TATERS

I will report on a project to study the apolarity of general hyperplane arrangements, including joint work with Jarosław Buczyński, Ștefan Tohǎneanu, and Alexander Woo. Combinatorial and matroid-theoretic results yield detailed information about the derivatives and annihilators of polynomials that factor as products of general linear forms. I will conclude with some remaining questions.

The process of cutting a regular Euclidean tessellation of R^n, along tile boundaries, in finitely many pieces and rearranging them to a new covering of R^n, compatible with the original tile positions, defines a permutation of set S of all tile centers. The group pei(T) of all such permutations is an infinite permutation group, the piecewise-isometry group pei(S) ≤ Sym(S). Particularly interesting is the fact that pei(S) has prominent relatives: In the analogous situation, when T is the standard tessellation of the hyperbolic plane by ideal triangles, then the corresponding group phi(T) contains Richard Thompson's groups V, T, and F.

pei(T) was introduced in the simple case when T is the standard unit-cube tessellation with S = Z^n (joint work with with H. Sach 2015). We showed, among other things, that pei(Z^n) is of type F(2^n-1) – a strong finiteness property. The core of my talk is a more recent tool: pei(Z^n) acts on a structure at infinity of Z^n which can be used to completely determine its normal subgroups. The concept is also available for more general Euclidean tessellations.

A Möbius group is generated by a matrix  [[1,x;0,1]] and its transpose. It is known that these groups are free for |x| greater than or equal to 2 and the (non-)freeness for 0<x<2 is a long-standing question. We show that for these groups arithmeticity implies non-freeness. This motivates an investigation for which rational values 0<x<2 one can prove arithmeticity. Our approach is group theoretic and involves the construction of a presentation and coset enumeration. Results indicate interesting patterns.

This is joint work with Alla Detinko (Huddersfield, UK) and Dane Flannery (Galway, Ireland).

Arboreal Galois representations record the action of the absolute Galois group of a field on the backward orbit of a point under a rational map. When the backward orbit is identified with the vertices of a rooted tree graph, elements of the Galois group act as automorphisms of the tree. We can study these groups from a more geometric perspective by considering the special case of iterated monodromy groups, where we consider the action of Galois on the backward orbit of a generic or transcendental point.

In this talk, I will introduce the (quantum) Yang-Baxter (YB) equation and describe some of its applications to low-dimensional topology (quantum invariants) and theoretical physics (Chern-Simons theory). Then, I will discuss the cohomology of YB operators, and its relation to deformation theory. Various important families of examples arising from Lie algebras and Hopf algebras will be discussed, along with quandle theory. I will present recent work on the study of YB cohomology of Lie algebras, and some results on perturbative expansions of YB operators associated with it. 

Harm Derksen, joint work with Igor Klep, Visu Makam,  and Jurij Volcic

In this talk we will consider the action of GL(n) on m-tuples of n x n matrices by simultaneous conjugation. For this action, orbits correspond to isomorphism classes of n-dimensional modules over the free algebra with m generators.   Hadwin and Larson in 2003 conjectured a criterion for two matrices  to lie in the same orbit, and another criterion for one matrix lying in the closure of the other. We show that the first conjecture is true and the second is false.

This talk is two short talks! 

First, the speaker will talk about some joint work on semigroups with the late Nick Baeth, leading to some research questions she's still working on. The starting point is the idea of a divisor sequence for an element $x$ in a semigroup $S$. The $n$-th term in the sequence counts the number of distinct irreducible divisors of the $n$-th power of $x$. Given some desirable divisor sequences, the authors developed a technique to create new desirable divisor sequences (in new semigroups), which led to questions about tensor products of semigroups more generally. 

Second, the speaker will describe her time working as a science policy fellow for the U.S. Senate Homeland Security and Governmental Affairs Committee (as part of Chairman Gary C. Peters' majority staff). Her portfolio started with federal information systems but grew to include AI when ChatGPT blew Congress's collective mind in the fall of 2022.

Since democracy is the foundation of at least one of the mini-talks, the audience may request that the speaker skip one to focus on the other.

This talk introduces Möbius homology, a novel homology theory for representations of finite posets into abelian categories, establishing a direct connection between poset topology and Möbius inversions. Möbius homology categorifies Möbius inversion by ensuring its Euler characteristic equals the inversion of the representation's dimension function. 

Many years ago, I proved that an abelian variety over a number field has only a finite number of torsion points defined over the maximal cyclotomic extension of the number field.  I am interested in computing examples: What sort of cyclotomic torsion group can you get if you take an abelian variety over the field of rational numbers, for example?  In 2019, Michael Chou classified all possibilities for an elliptic curve over $\bf Q$.  I will discuss mainly the example where the abelian variety is the modular Jacobian $J_0(N)$ with $N$ a prime number; this is the abelian variety in B. Mazur's famous "Eisenstein ideal" article.  I believe that it would be interesting to prove other general theorems and to do more computations.

A classic way to approach some long-standing questions in low-dimensional manifolds is to study embeddings of lower-dimensional manifolds into slightly higher dimensional ones; for example, much information can be gleaned about 3-manifolds by studying the knots and links therein.     I’ll talk about an approach to a long-standing conjecture in graph theory, the double-cycle cover conjecture,  which in some respects works in the opposite direction.   That is, I propose to understand some fundamental properties of graphs by studying properties of their embeddings in 3-dimensional space.

In this talk, I will begin with a brief introduction to graph products and a history of known results regarding the existence of unique prime factorizations and current bounds on prime counting functions. After listing some motivating conjectures, I’ll focus on my current research regarding exponential maps between the Cartesian and direct products (of graphs). In particular, I’ll discuss functions on the adjacency matrices of integral graphs, which give us an exponential map between the Kronecker sum and product. This talk will be mostly algebra and spectral theory, however I may throw in some interesting remarks on topology.

This talk explores the interplay between self Pontryagin dual groups and the Volume Conjecture in the context of quantum invariants of knots. We will begin by defining self dual groups and discussing the open problem of characterizing these mathematical structures. Moving forward, we will introduce Gaussian groups and the assertion that every Gaussian Group is self dual. A critical component of our discussion will be the quantum dilogarithm, which serves as a bridge linking these groups to quantum invariants of knots. We will delve into hyperbolic knots and their complements, addressing quantum invariants, particularly the Jones polynomial and the Kashaev invariant, in knot theory. Additionally, we will discuss the 6j symbol and its relevance to our topic and reference Kashaev's 1997 formula of the volume conjecture in the case of hyperbolic knots. Finally, we will conclude with the Volume Conjecture and its relationship to the colored Jones polynomial, framing an open question that invites further exploration in this area of research.

We consider a natural notion of positive definiteness for matrices over finite fields and prove an algebraic version of Schoenberg’s celebrated theorem characterizing the functions that preserve positive definiteness when applied entrywise to positive definite matrices. Our proofs build on several novel connections between positivity preservers and field automorphisms via the works of Weil, Carlitz, and Muzychuk–Kovacs, and via the Erdős–Ko–Rado theorem for Paley graphs. This is joint work with Dominique Guillot, Prateek Kumar Vishwakarma, and Chi Hoi Yip. The talk is based on the paper: https://arxiv.org/abs/2404.00222.

I will discuss some results about the complexity of polynomials in n variables, as when n is much larger than the degrees of the polynomials.  Perhaps surprisingly, in this asymptotic case, simple uniform behavior arises.