 from 16:00 to 16:00 (Montreal/EST time) On location

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.

**Address**

January 29, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Thomas Ransford (Université Laval)**

Eigenvalues are amongst the most useful tools of mathematics: they permit diagonalization of matrices, they describe asymptotics and stability, they give a matrix personality. However, when the matrix in question is not normal, standard eigenvalue analysis is only partially applicable and can even be misleading. This talk will be an introduction to the theory of pseudospectra, a refinement of standard spectral theory which has proved successful in applications concerning non-normal matrices. In particular I shall focus on the question: do pseudospectra determine matrix behavior?

**Address**

January 22, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Hansjoerg Albrecher (HEC, Lausanne)**

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.

**Address**

January 15, 2015 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Fang Yao (University of Toronto - Lauréat du Prix CRM-SSC)**

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.

**Address**

December 4, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

November 20, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Martin Wainwright (University of California, Berkeley)**

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.

**Address**

November 13, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Kartik Prasanna (University of Michigan, Ann Arbor)**

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.

**Address**

November 6, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Dani Wise (McGill University)**

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.

**Address**

October 30, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Georgia Benkart (University of Wisconsin-Madison)**

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.

**Address**

October 9, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alex Kontorovich (Rutgers University)**

**Address**

May 2, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Eric Urban (Columbia University)**

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.

**Address**

April 11, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alexey Kobotov (Concordia University)**

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.

**Address**

April 4, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

March 21, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **James Maynard (University of Oxford)**

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.

**Address**

March 14, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Dimitris Koukoulopoulos (Université de Montréal)**

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.

**Address**

February 14, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Vincent Borrelli (Université Claude Bernard-Lyon 1)**

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.

**Address**

February 7, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Charles Epstein**

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.

**Address**

January 17, 2014 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Boris Khesin (University of Toronto)**

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.

**Address**

December 13, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Yuji Kodama (Ohio State University)**

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.

**Address**

November 29, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Michael Gekhtman (Université of Notre-Dame)**

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.

**Address**

November 22, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jeremy Quastel (University of Toronto)**

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.

**Address**

November 15, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Henri Gillet (University of Illinois, Chicago)**

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.

**Address**

October 25, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

October 18, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Ram Murty (Queen's University)**

**Address**

September 20, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

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.

**Address**

April 12, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Narutaka Ozawa (RIMS, Kyoto University)**

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.

**Address**

April 5, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Ehud de Shalit (Hebrew University of Jerusalem)**

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

**Address**

March 28, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Victor Guillemin (Massachusetts Institute of Technology)**

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.

**Address**

March 1, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Frithjof Lutscher (Université d'Ottawa)**

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.

**Address**

February 15, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Nilima Nigam (Simon Fraser University, Canada)**

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.

**Address**

February 8, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Sergei Tabachnikov (Pennsylvania State University)**

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.

**Address**

February 1, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Elliott Lieb (Princeton University)**

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.

**Address**

January 25, 2013 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Sheila Margherita Sandon (CNRS, Nantes and CRM)**

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.

**Address**

December 7, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Yuri Tschinkel (New York University and Simons Foundation)**

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).

**Address**

November 16, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Konstantina Trivisa (University of Maryland)**

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.

**Address**

November 2, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jürg Fröhlich (ETH Zurich)**

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.)

**Address**

October 12, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Rupert Frank (Princeton University and Caltech)**

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.

**Address**

September 21, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Walter Neumann (Barnard College, Columbia University)**

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.

**Address**

September 14, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Robert McCann (University of Toronto)**

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.

**Address**

February 3, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Vadim Kaimanovich**

**Address**

January 27, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alex Eskin**

**Address**

January 20, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jason Starr**

**Address**

January 13, 2012 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Louis-Pierre Arguin**

**Address**

December 16, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alan Huckleberry**

**Address**

December 9, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Gilbert Strang**

**Address**

November 25, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Alex Furman**

**Address**

November 18, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Michael Levitin**

**Address**

November 11, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Bun Wong**

**Address**

November 4, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Leonid Chekhov**

**Address**

October 21, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Sergey Norin**

**Address**

September 30, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jean-Pierre Serre**

**Address**

September 23, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Jayce Getz**

**Address**

September 16, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

**Address**

September 15, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

**Address**

September 9, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Fedor Nazarov**

**Address**

June 10, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Claude Viterbo**

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.

**Address**

May 6, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Dusa McDuff**

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.

**Address**

April 15, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Morley Davidson**

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.

**Address**

April 8, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Mark Lewis**

"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.

**Address**

April 1, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Joseph Silverman**

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.

**Address**

March 25, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Leonid Polterovich**

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.

**Address**

March 18, 2011 from 16:00 to 18:00 (Montreal/EST time) On location

Colloquium presented by **Tatiana Toro (University of Washington)**

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.

**Address**