Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Gibson 126 A

Organizers: Daniel Bernstein, Mahir Can, Tài Huy Hà, Kalina Mincheva.

Schedule: Fall 2023

• Dermot McCarthy, Texas Tech University
  The number of $\mathbb{F}_q$-points on diagonal hypersurfaces with monomial deformation. abstract.
  In this talk, we consider the problem of counting the number of solutions to equations over finite fields using character sums. We start with a review of standard techniques and discuss Weil's seminal 1949 paper, which gives an exposition on the topic up to that point by examining diagonal hypersurfaces. We then consider the family of diagonal hypersurfaces with monomial deformation $D_{d, \lambda, h}: x_1^d + x_2^d \dots + x_n^d - d \lambda \, x_1^{h_1} x_2^{h_2} \dots x_n^{h_n}=0$, where $d = h_1+h_2 +\dots + h_n$ with $\gcd(h_1, h_2, \dots h_n)=1$, which was studied by Koblitz over $\mathbb{F}_{q}$ in the case ${d \mid {q-1}}$. We outline recent results where we provide a formula for the number of $\mathbb{F}_{q}$-points on $D_{d, \lambda, h}$ in terms of Gauss and Jacobi sums, which generalizes Koblitz's result. We then express the number of $\mathbb{F}_{q}$-points on $D_{d, \lambda, h}$ in terms of a $p$-adic hypergeometric function previously defined by the speaker. The parameters in this hypergeometric function mirror exactly those described by Koblitz when drawing an analogy between his result and classical hypergeometric functions.

• Karl Hofmann, Tulane and TU Darmstadt.
  The Left Half and the Right Half of the Brain: Mathematics and Art. abstract.
  One area in which mathematics and art approach each other is the field of advertising mathematics in posters for colloquium lectures or seminars, or in providing illustrations in books or articles. As I have been active in this direction I propose a lecture on the difficulties one encounters in the attempt to advertise mathematical contents pictorially. Accordingly, I shall guide the audience through a short tour of recent colloquium posters for the Mathematics Department of the Technical University of Darmstadt and explain some of the problems one encounters in illustrating and advertising mathematics.

• Daniel Bernstein, Tulane University.
  Matroid lifts and representability. abstract.
  Matroid theory is an area of combinatorics that is becoming increasingly relevant in a broad range of research areas, from optimization to algebraic geometry. A matroid is a combinatorial structure meant to abstract the notion of linear independence in a vector space. In particular, given a vector space and a finite subset $E\subseteq V$, the subsets $S\subseteq E$ that are linearly independent are an example of a matroid. Such matroids are called representable. Not all matroids are representable, but certifying that a given matroid is not representable can be a difficult task. In this talk, I will give an introduction to matroids and discuss a new certificate of non-representability that arises out of a matroid construction called lifts. This is joint work with Zach Walsh.

• Arvind Kumar, New Mexico State University.
  Regularity Bound of Generalized Binomial Edge Ideal of Graphs. abstract.
  I will discuss about the Castelnuovo-Mumford regularity of generalized binomial edge ideals. This class of ideals arises in the study of conditional independence ideals and was introduced by Johannes Rauh in 2011. I will cover Saeedi Madani and Kiani's conjecture about the regularity of this class of ideals.

• Abeer Al Ahmadieh, Georgia Tech.
  The Principal Minor Map and Its Tropicalization. abstract.
  The principal minor map takes an $n \times n$ square matrix and maps it to the $2^n$-length vector of its principal minors. In this talk, I will describe both the fiber and the image of this map. In 1986, Loewy proposed a sufficient condition for the fiber to be a single point up to diagonal equivalence. I will provide a necessary and sufficient condition for the fiber to be a single point. Additionally, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques, I will present equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. I will also provide examples to demonstrate that, for general matrices, no finite characterization is possible. Finally, I will describe the tropicalization of the image of the cone of positive semidefinite matrices under this map. This is based on joint research with Felipe Rinc\'on, Cynthia Vinzant and Josephine Yu.

• Mike Hanson, University of North Texas.
  Ramanujan congruences for partition functions. abstract.
  Integer partitions have been studied by mathematicians for many years and have a wide array of applications in and outside of mathematics. Ramanujan was the first to discover surprising divisibility properties of the classical partition-counting function $p(n)$, known as the Ramanujan congruences. Mathematicians later used the connection between partitions and modular forms to fully classify the Ramanujan congruences for $p(n)$. In this talk, we make use of this connection to conclude something more general about divisibility properties of a special class of modular forms called eta-products, and we explore applications to other types of partition-counting functions.

• Elizabeth O'Riley, Johns Hopkins University.
  Spectrahedral Regression. abstract.
  Convex regression is the problem of fitting a convex function to a collection of input-output pairs, and arises naturally in applications such as economics, engineering and computed tomography. We present a new approach to this problem called spectrahedral regression, in which we fit a spectrahedral function to the data, i.e. a function that is the maximum eigenvalue of an affine matrix expression of the input. This method generalizes polyhedral (also called max-affine) regression, in which a maximum of a fixed number of affine functions is fit to the data. We first provide bounds on how well spectrahedral functions can approximate arbitrary convex functions via statistical risk analysis. Second, we analyze an alternating minimization algorithm for the non-convex optimization problem of fitting a spectrahedral function to data. Finally, we demonstrate the utility of our approach with experiments on synthetic and real data sets. This talk is based on joint work with Venkat Chandrasekaran.

• Signe Lundqvist, Umeå University.
  Realizations of hypergraphs and their motions. abstract.
  The mathematical theory of structural rigidity has a long history. In the nineteenth century, Cauchy studied rigidity of polyhedra, and Maxwell studied graph frameworks. The rigidity theory of graph frameworks has since been studied extensively. Pollaczek-Geiringer, and later Laman, proved a combinatorial characterization of the minimally rigid graphs in the plane. Combinatorial rigidity theory is also concerned with geometric realizations of other combinatorial structures. In this talk, we will focus on rigidity of realizations of hypergraphs as points and straight lines. We will discuss how to determine whether a realization of a hypergraph is rigid, in the sense that there are no motions of the realization that preserve the incidences of points and lines, and the distance between any pair of points that lie on a line. We will also discuss motions of realizations of hypergraphs that preserve only the incidences between points and lines. We will see that classical theorems in incidence geometry, such as Pascal's theorem, make determining rigidity with respect to such motions a difficult problem. The talk will be based on joint work with K.Stokes and L-D. Öhman, as well as work in progress, joint with L.Berman, B.Schulze, B.Servatius, H.Servatius, K.Stokes and W.Whiteley.

• Rebecca R.G., George Mason University
  Closure operations in commutative algebra. abstract.
  Closure operations appear throughout commutative algebra, including tight closure, integral closure, and many others. In this talk I will give an overview of closure operations, test ideals, and their applications to studying singularities, then demonstrate how to construct closure operations that are compatible with passing to submodules or quotient modules. This work is joint with Neil Epstein and Janet Vassilev.