Spring 2024

Title: Some properties of Lovász-Saks-Schrijver Ideals

Abstract:  Lovász-Saks-Schrijver ideals, LSS ideals for short, are a family of ideals associated to graphs that were introduced in the context of orthogonal representations of graphs and studied for the first time in 1989 by Lovász, Saks and Schrijver. The study of such ideals lie in the intersection between algebraic geometry, commutative algebra and combinatorics as some geometric and algebraic properties can be exhibited from combinatorial invariants of the graph and vice versa. Our goal is to study this relationship, focusing on some algebraic properties of LSS ideals such as defining a unique factorization domain.

Prerequisites: Familiarity with the definition of rings, ideals, unique factorization domains, localization;  basic definitions in graph theory. Basics of Gröbner basis theory will be recalled during the talk.

Title: Sums of Squares: a real projective story

Abstract:  A multivariate real polynomial is nonnegative if its value at any real point is greater than or equal to zero.  These special polynomials play a central role in many branches of mathematics including algebraic geometry, optimization theory, and dynamical systems.  However, it is very difficult, in general, to decide whether a given polynomial is nonnegative.  In this talk, we will review some classic methods for certifying that a polynomial is nonnegative.  We will then present novel certificates in some important cases.  This talk is based on joint work with Grigoriy Blekherman, Rainer Sinn, and Mauricio Velasco.

Prerequisites:  polynomials, real numbers, and some intuition about projective space

Title:  Frieze patterns - from combinatorics to representation theory

Abstract:  Friezes are infinite arrays of numbers, in which every four neighbouring vertices arranged in a diamond satisfy the same arithmetic rule. Introduced in the late 1960s by Coxeter, and further studied by Conway and Coxeter in their remarkable papers from 1973, this topic has been nearly forgotten for over thirty years. But recently, frieze patterns have attracted a lot of interest and appeared in many areas of mathematics, like combinatorics, geometry, and representation theory.

In this talk I will review the beautiful Conway-Coxeter theorem relating Coxeter's frieze patterns to triangulations of polygons and then focus on recent developments in representation theory, in particular in connection with cluster algebras and cluster categories.

Prerequisites: Linear algebra, determinants

Title: Introduction to Castelnuovo-Mumford regularity

Abstract:  Suppose we want to find all polynomials f(x)  such that

(*)  f(1)=1,   f(2)=3,   f(3)=6 and f(4)=2.

We know what to do: we find one specific f_0(x)  satisfying (*) by  solving a linear system or with Lagrange interpolators. Then the general polynomial  that satisfies (*) is  f(x)=f_0(x)+(x-1)(x-2)(x-3)(x-4)h(x) with h(x) any polynomial. The point here is that  the polynomial (x-1)(x-2)(x-3)(x-4) generates the ideal of polynomials vanishing at 1,2,3 and 4.

When we study more general interpolation problems, with more variables and with  more complicated conditions   (e.g. polynomials that vanish on a finite union of lines in a 3 dimensional space) finding the ideal of polynomials that vanish on the corresponding loci becomes more and more difficult. Already bounding the degrees of generators of that ideal is a complicated process.

The Castelnuovo-Mumford regularity of an ideal is an invariant that bounds from above the largest degree of the  generators and, because of its multifaceted and versatile nature, is often easier to compute or to bound. The goal of the talk is to introduce this notion  and to explain what we know and do not know about it especially in the case of subspace arrangements.

Prerequisites:  polynomial rings

Title:  Principal ideals with respect to a group action

Abstract:  We delve into the fascinating interplay between groups and polynomials. There are many different ways one can transform a polynomial according to an action of some group: for example, the symmetric group acts on polynomials by permuting their variables. We consider ideals generated by the polynomials resulting from a specified input under a group action, dubbing them principal ideals with respect to the group action.

In this talk we focus on two classes of principal ideals with respect to two different groups and we determine their noteworthy properties. This is based on joint work with Alessandra Constantini and separately with Megumi Harada and Liana Sega.

Prerequisites: polynomials, ideals, the symmetric group.

Title: Short virtual resolutions

Abstract:  Free resolutions are a tool in commutative algebra for representing

algebraic objects in terms of simpler pieces.  Over a polynomial ring

their maximum length is determined by the number of variables.  I will

discuss methods for reducing the length of a resolution, depending on

what information is geometrically important.

Prerequisites: polynomials, rings, ideals, matrices

Title: 

Abstract:  

Prerequisites: 

Title: Classifying numerical semigroups using polyhedral geometry

Abstract:  A numerical semigroup is a subset of the natural numbers that is closed under addition.  There is a family of polyhedral cones $C_m$, called Kunz cones, for which each numerical semigroup with smallest positive element $m$ corresponds to an integer point in $C_m$.  It has been shown that if two numerical semigroups correspond to points in the same face of $C_m$, they share many important properties, such as the number of minimal generators and the Betti numbers of their defining toric ideals.  In this way, the faces of the Kunz cones naturally partition the set of all numerical semigroups into "cells" within which any two numerical semigroups have similar algebraic structure.  

Prerequisites: Basic proof concepts

Title: αβ=2

Abstract:  Some geometric objects X have a “Chow ring” Ch(X) — an algebraic device that keeps track of how the subobjects of X intersect with each other. When X is polyhedral (or toric), the Chow ring Ch(X) features a rich, beautiful, powerful structure.  In this interactive talk we will learn about four *very* different ways of thinking about Ch(X). Our shared goal will be to give four different proofs of the equation αβ=2 (we’ll see what α and β mean), and learn something about Chow rings along the way. If there is time at the end, I will say a few words about what this has to do with the solution of Rota's 1970 conjecture about colorings of maps.

Prerequisites: Pencil and paper. My goal / hope is that everyone interested can engage with this talk and learn something new.