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Math & Stats Department Colloquium
Friday, January 17, 2025

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Jean-Christophe Nave (McGill University)

Title: Simulating problems with a wide range of scales without paying (much) for it

Abstract: In this talk I will present a new strategy to solve evolution PDEs which solutions develop a wide range of scales. Typical approaches to capture fine scales include mesh refinement (h-adaptivity) and polynomial adaptivity. I will show that by geometrizing the problem, one may leverage the semigroup structure of the solution operator. As a result, one may discretize this operator as a composition of submaps over finite time intervals. This leads to a new type of space-time adaptivity: compositional adaptivity. This approach possesses exceptional resolution properties while remaining computationally efficient. I will illustrate this technique with problems from fluid dynamics.

Math & Stats Department Colloquium
Friday, January 10, 2025

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Sinan Gezici (Bilkent University)

Title: Optimal Decision Rules for Hypothesis Testing Problems under General Criterion Involving Error Probabilities

Abstract: In this talk, we consider M-ary hypothesis-testing problems and characterize optimal decision rules under general criterion involving pairwise error probabilities, where likelihood ratios are employed as sufficient statistics. It is proved that an optimal decision rule can be obtained as a randomization between at most two deterministic decision rules in certain cases while a randomization among at most M(M-1)+1 deterministic decision rules can be required in other cases for optimality. We focus on some special cases of this generic approach such as prospect theory based hypothesis testing, where behavioral decision-makers distort probabilities and costs. We investigate the optimality of the likelihood ratio test (LRT) for prospect theory based binary hypothesis testing, and show that randomization of LRTs can be required for optimality. We provide various detection examples to illustrate the results.

Math & Stats Department Colloquium
Friday, November 29th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Sina Sanjari (Royal Military College of Canada)

Title: Optimality of Symmetric Policies under Decentralized Information and Controlled Measure

Abstract: We consider multi-agent stochastic systems comprising a finite number of decision-makers and their mean-field limits. We focus on stochastic exchangeable systems with centralized, mean-field sharing, and fully decentralized information structures. For finite population systems, we show that an optimal policy exists that is exchangeable (permutation invariant). We then show that a sequence of exchangeable optimal policies for a finite population setting converges to an optimal policy for the infinite population problem, which is conditionally symmetric (identical), independent, and decentralized. This symmetry in optimal decision-making proves the optimality of a representative single-agent problem with controlled probability measures. Finally, we discuss connections to the measure-valued Markov decision processes and a dynamic programming approach.

Algebra & Geometry Seminar
Tuesday, November 26th, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker:  Rikuto Ito (Nagoya University)

Title: Potential modularity of K3 surfaces with complex multiplication

Abstract: Let X be a K3 surface defined over a number field k. Let T(X) be the transcendental lattice of H^2(X, Q(1)) of rank 22-ρ(X) where ρ(X) is the Picard number of X. We prove that T_ℓ(X, k) is modular if X has complex multiplication over k.

PDEs & Applications Seminar

Friday, November 22nd, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Catherine Sulem (University of Toronto)

Title: Bloch-Floquet band gaps for linearized water waves over a periodic bottom

Abstract: A central object in the analysis of the water wave problem is the Dirichlet-Neumann operator. This work concerns the study of its spectrum in the context of the water wave system linearized near equilibrium in a domain with a variable bottom, assumed to be a smooth periodic function. We use the analyticity of the Dirichlet-Neumann operator with respect to the bottom variation and combine it with general properties of elliptic systems and spectral theory for self-adjoint operators to develop a Bloch-Floquet theory and describe the structure of its spectrum. We find that under some conditions on the bottom variations, the spectrum is composed of bands separated by gaps, with explicit formulas for their sizes and locations. This is a joint work with Christophe Lacave and Matthieu Ménard.

Math & Stats Department Colloquium
Friday, November 22nd, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Catherine Sulem (University of Toronto)

Title: A Hamiltonian approach to nonlinear modulation of surface water waves in the presence of linear shear currents

Abstract: This is a study of the water wave problem in a two-dimensional domain in the presence of constant vorticity. The goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear surface waves. Starting from the Hamiltonian formulation of the water wave problem and using techniques of Hamiltonian transformation theory,  we derive a Hamiltonian, high-order Nonlinear Schrödinger equation (often referred to as Dysthe equation) for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation recently derived by Curtis, Carter and Kalisch (2018).  This is a joint work with Philippe Guyenne and Adilbek Kairzhan.