Kalina Mincheva

Algebra and Combinatorics Seminar

We meet weekly on Wednesday at 3:00 PM (Central time), Gibson 310

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

Schedule: Fall 2021

• Karl H. Hofmann, Tulane University (Zoom).
  Group Algebras of Compact Groups and Enveloping Algebras of Profinite-Dimensional Lie Algebras, abstract.
  We reintroduce elementary Linear Algebra over $\mathbb{R}$ and $\mathbb{C}$ with a distinct emphasis on weakly complete vector spaces as the duals of ordinary vector spaces with an emphasis on the definition of weakly complete topological algebras. Their category we call $\mathcal{WA}$. We describe some basic properties of a $\mathcal{WA}$-algebra $A$ and its group $A^{-1}$ of units (i.e. invertible elements) and its exponential function. We also need the concept of a (symmetric) Hopf algebra, notably within $\mathcal{WA}.$ This will allow us to define, for a topological group $G$, its (topological) group algebra $\mathbb{K}[G]$ over $\mathbb{K}$. The goal is to illustrate its significance in the general theory of compact groups. As time permits we shall touch upon a parallel concept: The weakly complete universal enveloping algebra $U_{\mathbb{K}}(\mathfrak{g})$ of a (profinite-dimensional) Lie algebra over $\mathbb{K}$ and the relationship to $\mathbb{K}[G]$ in view of the Lie algebra $\mathfrak{g}=\mathfrak{L}(G)$ of $G$. (This continues lectures given to the Tulane Algebra Seminar on Sep 10-2018, Mar 7-2019, Oct 2-2019; a lecture scheduled for Mar 18-2020 was cancelled due to the breakout of the Covid 19-pandemic. Main Reference: K.H.Hofmann and S.A.Morris: The Structure of Compact Groups, DeGruyter Berlin, 4th Edition 2020 [in the Tulane Library].)

• Olivia Beckwith, Tulane University (GI-310).
  Modular forms and divisibility properties of partition numbers, abstract.
  My research focuses on elliptic modular forms and their connections to different areas of number theory. Two of my favorite areas are the study of integer partitions and quadratic number fields. For Part 1 of this series, I will start with a brief crash course defining modular forms. Then I will describe some of my work studying the divisibility properties of numbers which count integer partitions. This includes joint work with Scott Ahlgren and Martin Raum, and may briefly mention ongoing work with Jack Chen, Maddie Diluia, Oscar Gonzales, and Jamie Su.

• Olivia Beckwith, Tulane University (GI-310).
  Real analytic modular forms and quadratic number fields, abstract.
  For Part 2 of my introduction to my research, I will focus on my other favorite area: quadratic number fields. First I will define a class of real-analytic modular forms. Then I will show how they can be used in the study of class numbers of imaginary quadratic number fields, as well as Hecke series for real quadratic number fields. This includes joint ongoing work with Gene Kopp.

• Daniel Bernstein, Tulane University (GI-310).
  Toric ideals from statistics. abstract.
  Toric ideals are prime polynomial ideals that are generated by monomial differences. They have a rich theory with connections to polyhedral combinatorics, optimization, and statistics, which I will discuss in this talk.

• Daniel Bernstein, Tulane University (GI-310).
  Algebraic matroids in rigidity theory and matrix completion. abstract.
  A set of quantities is algebraically independent over a field $F$ if they satisfy no polynomial equations with coefficients in $F$. Matroids are a combinatorial generalization of (algebraic) independence. In this talk, I will give an introduction to matroids from an algebraic perspective, and explain how they arise in rigidity theory and matrix completion. Time permitting, I will introduce tropical geometry and discuss how it can be used to understand particular algebraic matroids.

• Tài Huy Hà, Tulane University (GI-310).
  Saturation bounds for smooth varieties. abstract.
  Let $X$ be a smooth variety which is ideal-theoretically defined by an ideal $J$. We discuss linear bounds for the saturation degrees of powers of $J$ in terms of its generating degrees. Our work extends a classical result of Macaulay, and fills the gap between studies on algebraic and geometric notions of the Castelnuovo-Mumford regularities of smooth varieties. This is a joint work with L. Ein and R. Lazarsfeld.

• Netanel Friedenberg, Tulane University (GI-310).
  Completions of degenerations of toric varieties. abstract.
  After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a new result about the existence of normal equivariant completions of these degenerations. Prior knowledge of toric varieties will not be necessary for understanding this talk.

• Thái Nnguễn, Tulane University (GI-310).
  Newton-Okounkov Bodies, Rees Algebra, and Analytic Spread of Graded Systems of Monomial Ideals. abstract.
  Newton-Okounkov bodies are convex sets associated to algebro-geometric objects, that was first introduced by Okounkov in order to show the log-concavity of the degrees of algebraic varieties. In special cases, Newton-Okounkov bodies associated to graded systems of ordinary powers and symbolic powers of a monomial ideal are Newton polyhedron and symbolic polyhedron of the ideal. Studying these polyhedra can be beneficial to the study of relation between ordinary powers, integral closure powers and symbolic powers of a monomial ideal as well as its algebraic invariants. In this talk, I will survey some known results in this subject and present our results on computing and bounding the analytic spread of a graded system of monomial ideals and some related invariants through the associated Newton-Okounkov body. This is joint work with Tài Huy Hà.

• Erin Chambers, Saint Louis University
  TBA. abstract.