# Southern California Number Theory Day

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# Primitive elements in number fields and Diophantine avoidance

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The famous primitive element theorem states that every number field K is of the form Q(a) for some element a in K, called a primitive element. In fact, it is clear from the proof of this theorem that not only there are infinitely many such primitive elements in K, but in fact most elements in K are primitive. This observation raises the question about finding a primitive element of small “size”, where the standard way of measuring size is with the use of a height function. We discuss some conjectures and known results in this direction, as well as some of our recent work on a variation of this problem which includes some additional avoidance conditions. Joint work with Lenny Fukshansky at Claremont McKenna College.

# 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.