UTRGV Algebra & Number Theory Seminar

The Algebra & Number Theory Seminar meets on Tuesdays or Fridays at 1:00pm in EMAGC 3.502 during Spring '25. Talks will be given in a hybrid format. If you would like to give a talk in the seminar please contact Debanjana Kundu or Luigi Ferraro.

Souvik Dey

Charles University

Complexity and curvature of (pairs of) Cohen-Macaulay modules, and their applications

The complexity and curvature of a module, introduced by Avramov, measure the growth of Betti and Bass numbers of a module, and distinguish the modules of infinite homological dimension. The notion of complexity was extended by Avramov-Buchweitz to pairs of modules that measure the growth of Ext modules. The related notion of Tor complexity was first studied by Dao. Inspired by these notions, we define Ext and Tor curvature of pairs of modules. The aim of this talk is to study (Ext and Tor) complexity and curvature of pairs of certain CM (Cohen-Macaulay) modules, and establish lower bounds of complexity and curvature of pairs of modules in terms of that of a single module. It is well known that the complexity and curvature of the residue field characterize complete intersection local rings. As applications of our results, we provide some upper bounds of the curvature of the residue field in terms of curvature and multiplicity of any nonzero CM module. As a final upshot, these allow us to characterize complete intersection local rings (including hypersurfaces and regular rings) in terms of complexity and curvature of pairs of certain CM modules. In particular, under some additional hypotheses, we characterize complete intersection or regular local rings via injective curvature of the ring or that of the module of Kähler differentials.

Alexey Glazyrin

University of Texas Rio Grande Valley

Energy minimization problems

Given a continuous 2-point kernel $K$, the total energy of a point configuration is defined as the sum of values of $K$ over all pairs of points. Minimizing energy defined this way may be interpreted as finding the optimal distribution of particles under the interaction defined by the kernel $K$. Problems of this type have a rich history and arise in various areas of mathematics and science. We are particularly interested in a measure-theoretic extension of such problems. Often optimal configurations tend to cluster and optimal measures appear to be discrete. In this talk, we will cover this phenomenon in the case of $p$-frame potentials.

Andrew Alaniz

University of Texas Rio Grande Valley

Towards trace and determinant formulas for generalized cubic dissection matrices

Dissecting the $q$-series of a modular form is a common tool used to study its Fourier coefficients. In a previous work, Huber--Alaniz introduced generalized cubic dissection operators along with their matrix representations on certain spaces of modular forms. In this talk, we discuss progress towards a problem that extends a pair of conjectures posed by Huber--Alaniz via linear algebraic methods. We also discuss some combinatorial problems arising upon reducing the matrices modulo a prime $p$.

Pavel Snopov

University of Texas Rio Grande Valley

An Algebraic Perspective on Persistent Homology

Persistent homology has emerged as a fundamental tool in topological data analysis, providing a robust framework for capturing the shape of data. At its core, persistent homology is governed by the structure of persistence modules, which can be studied through an algebraic lens. In this talk, we will introduce the general setting of persistent homology and discuss the structure theorem for finitely generated 1-parameter persistence modules. Moving beyond the 1-parameter case, we will explore multiparameter persistence, where classification becomes significantly more complex. Time permitting, we will also touch on recent developments in the algebraic study of persistence modules.

Alexey Garber

University of Texas Rio Grande Valley

Geometric lattices and parallelohedra

In this talk I am going to discuss a well-known connection between lattices in $\mathbb{R}^d$ and convex polytopes that tile $\mathbb{R}^d$ with translations only. My main goal will be to highlight several geometric, algebraic, and number-theoretic settings where lattices and such polytopes appear. Particularly, I’ll discuss my recent results on the Voronoi conjecture on parallelohedra, a century old conjecture that while stated in simple terms, is still open in full generality.

