01/27/2025 – Eric Riedl and Neil Nicholson: Academic Job Panel
Abstract: Eric Riedl and Neil Nicholson will give a brief introduction to their respective path to an academic job. After this there will be time for questions and open-ended discussions regarding the job-finding process.
02/03/2025 – Chase Bender: A General Polarization Identity
Abstract: We prove a generalization of the polarization identity of linear algebra expressing
the inner product of a complex inner product space in terms of the norm, where the
field of scalars is extended to an associative algebra equipped with an involution,
and polarization is viewed as an averaging operation over a compact multiplicative
subgroup of the scalars. Using this we prove a general form of the Jordan-von
Neumann theorem on characterizing inner product spaces among normed linear
spaces, when the scalars are taken in an associative algebra.
02/10/2025 – Fuxiang Yang: Syzygy and Geometry
Abstract: Algebraic Geometry is a subject that studies the set of solutions of a system of polynomial equations. We call such a set of solutions a variety. It is natural to ask: Given a variety, how can we recover the system of equations? Moreover, how can we measure the complexity of a variety by looking at its defining equations, relations between those equations, relations between the relations, etc. This collection of equations, relations, and higher relations is what we call syzygies. In this talk, we are going to talk about some connections between syzygy and geometry.
02/24/2025 – Juan Ramirez: Rees Algebras and their Defining Equations
Abstract: In this talk we will give a friendly introduction to the Rees Algebras of Ideals with the main focus being on the study of their defining equations. We will survey some history on the study of Rees Algebras and introduce the type of questions interesting to a commutative algebraists. In the last part of the talk we will consider the case of Ideals presented by $3 \times 2$ matrices in the polynomial ring $R = k[x, y]$ and present some known results, works in progress, and future directions.
03/03/2025 – Joshua Lehman: The Arf Invariant
Abstract: Let $(V,\langle -,-\rangle)$ be a finite dimensional symplectic vector space over $\mathbb{F}_2$. A function $q : V \to \mathbb{F}_2$ is called a quadratic form if $q(x+y) = q(x)+q(y)+\langle x,y\rangle$ for all $x,y \in V$. Elements of $V$ vote for $0$ or $1$ according to their value under $q$. The winner of this election is called the Arf invariant of $q$. We show this is an invariant of the quadratic form $q$ and discuss applications in topology. We’ll finish with a detour into algebraic geometry and show, using a theorem of Dennis Johnson, that a smooth quartic has exactly 28 bitangents (28 is the number of quadratic forms on the first homology of a quartic with Arf invariant 1).
03/17/2025 – Samuel Heard: TBA
Abstract: