Abstract: Let $X$ be a smooth toric variety with dense torus $T$. It is known that the syzygy order of equivariant cohomology $H_T^*(X)$ as an $H^*(BT)$-module can be computed from the combinatorics of the underlying fan. Suppose $S\subset T$ is a subtorus, we will compute the syzygy order of equivariant cohomology $H_S^*(X)$ with respect to $S$ as an $H^*(BS)$-module in some cases.

Abstract: I'll explain a conjectural characterization of algebraic solutions to (possibly non-linear) algebraic differential equations, in terms of the arithmetic of the coefficients of their Taylor expansions, strengthening the Grothendieck-Katz p-curvature conjecture. I'll give some evidence for the conjecture coming from algebraic geometry: in joint work with Josh Lam, we verify the conjecture for algebraic differential equations (both linear and non-linear) and initial conditions of algebro-geometric origin. In this case the conjecture turns out to be closely related to basic conjectures on algebraic cycles, motives, and so on.

Abstract: Polar convex sets can be seen as images of convex sets under some M\"obius transform. The notion of polar convex sets in the complex plane has been used to analyze the behavior of critical points of polynomials. In this talk, we go over the not too long history of polar convex sets in the plane. Following which, we extend the notion to finite dimensional Euclidean spaces. The goal of the talk is to build a theory of polar convexity and to show that the introduction of a pole creates a richer geometry compared to classical convex sets. Polar convexity enjoys a beautiful duality that does not exist in classical convexity. We will also formulate polar analogues of several classical results in convex optimization. Finally, we give a full description of the convex hull of finitely many points with respect to finitely many poles.