ABSTRACTS
TREVOR ARRIGONI
Singularity Theory in Positive Characteristic
Algebraic Geometry has historically been motivated by trying to understand the zeros of polynomials. In general, the zero set of a polynomial will graph a curve or surface in space. Thus, we aim to learn more about this object by studying the defining polynomial. For example, consider the polynomial f(x,y) = x^2 + y^2 - 1. The zero set of f is the unit circle, an object that we know a lot about. But what if the zero set is more complicated or is a more obscure object? In this talk, we will discuss one of the ways of studying such zero sets. In particular, we will examine the singularities of a given zero set. Seen as cusps or places where the zero set intersects itself, singularities were historically studied by seeing when certain integrals associated to the defining polynomial converge, which results in an invariant called the log canonical threshold. But, what if we try to tackle this question in more of an algebraic way? In this talk, we will examine the history and current directions of the algebraic approach to understanding singularities, which will involve passing to positive characteristic where we have an analogue of the log canonical threshold called the F-pure Threshold. By the end of this talk, we will discuss some current research being done on computing the F-pure Threshold of homogeneous polynomials.
LIAM KAHMEYER
A homotopy invariant of image simple fold maps to oriented surfaces
The set of singular points of a generic map $f: M \rightarrow F$ of a manifold of dimension $m \geq 2$ to an oriented surface $F$ is a closed smooth curve, denoted by $\Sigma(f)$. If $f|_{\Sigma(f)}$ is a topological embedding, we say $f$ is an \emph{image simple map}. The goal of this talk is to show that the number of connected components of the singular set, denoted by $\#|\Sigma(f)|$, is a homotopy invariant of image simple fold maps provided one of the following conditions is satisfied: (a) $\textrm{dim}(M) = 2q$ for $ q \geq 1$, (b) the singular set of the homotopy is an orientable manifold, or (c) the image of the singular set of the homotopy does not have triple self-intersection points.
I will define and invoke so-called chessboard functions to equip the image of the singular set $f(\Sigma)$ with a canonical local orientation and co-orientation. This canonical orientation gives rise to the cumulative winding number $\omega(f) \in \frac{1}{2} \Z$ of $\Sigma(f)$, which I will use to prove the result for cases (a) and (c). To prove the result for case (b), I will use the invariant $I(f) = 2 \#|\Sigma(f)| +2 \Delta(f) + |A_2(f)|$, where $\Delta(f)$ is the modulo two residue class of double points of $\Sigma(f)$ and $|A_2(f)|$ is the number of cusp singular points of $\Sigma(f)
RYAN FRIER
Extremizers for the Strichartz Inequality for a Fourth-Order Schrödinger Equation
In this paper, we consider the Strichartz inequality for a fourth-order Schrödinger equation on $\mathbbm{R}^3$. We show that extremizers exist using a linear profile decomposition which follows from the endpoint version decomposition and the stationary phase method. Based on the existence of extremizers, we use the associated Euler-Lagrange equation to show that the extremizers have exponential decay and consequently must be analytic.
This is a joint work with Boning Di.
JOSH VENICK
Ricci Solitons and the Hitchin-Thorpe Inequality
Einstein Manifolds arise as solutions to Einstein's vacuum field equations. In the compact 4-dimensional case, we have the Hitchin-Thorpe inequality, which gives a relationship between the signature and the Euler characteristic of an Einstein manifold. A slight generalization to this is the notion of a Ricci soliton or a gradient Ricci soliton. In this talk, I will review some fundamental results from differential geometry and discuss some conditions under which compact 4-dimensional Ricci solitons satisfy the Hitchin-Thorpe inequality.
SIEN GONG
An introduction to $G_2$-flow
The past few decades have witnessed great triumphs of geometric flows, of which the most remarkable is Perelman’s proof of Poincaré conjecture using Ricci flow. Now we are interested in geometric flows of $G_2$ structure on a 7-manifold. This expository talk will begin with an overview of geometric flows. Then the concept of $G_2$ structure will be introduced. Finally, I will conclude with a discussion on $G_2$ flow.
KOJI SUNAMI
Intro to Langlands Program
Langlands dual is a non-abelian version of class field theory, a theory that corresponds reductive algebraic groups of global fields and local fields with Galois extension, and it is used for mirror symmetry of moduli space of a Higgs bundle, for example. My talk is an elementary part of Shimura variety, a generalization of modular form, and Satake isomorphism, which helps the construction of Langlands dual group.
RAY ZHANG
Asymptotics of the one-point distribution of the KPZ fixed point conditioned on a large height at an earlier point
We consider the Kardar-Parisi-Zhang (KPZ) fixed point
H(x,τ) with the step initial condition and investigate the distribution of H(x,τ) conditioned on a large height at an earlier space-time point H(x′,τ′). When the height H(x′,τ′) tends to infinity, we prove that the conditional one-point distribution of H(x,τ) in the regime τ>τ′ converges to the Gaussian Unitary Ensemble (GUE) Tracy-Widom distribution, and the next three lower order error terms can be expressed as the derivatives of of the GUE Tracy-Widom distribution. These KPZ-type limiting behaviors are different from the Gaussian-type ones obtained in [Liu-Wang22] where they study the finite dimensional distribution of H(x,τ) conditioned on a large height at a later space-time point H(x′,τ′). They prove, with the step initial condition, the conditional random field H(x,τ) in the regime τ<τ′ converges to the minimum of two independent Brownian bridges modified by linear drifts as H(x′,τ′) goes to infinity. The two results stated above provide the phase diagram of the asymptotic behaviors of a conditional law of KPZ fixed point in the regimes τ>τ′ and τ<τ′ when H(x′,τ′) goes to infinity.
ELIZABETH HALE
Fractional Leibniz Rules in the Setting of Banach Function Spaces
Fractional Leibniz rules are reminiscent of the product rule learned in calculus classes, offering estimates in the Lebesgue norm for fractional derivatives of a product of functions in terms of the Lebesgue norms of each function and its fractional derivative. Â In this talk, I will give an introduction to fractional differential operators and the associated Leibniz rules, as well as an overview on the tools in Fourier analysis we use to prove such estimates.
