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Department Colloquium

Speaker: Peyman Mohajerin Esfahani (University of Toronto)

Title: From Optimization to Control: An Algorithmic Perspective

Abstract:
In this talk, we draw an explicit analogy across four problem classes in optimization and control with a unified solution characterization. This viewpoint allows for a systematic transformation of algorithms from one domain to the other. With this in mind, we exploit two linear structural constraints specific to control problems with finite state-action pairs to approximate the Hessian in a second-order-type algorithm from optimization. This leads to novel first-order control algorithms with the same computational complexity as (model-based) value iteration and (model-free) Q-learning, while they exhibit an empirical convergence behavior similar to (model-based) policy iteration and (model-free) Zap Q-learning with very low sensitivity to the discount factor. If time permits, we also discuss how an interesting analogy between the convex conjugate operator and the Fourier transform can reduce the typical time complexity of the dynamic programming operation from O(XU) to O(X + U) where X and U denote the size of the discrete state and input spaces, respectively.

Calabi-Yau Manifolds Seminar

Speaker: Rikuto Ito (Nagoya University)

Title: Modularity for rank-21 cubic fourfolds over Q

Abstract: A cubic fourfold is a Fano variety defined by a homogeneous cubic equation in five-dimensional projective space. Cubic fourfolds share many structural similarities with K3 surfaces; in particular, when the second Picard number is at least two, Hassett (2000) showed that a cubic fourfold is cohomologically associated with a K3 surface. Under additional hypotheses, this correspondence is motivic in the sense of Chow motives (B\”ulles 2018). Consequently, one expects cubic fourfolds to exhibit arithmetic phenomena closely parallel to those of K3 surfaces.

In this talk, I will explain a proof of the modularity of cubic fourfolds of rank 21 defined over \mathbf{Q}. The corresponding statement for singular K3 surfaces was obtained by Livné, but under the very restrictive assumption that the Picard number is 20 over \mathbf{Q}. Our approach does not require any assumption on the Picard number over \mathbf{Q}; we only assume that the cubic fourfold has rank 21 over \mathbf{C}.

We also obtain the modularity of singular K3 surfaces without Livné’s Picard-number hypothesis, using a method that is essentially identical to the one applied to cubic fourfolds. In both settings, the resulting Galois representations are shown to arise from the same type of weight-3 CM newform. I will focus on the construction of this modular form and the shared mechanism underlying both proofs.

Curves Seminar

Speaker: Trevor Shillington

Affiliation: Queen's University

Title: Symmetric Polynomials and Functions

Abstract: In this talk, we will attempt to wrap up the combinatorics and pave the way to representation theory by investigating symmetric polynomials. This will include the interconnectedness of different types of symmetric polynomials, as well as prove one of the Jacobi-Trudi identities that has been teased before: Jacobi’s original definition of the Schur polynomials. We will wrap-up with defining the ring of symmetric functions and investigating some of the geometry they can be given and how this ring unifies identities about symmetric polynomials.

Curves Seminar

Speaker: Luke Steverango

Affiliation: Queen's University

Title: Littlewood-Richardson Numbers and Reverse Lattice Words

Abstract: We will continue our exploration of Littlewood Richardson numbers by considering many different interpretations of these numbers. This will lead us to define reverse lattice words which will help explore these interpretations.

Dynamics, Geometry and Groups Seminar

Speaker: Liangbing (Luo)

Title: Logarithmic Sobolev inequalities on sub-Riemannian manifolds

Abstract:
The logarithmic Sobolev inequality has been first introduced and studied by L. Gross on a Euclidean space, and since then it found many applications. In particular, many existing results concern the question on how the constant in the logarithmic Sobolev inequality depends on the geometry of the underlying space. As for many of sub-Riemannian manifolds, curvature bounds (or classical Bakry-Emery estimates) are not available. We use different techniques, such as symmetry of the space, to study the inequality and its constant, especially its dimension-free property. Examples in both finite and infinite dimensional settings are provided.