Lawrence Washington

University of Maryland

Sums of primes

The sum of the primes up to $x$ is asymptotic to the number of primes up to $x^2$, namely $\pi(x^2)$. However, the sum tends to be less than $\pi(x^2)$. We quantify this and give an equivalent form of the Riemann Hypothesis.

Raul Alvarez

University of Texas Rio Grande Valley

A DG algebra structure with divided powers on the generalized Taylor resolution

Free resolutions of ideals in commutative rings provide valuable insights into the complexity of these ideals. In 1966, Taylor constructed a free resolution for monomial ideals in polynomial rings, which Gemeda later showed in 1976 to admit a differential graded (DG) algebra structure—allowing for a way to multiply elements in the resolution. This additional structure enables the derivation of stronger results. In 2002, Avramov demonstrated that the Taylor resolution admits a DG algebra structure with divided powers, meaning that for any element $x$ of even degree in the resolution, one can associate a sequence behaving like $\frac{x^k}{k!}$ for all $k\geq0$. In 2007, Herzog introduced the Generalized Taylor resolution, which is usually smaller than the original Taylor resolution. Recently, in 2023, VandeBogert established that the Generalized Taylor resolution also admits a DG algebra structure. In this talk, based on joint work with Ferraro, we extend Avramov's result by proving that the Generalized Taylor resolution admits a DG algebra structure with divided powers.

Linoy Utkina

University of Texas Rio Grande Valley

The Herzog-Takayama resolution over a skew polynomial ring

Let $k$ be a field, and let $I$ be a monomial ideal in the polynomial ring $R=k[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex that provides a finite free resolution of $R/I$ as an $R$-module. Building on this, Ferraro, Martin and Moore extended this construction to monomial ideals in skew polynomial rings. Since the Taylor resolution is generally not minimal, significant effort has been devoted to identifying classes of ideals with minimal free resolutions that are relatively straightforward to construct. In a 1987 paper, Eliahou and Kervaire developed a minimal free resolution for a class of monomial ideals in $R$ known as stable ideals. This result was later generalized to stable ideals in skew polynomial rings by Ferraro and Hardesty. In a 2002 paper, Herzog and Takayama constructed a minimal free resolution for monomial ideals with linear quotients, a broader class of ideals containing stable ideals. Their resolution reduces to the Eliahou-Kervaire resolution in the stable case. In this talk, based on a joint work with Ferraro, we will show how to generalize the Herzog-Takayama resolution to skew polynomial rings.

Justin Lyle

Auburn University

Vanishing of Tor and Depth of Tensor Products Over Cohen-Macaulay Rings

Let $R$ be a commutative Noetherian local ring and let $M,N$ be finitely generated $R$-modules. We say $R$ satisfies the condition (dep) if the depth formula $$\mathrm{depth}_R(M \otimes_R N)+\mathrm{depth}(R)=\mathrm{depth}_R(M)+\mathrm{depth}_R(N)$$ holds wheneve $\mathrm{Tor}^R_i(M,N)=0$ for all positive $i$. Recent work of Kimura-Lyle-Otake-Takahashi shows this condition need not hold in general, and in fact characterizes the so-called AB property for Gorenstein rings of positive dimension. Replacing the equality in the above formula by either the inequality $\ge$ or the inequality $\le$, the (dep) condition bifurcates into two conditions which we call (ldep) and (rdep), respectively. We show when $R$ is Cohen-Macaulay that (ldep) is tied to the so-called uniform Auslander condition, and that (ldep) implies (rdep) for Gorenstein rings of positive dimension. We also describe how these conditions behave under standard operations in commutative algebra, such as localization, completion, and modding out by a regular sequence. This talk is based on joint work with Kaito Kimura and Andrew Soto-Levins.

