UTRGV Algebra & Number Theory Seminar

The Algebra & Number Theory Seminar meets on Thursdays at 4:00pm in EMAGC 3.502 during Spring '24. 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.

Jeffrey Opoku

University of Texas, Rio Grande Valley

Ramanujan type congruences for quotients of Klein Forms

In this work, Ramanujan type congruences modulo powers of primes $p \ge 5$ are derived for a general class of products that are modular forms of level p. These products are constructed in terms of Klein forms and eta quotients and are generating functions for certain important arithmetic functions known to satisfy Ramanujan type congruences for $p = 5, 7, 11$. The vectors of exponents corresponding to these products that are modular forms for $\Gamma _{1} (p)$ are subsets of bounded polytopes with explicit parameterizations. This allows for the derivation of a complete list of products that are modular forms for $\Gamma_{1} (p)$ of weights $k$ for primes $5 \le p \le 19$ and whose Fourier coefficients satisfy Ramanujan type congruences for all powers of the primes. Here, we provide a comprehensive characterization of these products modulo powers of $5$ for the level $5$ products, and a characterization modulo $7$ for the level $7$ products.

Souvik Dey

Charles University

Projective dimension of tensor product of modules

Given two non-zero finitely generated modules over a commutative Noetherian local ring, the derived tensor product has finite projective dimension if and only if so does each of the modules. This no longer remains true if "derived tensor product" is replaced by ordinary tensor-product. In this talk, we discuss several results illustrating certain hypothesis on the modules or the ring under which finiteness of projective dimension of tensor product two modules implies or is implied by the finiteness of projective dimension of the individual modules. This is based on joint work (some ongoing) with Olgur Celikbas, Toshinori Kobayashi and Hiroki Matsui.

Joselyne Aniceto

University of Texas, Rio Grande Valley

Congruence properties of consecutive coefficients in arithmetic progression of Gaussian polynomials

A 2023 result of Eichhorn, Engle, and Kronholm describes an interval of consecutive congruences for $p(n,m,N)$, the function that enumerates the partitions of $n$ into at most $m$ parts, none of which are larger than $N$, in arithmetic progression. This function is the partition theoretic interpretation of the coefficient on $q^n$ of the Gaussian polynomial, ${{N+m \brack m}}$, otherwise known as the $q$-binomial coefficient. In this talk we will considerably expand their result to capture a much larger family of congruences. We will consider known infinite families of congruences for $p(n,m)$, the function that enumerates the partitions of $n$ into at most $m$ parts, and introduce a related infinite family of congruences for a two-colored partition function. The result of Eichhorn, Engle, and Kronholm becomes a special case of our expanded theorem.

Luigi Ferraro

University of Texas, Rio Grande Valley

Trimming five generated Gorenstein ideals

Let $R$ be a regular local ring of dimension 3 with maximal ideal $\mathfrak{m}$. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. In a previous work, A. Hardesty and I constructed an explicit free resolution of $R/J$ and computed a DG algebra structure on this resolution. We used the products on this resolution to study the Tor algebra of such trimmed ideals. Missing from our result was the case where $I$ is five generated. In this talk, which is based on a joint work with F. Moore, we will address this case after covering the necessary background information.

Nina Zubrilina

Princeton University

Root Number Correlation Bias of Fourier Coefficients of Modular Forms

In a recent machine learning-based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number. In my talk, I will discuss this root number correlation bias when the average is taken over all weight k modular newforms. I will point to a source of this phenomenon in this case and compute the correlation function exactly.

Elena Poletaeva

University of Texas, Rio Grande Valley

On finite $W$-algebras and Yangians

Finite $W$-algebras are certain associative algebras attached to a pair $(\mathfrak{g}, e)$, where $\mathfrak{g}$ is a complex semi-simple Lie algebra and $e\in \mathfrak{g}$ is a nilpotent element. They are generalizations of the universal enveloping algebra $U(\mathfrak{g})$ and have many interesting applications. They have been extensively studied by mathematicians and physicists. It is a result of B. Kostant that if $\mathfrak{g}$ is a reductive Lie algebra and $e\in \mathfrak{g}$ is a regular nilpotent element, then the finite $W$-algebra coincides with the center of $U(\mathfrak{g})$. The Yangian of $\mathfrak{g}$ is an infinite-dimensional Hopf algebra $Y(\mathfrak{g})$. It is a deformation of the universal enveloping algebra of the Lie algebra of polynomial currents of $\mathfrak{g}$: $\mathfrak{g}[t] = \mathfrak{g}\otimes \mathbb{C}[t].$ J. Brundan and A. Kleshchev described finite $W$-algebras for the general linear Lie algebra as truncations of shifted Yangians of $\mathfrak{gl}(n)$. The queer Lie superalgebra $Q(n)$ is a super analogue of $\mathfrak{gl}(n)$. The super-Yangian $YQ(n)$ of $Q(n)$ was defined by M. Nazarov. We show that there exists a relationship between the super-Yangian $YQ(1)$ and principal finite $W$-algebra for $Q(n)$. We use it to classify simple finite-dimensional modules over these superalgebras. This is a joint work with V. Serganova.

