Introduction: Topics of these seminars include differential equations (ODEs, PDEs, DDEs, FDEs, etc.), dynamical systems theory, and their applications (often in mathematical biology). To get the brain gears turning, each session will kick off with a fun trivia!

Stabilizer codes offer a robust and efficient framework for encoding quantum information and detecting errors. This family includes a large class of codes such as CSS codes, surface codes and Toric codes. In this talk, we will focus on the fundamental mathematical principles of stabilizer codes. Using the aspects of subgroups of Pauli groups, this family of the codes offers a unified scheme for detecting and correcting errors in quantum world. This unification simplifies both error detection and error correction for these family of codes.

As implied by Bezout's theorem, n generic polynomials of degrees d1,...,dn in C^n have exactly d1*d2*...*dk common roots. Here, the degrees of each polynomial are specified, but they are otherwise generic. Adding constraints, one may impose which *monomials* are involved in each polynomial, resulting in a 'sparse polynomial system'. The analogue of Bezout for sparse systems is the celebrated Bernstein-Kouchnirenko-Khovanskii (BKK) theorem.

The BKK theorem relates a solution count to the polyhedral geometry of the monomial support of a sparse system. Relating other algebro-geometric features to combinatorial data of sparse systems is an active area of research. I will give a survey of what is known about some of these connections, how the associated theorems are used in practice, and what has yet to be discovered.

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.

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.

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.