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: Web Diagrams, Specht Modules, and Positivity
Abstract: Webs were first introduced as diagrams encoding the representation theory of quantum groups. They have since been applied to various fields of mathematics, such as knot theory, geometry, and combinatorics. In this talk, we discuss an application to representations of the Hecke Algebra $H_d(q)$. For the Specht Module $S^{\lambda}$ corresponding to $\lambda = (n+r,n)$, we introduce the web basis. We will show that the transition matrix between the web basis and the standard basis of $S^{(n+r,n)}$ is unitriangular and satisfies a strong positivity property.
03/24/2025 – Tan Özalp: What is an Ultrafilter?
Abstract: Is there a non-Lebesgue-measurable set? Does every compact semigroup have an idempotent element? Is there a countably additive measure on the real numbers? Why is the additive group of integers amenable? Does Ramsey’s theorem fail for infinite subsets? How many null sets do you need to cover the real line? Is there a non-sofic group? Can we linearly order candidates in a political election? Whenever you color the natural numbers with two colors, can you always find arbitrarily long monochromatic arithmetic progressions? Does God exist? We shall answer none of these questions in this talk. Instead, we will try to motivate the concept of \emph{ultrafilters} for a general audience.
03/31/2025 – Gavin Dooley: Reduction: An Introduction
Abstract: According to a classic joke, a mathematician can boil a kettle of water by emptying the kettle, reducing the problem to one that has already been solved. (See speaker for more details.) What does it mean to reduce one problem to another? In this talk, we discuss computable reducibility, Weihrauch reducibility, strong computable reducibility, and strong Weihrauch reducibility, four related formalizations of what it means to reduce one problem to another. As an example, we explain how versions of Ramsey’s theorem with different dimensions and different numbers of colors relate to each other under these reducibilities. Finally, we ask a question.
04/14/2025 – Emma Dooley: What does it mean to be “rational(ly connected)”?
Abstract: Unfortunately, if you were hoping this talk would help you figure out what it means to be a rational person, you are out of luck. However, in this talk we will discuss what it means to be rational(ly connected) in the world of algebraic geometry. We will see when these two notions coincide and when they diverge. In particular, we will see some classical examples of rationally connected varieties and some techniques one can use in order to determine whether a variety is rationally connected. Maybe you can take some meaning away from this talk as to what a “rational” person is, but I make no promises.
04/28/2025 – Jaziel Torres: Entire Curves, Hyperbolicity, and Varieties of General Type: A Glimpse into the Green-Griffiths-Lang Conjecture
Abstract: In this talk, I will introduce the class of complex varieties of general type from the perspective of the Minimal Model Program and the theory of canonical models, as a higher dimensional generalization of curves of genus at least 2. I will then motivate the study of entire curves and Kobayashi hyperbolicity, leading to the formulation of the Green-Griffiths-Lang conjecture. Time permitting, I will sketch some techniques involved in approaching this conjecture, including the theory of jet differentials. Throughout the talk I will highlight examples, the interplay between complex, differential, arithmetic and algebraic geometry, and those who have shaped our understanding of these topics.