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

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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

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# De Rham cohomology on Berkovich curves

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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

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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

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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

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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

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# Two dimensional delta symbol and applications to quadratic forms

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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

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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.