Jeff Hatley

Union College

Ranks of elliptic curves in quadratic twist families

Fix an elliptic curve $E/\mathbb{Q}$. given by the equation $y^2 = f(x)$, where $f$ is a smooth cubic. For each squarefree integer $d$, one may form the associated \textit{quadratic twist} $E^{(d)}$, which is given by $dy^2 = f(x)$. These two elliptic curves are distinct, but they become isomorphic over the field extension $\mathbb{Q}(\sqrt{d})$. Varying over all $d$, it is natural to wonder about the distribution of the Mordell-Weil ranks of the elliptic curves in this quadratic twist family. A conjecture due to Goldfeld predicts that half of the curves in this family should have rank 0, and half should have rank 1. In this talk, we study this problem using techniques from Iwasawa theory to obtain effective results in the direction of Goldfeld's conjecture. This is joint work with Anwesh Ray.

Jena Gregory

University of Texas Rio Grande Valley

Combinatorial statistics witnessing an infinite family of congruences for a sum of partition functions

In 2007, Kronholm established infinite families of congruences in arithmetic progression, modulo any prime $\ell$, for $p(n,m)$, the function enumerating the partitions of $n$ into parts whose sizes come from the set $[m]$. In 2022, Eichhorn, Kronholm, and Larsen proved there are combinatorial statistics, known as cranks, that witness Kronholm's infinite families of congruences. In this talk, we explore an extension of these results and consider cranks witnessing a sum/difference congruence of the form $$p(n,m) \pm p(n',m)\equiv 0 \pmod{\ell},$$ where $n'$ is determined by $n$. By an analysis of Ehrhart's $h^{*}$-vector, we have established that for certain primes and small values of $m$, there are cranks witnessing this sum/difference congruence.

Santanu Chakraborty

University of Texas Rio Grande Valley

A study on binary representation of odd numbers in a Collatz sequence

In the 88 year old Collatz Conjecture, one talks about a sequence consisting of positive integers where if a term $n$ in the sequence is even, the next term is $\frac{3}{2}$ and if it is odd, the next term in the sequence is $3n+1$. The conjecture states that such a sequence is a terminating sequence with the last term in the sequence being 1. Here we do not claim to prove the conjecture, but we prove some interesting results that may help future researchers. For this, we classified the odd positive integers in to two broad groups, $G_1$ and $G_2$: $G_1$ consists of numbers of the form $4m+1$ and $G_2$ consists of numbers of the form $4m+3$. One preliminary and well known observation about these two groups is, if an odd number in the Collatz sequence belongs to $G_1$ ($G_2$), the next odd number in the sequence is smaller (bigger). One of our initial results studies that if an odd number in a Collatz sequence belongs to $G_2$, then assuming that number of consecutive ones starting from the units' position in the binary representation is $\ell$ ($\ell \geq 2$), the next $\ell-2$ odd numbers in the sequence also belongs to $G_2$ but the ($\ell-1$)st number must belong to $G_1$. And our main result thoroughly investigates how the binary representation of an odd number in a Collatz sequence determines the group for the the next odd number in the sequence - $G_1$ or $G_2$.

Jeffery Opoku

University of Texas Rio Grande Valley

Ramanujan-Fine Integrals for level 10

We investigate the question of when an eta quotient is a derivative of a formal power series with integer coefficients and present an analysis in the case of level 10. As a consequence, we establish and classify an infinite number of integral evaluations such as \[ \int_0^{e^{-2\pi/\sqrt{10}}} q\prod_{j=1}^{\infty} \frac{(1-q^j)^3(1-q^{10j})^8}{(1-q^{5j})^7} \textrm{d}q = \frac14\left(\sqrt{10-4\sqrt{5}}-1\right). \] We describe how the results were found and give reasons for why it is reasonable to conjecture that the list is complete for level 10.

Danny Lara

University of Texas Rio Grande Valley

General Representation Type Algebras

We will introduce the notation of finite general representation type for finite dimensional algebras. This is related to the Dense Orbit Property introduced by Chindris, Kinser, and Weyman. We will use a combination of geometric, combinatorial, and geometric methods to produce a family of algebras that are of wild representation type but of finite general representation type.