Algebra, Geometry, and Number Theory Seminar

Abstracts

Adam Topaz - An overview of the Liquid Tensor Experiment

The liquid tensor experiment (LTE) was initiated in December 2020 with a challenge by P. Scholze to formalize and verify a foundational result from his work with D. Clausen on analytic geometry in the context of condensed mathematics. This challenge was immediately taken up by several members of the lean interactive theorem prover community, and, six months later, the key technical core of the Clausen-Scholze result was formally verified by this group. In this talk, I will give an overview of LTE, lean, and its mathematics library. No previous experience with condensed mathematics and/or interactive theorem provers will be necessary for this talk. There will be a pretalk at 2:30 introducing the Lean Theorem Prover.

Frank Thorne - Arithmetic Statistics via Fourier Analysis

I will give an overview of "Bhargava's averaging method", a lattice point counting method developed by Manjul Bhargava which yields a variety of results in arithmetic statistics. I will then outline a variant which incorporates Fourier analysis and improves the resulting error terms. This is joint work with Tess Anderson and Manjul Bhargava.

Shaoyun Yi - On dimensions of the spaces of Siegel modular forms

In this talk, we present new dimension formulas of Siegel cusp forms of degree 2 for certain congruence subgroups of level 4. To obtain these desired dimension formulas, we will explore the connection between Siegel cusp forms of degree 2 and cuspidal automorphic representation of $GSp(4)$. This is joint work with Manami Roy and Ralf Schmidt.

Matthew Ballard - Orlov’s Conjecture via reduction to characteristic $p$

Orlov's Conjecture on derived dimension states that a classical invariant, the Krull dimension, coincides with a novel invariant, Rouqiuer's dimension of a triangulated category. Since the spectacular work of Deligne and Illusie, reduction to characteristic $p$ has been a prominent tool in the algebraic geometry toolbox. In this talk, we will discuss how to use it to reduce Orlov's Conjecture to the case of varieties over closed fields of characteristic $p$. Next we will focus on a special candidate: $F^e_{\ast} \mathcal O$ the iterated Frobenius pushforward of the structure sheaf. In the immortal words of the Most Interesting Man in the World${}^{\text{TM}}$: This doesn't always generate but, when it does, it does it well. As a special case, we will see how to resolve a 15 year old question of Bondal. This is joint work with Alexander Duncan (UofSC) and Patrick McFaddin (Fordham). There will be pre-talk at 2:30 on Rouquier Dimension.

Kyle Maddox - Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we also show that nearly all pinched Veronese rings are F-nilpotent, a singularity type of recent interest. For such rings we also compute upper bounds on their Frobenius test exponents, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

Pan Yan - L-function for Sp(4)xGL(2) via a non-unique model

We prove a conjecture of Ginzburg and Soudry (2020 IMRN) on an integral representation for the tensor product partial L-function for Sp(4)xGL(2), which is derived from the twisted doubling method of Cai, Friedberg, Ginzburg, and Kaplan. We show that the integral unfolds to a non-unique model and analyze it using the New Way method of Piatetski-Shapiro and Rallis.

Last semester's seminar.