Primitive elements in number fields and Diophantine avoidance

Speaker: 

Sehun Jeong

Institution: 

Claremont Graduate University

Time: 

Thursday, October 10, 2024 - 3:00pm to 4:00pm

Location: 

RH 306

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

Speaker: 

Kelly Isham

Institution: 

Colgate University

Time: 

Friday, May 17, 2024 - 3:00pm to 4:00pm

Location: 

RH 340P
Subgroups in $\mathbb{Z}^n$ are well-understood. For example, the growth rate of the number of subgroups in $\mathbb{Z}^n$ is known, and futher, for any $k$, a positive proportion of subgroups have corank $k$, though subgroups grow sparse as $k$ increases. Much less is known about subrings in $\mathbb{Z}^n$. There is not even a conjecture about what the growth rate of the number of subrings in $\mathbb{Z}^n$ should be. In this talk, we compare subgroup growth and subring growth. We then focus on subrings of corank $k$ and show that while the proportion of subgroups of any fixed corank is always positive, the proportion of subrings of any fixed corank is not. This is joint work with Nathan Kaplan.

Hodge-Tate prismatic crystals and Sen theory

Speaker: 

Hui Gao

Institution: 

SUSTech, Shenzhen

Time: 

Thursday, April 25, 2024 - 3:00pm to 3:50pm

Host: 

Location: 

RH306

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

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.

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