Tuesday, October 4, 2022

Logic Seminar

Alex Van Abel, Wesleyan

Asymptotics of the Spencer-Shelah Random Graph Sequence

Abstract:  This talk is based on work in progress. In combinatorics, the Spencer-Shelah random graph sequence is a variation on the independent-edge random graph model. We fix an irrational number $a \in (0,1)$, and we probabilistically generate the n-th Spencer-Shelah graph (with parameter $a$) by taking $n$ vertices, and for every pair of distinct vertices, deciding whether they are connected with a biased coin flip, with success probability $n^{-a}$. On the other hand, in model theory, an $R$-mac is a class of finite structures, where the cardinalities of definable subsets are particularly well-behaved. In this talk, we will introduce the notion of "probabalistic $R$-mac" and present an incomplete proof that the Spencer Shelah random graph sequence is an example of one.

 

 

 

Friday, September 30, 2022

Algebra Seminar

John Cullinan, Bard College

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions.

Abstract:  Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$.  Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$.  We have previously shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$.
In this talk we determine the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over $\mathbf{Q}$ and asking for what proportion of primes $p$ do we have $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.  We also determine how the discriminants of the endomorphism rings are distributed at these primes.  This is joint work with Nathan Kaplan of UC Irvine.

 

Wednesday, September 28, 2022

Topology Seminar

Yu-Chan Chang, Wes

The RAAG Recognition Problem for Bestvina–Brady Groups.

Abstract: Given a finite simplicial graph, the associate right-angled Artin group (RAAG) is generated by the vertex set of the graph, and two generators commute if they are connected by an edge. The RAAG Recognition Problem asks whether a given group is (isomorphic to) a RAAG. Some algebraic structures of the given group are solutions to this problem. For example, if a group is not finitely presented, then it cannot be a RAAG. Some quasi-isometry invariants of groups also provide solutions, such as Dehn functions. However, this recognition problem is difficult in general, even when the given group has a very simple presentation.

 

Bestvina–Brady groups (BBGs) are subgroups of RAAGs, and they provide important (counter)examples in group theory and topology. In this talk, we will discuss the RAAG Recognition Problem for BBGs. In particular, we will see a complete solution to this problem for the BBGs associated to 2-dimensional flag complexes. This is joint work with Lorenzo Ruffoni.

 

Friday, September 23, 2022

Algebra Seminar

Jeff Yelton, Wesleyan

Clusters, semistable models, and inertia actions for hyperelliptic curves

Abstract:  Given a hyperelliptic curve C : y^2 = f(x) over a global field K, I will relate the “clustering” of its branch points with respect to a prime P of K to the construction of a semistable model of the curve at that prime.  This in turn sheds light on actions of inertia subgroups of the absolute Galois group Gal(K) which are associated to C.  All of this has a very nice description for primes P of residue characteristic not 2, and newer results suggest a somewhat analogous situation when the residue characteristic is 2.

 

Wednesday, September 21, 2022

Topology Seminar

Zvi Rosen, Florida Atlantic University

Neural Codes and Oriented Matroids

Abstract: Fix n convex open subsets of Euclidean space, representing stimuli that trigger specific neurons in the brain. Then, record all possible subsets of [n] whose intersection is nonempty, and not covered by other sets. This combinatorial object is called a "convex neural code." In this talk, we relate the emerging theory of convex neural codes to the established theory of oriented matroids, which generalize systems of signed linear relations. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.

 

Friday, September 16, 2022

Algebra Seminar

Wai Kiu Chan, Wesleyan

On the exceptional sets of integral quadratic forms

Abstract:  A collection S of equivalence classes of positive definite integral quadratic forms in n variables is called an n-exceptional set if there exists a positive definite integral quadratic form which represents all equivalent classes of positive definite integral quadratic forms in n variables except those in S.  In this talk, I will describe a recent joint work with Byeong-Kweon Oh which shows that for any given positive integers m and n, there is always an n-exceptional set of size m and there are only finitely many of them.

 

 

Tuesday, September 13, 2022

Logic Seminar

Scott Mutchnik, UC Berkeley

NSOP_2 Theories

Abstract:  Model theory has been described as a "geography of tame mathematics," creating a map of the universe of first-order theories according to various dividing lines, such as tree properties or order properties. While some regions of this map, such as the stable theories or simple theories, are well-understood to varying degrees, as we progress outward it even becomes open whether some regions are empty or not. Extending the NSOP_n hierarchy of Shelah [1995] defining an ascending chain of strong order properties for n > 2, Džamonja and Shelah [2004] introduce two further tree properties, NSOP_1 and NSOP_2, and ask whether the implications between NSOP_1 and NSOP_2 and between NSOP_2 and NSOP_3 are strict. We have answered the first of these questions, showing that the class NSOP_1 coincides with NSOP_2. We discuss this result and some aspects of its proof, which incorporates ideas from various other regions of the model-theoretic map such as the NSOP_1, NSOP_3 and NTP_2 theories.