Shellability and Homology of q-matroids with applications to Rank Metric Codes

Speaker: 

Sudhir Ghorpade

Institution: 

IITB

Time: 

Thursday, March 7, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. It is a classical result that matroid complexes, that is, simplicial complexes formed by the class of independent subsets in a matroid, are shellable. This has some bearing on the study of linear block codes, especially in regard to their Betti numbers and generalized weight enumerator polynomials. 

We now know that q-matroids have close connections with rank metric codes in a manner similar to the connection between matroids and codes. A recent result establishes shellability of q-matroid complexes and also determines the homology of these complexes in many cases. The determination of homology has now been completed for arbitrary q-matroid complexes. 

We will outline these developments whlie making an attempt to keep the prerequisites at a minimum. 

The contents of this talk are based on a joint work with Rakhi Pratihar and Tovohery Randrianarisoa, and also with Rakhi Pratihar, Tovohery Randrianarisoa, Hugues Verdure and Glen Wilson. 

New universal limits for cokernels of random matrix products

Speaker: 

Roger Van Peski

Institution: 

KTH

Time: 

Tuesday, April 2, 2024 - 3:00pm to 4:00pm

Location: 

RH 306
Since 1980s work of Cohen-Lenstra and Friedman-Washington, many (pseudo-)random groups in number theory, combinatorics and topology have been conjectured---and sometimes proven---to match certain universal distributions, which appear as large-N limits of cokernels of N x N random matrices over $\mathbb{Z}$ or $\mathbb{Z}_p$. In this talk I discuss a new such distribution, the cokernel of a product of k independent matrices. For each fixed k, it converges to a universal distribution, generalizing in a natural way the k=1 case of the Cohen-Lenstra distribution. As time permits I will discuss the case when the number of products k goes to infinity along with N. Then the groups do not converge, but the fluctuations of their ranks and other statistics still approach limit distributions related to a new interacting particle system, the 'reflecting Poisson sea'. Based on https://arxiv.org/abs/2209.14957v2 (with Hoi Nguyen) and https://arxiv.org/abs/2312.11702, https://arxiv.org/abs/2310.12275.

De Rham cohomology on Berkovich curves

Speaker: 

Andrea Pulita

Institution: 

Institut Fourier (IF), Universite Grenoble Alpes

Time: 

Thursday, February 22, 2024 - 3:00pm

Host: 

Location: 

RH 306

The talk is an invitation to the theory of p-adic differential equations and their de Rham cohomology. I will give an overview of the existing results, with an emphasis on de rham cohomology.

Distribution of even and odd integers in gaps of numerical semigroups

Speaker: 

Hayan Nam

Institution: 

Duksung Womens University

Time: 

Thursday, January 18, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

A numerical semigroup is a collection of nonnegative integers that includes zero, is closed under addition, and has a finite complement. The gap of a numerical semigroup is defined as the complement of the semigroup. In this talk, we observe the distribution of even and odd integers within the gaps of numerical semigroups.

A Chebotarev Density Theorem over Local Fields

Speaker: 

John Yin

Institution: 

University of Wisconsin

Time: 

Tuesday, January 23, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

I will present an analog of the Chebotarev Density Theorem which works over local fields. As an application, I will use it to prove a conjecture of Bhargava, Cremona, Fisher, and Gajović. This is joint work with Asvin G and Yifan Wei.

Random multiplicative functions: old and new results

Speaker: 

Max Xu

Institution: 

Stanford

Time: 

Thursday, February 8, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

Random multiplicative functions are probabilistic models for important arithmetic functions in number theory, e.g. Mobius function, Dirichlet characters. In this talk, I would like to introduce the topic and emphasize some recent developments. Part of the talk is based on joint works with Angelo, Harper, and Soundararajan. 

Lower bounds for the modified Szpiro ratio

Speaker: 

Alex Barrios

Institution: 

University of St. Thomas

Time: 

Thursday, March 14, 2024 - 3:00pm to 4:00pm

Location: 

RH 306
Let $a,b,$ and $c$ be relatively prime positive integers such that $a+b=c$. How does c compare to $\operatorname{rad}(abc)$, where rad(n) denotes the product of the distinct prime factors of $n$? According to the explicit $abc$ conjecture, it is always the case that $c$ is less than the square of $\operatorname{rad}(abc)$. This simple statement is incredibly powerful, and as a consequence, one gets a (marginal) proof of Fermat's Last Theorem for exponent $n$ greater than $5$. In this talk, we introduce Masser and Oesterlé's $abc$ conjecture and discuss some of its consequences, as well as some of the numerical evidence for the conjecture. We will then introduce elliptic curves and see that the $abc$ conjecture has an equivalent formulation in this setting, namely, the modified Szpiro conjecture. We conclude the talk by discussing a recent result that establishes the existence of sharp lower bounds for the modified Szpiro ratio of an elliptic curve that depends only on its torsion structure.

Two dimensional delta symbol and applications to quadratic forms

Speaker: 

Junxian Li

Institution: 

UC Davis

Time: 

Thursday, February 1, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

The delta symbol developed by Duke-Friedlander-Iwaniec and
Heath-Brown has played an important role in studying rational points on
hypersurfaces of low degrees. We present a two dimensional delta symbol
and apply it to establish a quantitative Hasse principle for a smooth
intersection of two quadratic forms defined over Q in at least ten
variables. The goal of these delta symbols is to carry out a (double)
Kloosterman refinement of the circle method. This is based on a joint
work with Simon Rydin Myerson and Pankaj Vishe.

Abelian covers of P^1 of p-ordinary Ekedahl-Oort type

Speaker: 

Yuxin Lin

Institution: 

Caltech

Time: 

Thursday, February 15, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

Given a family of abelian covers of P^1 branched at at least four points and a prime p of good reduction, by considering the associated Deligne--Mostow Shimura variety, we obtain lower bounds for the Ekedahl-Oort types, and the Newton polygons, at prime p of the curves in the family. In this paper, we investigate whether such lower bounds are sharp. In particular, we prove sharpeness when the number of branching points is at most five and p sufficiently large. Our result is a generalization under stricter assumptions of Irene Bouw, which proves the analogous statement for the p-rank, and it relies on the notion of Hasse-Witt triple introduced by Ben Moonen.  

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