# Subring growth in Z^n

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# Hodge-Tate prismatic crystals and Sen theory

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We discuss Hodge-Tate crystals on the absolute prismatic site of O_K, where K is a p-adic field. These are vector bundles defined over the Hodge--Tate structure sheaf. We first classify them by O_K-modules equipped with small endomorphisms. We then classify rational Hodge-Tate crystals by nearly Hodge--Tate C_p-representations. This is joint work with Yu Min and Yupeng Wang.

# 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.