Brandt Kronholm

University of Texas, Rio Grande Valley

Combinatorial witnesses for infinite families of partition congruences

Ramanujan's congruences for the general partition function are well known and well-studied. So too are Dyson's rank and the Andrews Garvan crank which combinatorically explain Ramanujan's congruences. In this talk I will discuss Ramanujan-style congruences for the restricted partition function $p(n,m)$ enumerating the partitions of $n$ into parts not larger than $m$ and newly discovered cranks witnessing them. Some proofs, some results, some conjectures, and some questions.

Alexis Hardesty

Texas Woman's University

Realizing Algebra Structures on Free Resolutions of Grade 3 Perfect Ideals

Perfect ideals of grade 3 can be classified based on algebra structures on their minimal free resolutions. The classification is incomplete in the sense that it remains open which of the possible algebra structures actually occur; this realizability question was formally posed by Avramov in 2012. Of five classes of algebra structures, the realizability question had previously been answered for one class. In this talk, we discuss a new insights on the realizability question, answering the question for two more classes and obtaining a partial answer for a third.

Jena Gregory

University of Texas, Rio Grande Valley

Ramanujan's Partition Congruences and Dyson's Rank: How is the Division to be Made?

A partition of a positive integer $n$ is a finite nonincreasing sequence of positive integers $\lambda_1, \dots, \lambda_m$ such that $\sum^m_{i=1} \lambda_i =n$. For example, the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, 1+1+1+1. The sequence of partitions numbers is $$\{p(n)\}_{n=0}^{\infty}=1,1,2,3,5,7,11,15,22,30,...$$ Mathematicians are interested in the patterns in this sequence. In 1919, Ramanujan proved the following congruences with $q$-series: For all nonnegative integers $k$, \begin{align} %\begin{split} p(5k+4)&\equiv 0\pmod 5\label{Ramanujan19195} \\ p(7k+5)&\equiv 0\pmod 7 \label{Ramanujan19197}\\ p(11k+6)&\equiv 0\pmod{11}. \label{Ramanujan191911} %\end{split} \end{align} In 1944, Dyson requested proofs of Ramanujan's congruences that ``will not appeal to generating functions but will demonstrate by cross-examination of the partitions themselves." Dyson proposed a statistic called the rank that would do just that. In this talk, we discuss Ramanujan’s partition congruences and Dyson’s rank. We extend these ideas to a certain restricted partition function and consider Dyson style witnesses for congruences.

Tim Huber

University of Texas, Rio Grande Valley

Eta Quotients with Identically Vanishing Coefficients

For a function $A(q)=\sum_{n\geq 0} a_n q^n$, define \[ A_{(0)}:=\{n\in \mathbb{N}: a_n=0\}. \] If $A(q)$ and $B(q)$ are two functions for which $A_{(0)}=B_{(0)}$, then we say that $A(q)$ and $B(q)$ have identically vanishing coefficients. We will discuss methods for proving pairs $(A(q),B(q))$ of eta quotients have identically vanishing coefficients. The eta quotients we consider are called lacunary since the proportion of vanishing coefficients tends to zero. Our work is motivated by the work of J. P. Serre on lacunary even powers of the Dedekind eta function.

Antonio Lei

University of Ottawa

Anticyclotomic Iwasawa theory of elliptic curves

Let $E$ be an elliptic curve, $p$ an odd prime and $K$ an imaginary quadratic field where $p$ is unramified. One may use Iwasawa theory to study the growth of Mordell-Weil ranks and the $p$-part of the Shafarevich-Tate groups inside the anticyclotomic $\mathbb{Z}_p$ extension of $K$. In this talk, we will discuss the case where $E$ has good supersingular reduction at $p$. In particular, we discuss generalizations of Kobayashi’s plus and minus Selmer groups and the corresponding Iwasawa main conjectures.

Mohsen Gheibi

Florida A&M University

Characteristic and co-characteristic modules of a module over a local ring

In this talk, I will introduce the characteristic and co-characteristic modules of a finitely generated module $M$ over a local ring $R$. First, we will see the motivations behind the definition of these modules, and then, I will explain their basic properties such as existence, uniqueness, and functorial property. Next, we will see some examples, and at the end, I will present some applications of these modules, including the characterization of Cohen-Macaulay and Gorenstein rings.