# Algebra and Algebraic Geometry Seminar

## @ McMaster University

This is the homepage of the Algebra and Algebraic Geometry Seminar at McMaster University. During Winter 2024, talks take place Tuesdays at 9:30-10:20am (Eastern) at HH 207. This seminar is primarily intended for McMaster University graduate students, postdocs, and faculty with an interest in algebra, algebraic geometry, number theory, or related areas. If you are interested in giving a talk, please contact one of the faculty members affiliated with the seminar: Cam Franc, Megumi Harada, Jenna Rajchgot, Adam Van Tuyl, or our postdoc, Thanh Thai Nguyen.

# Upcoming Talk:

Title: Symbolic Defect of Monomial Ideals

Abstract: Symbolic powers of ideals in a Noetherian commutative ring have long been a topic of interest in commutative algebra. They are directly related to the associated primes, and always contain the ordinary powers. The symbolic defect function is a numerical function designed to measure the ``closeness” of symbolic powers and the ordinary powers. In particular, symbolic defect measures the minimal number of generators of the quotient of symbolic powers with ordinary powers. In this talk, we will introduce tools to help us study minimal generation of ordinary powers, symbolic powers, and symbolic defect, particularly for monomial ideals. We compare the latter function to a related invariant, the integral symbolic defect, and then discuss some recent results.

# Schedule for Winter 2024:

Title: Symbolic Defect of Monomial Ideals

Abstract: Symbolic powers of ideals in a Noetherian commutative ring have long been a topic of interest in commutative algebra. They are directly related to the associated primes, and always contain the ordinary powers. The symbolic defect function is a numerical function designed to measure the ``closeness” of symbolic powers and the ordinary powers. In particular, symbolic defect measures the minimal number of generators of the quotient of symbolic powers with ordinary powers. In this talk, we will introduce tools to help us study minimal generation of ordinary powers, symbolic powers, and symbolic defect, particularly for monomial ideals. We compare the latter function to a related invariant, the integral symbolic defect, and then discuss some recent results.

# Schedule for Fall 2023:

Title: Alternating sign matrix varieties

Abstract: Matrix Schubert varieties, introduced by Fulton in the '90s, are affine varieties that "live above" Schubert varieties in the complete flag variety. They have many desirable algebro-geometric properties, such as irreducibility, Cohen--Macaulayness, and easily-computed dimension. They also enjoy a close connection with the symmetric groups.

Alternating sign matrix (ASM) varieties, introduced by Weigandt just several years ago, are generalizations of matrix Schubert varieties in two senses: (1) ASM varieties are unions of matrix Schubert varieties and (2) the defining equations of ASM varieties are determined by ASMs, which are generalizations of permutation matrices. ASMs have been important objects of study in enumerative combinatorics since at least the '80s and appear in statistical mechanics as the 6-vertex lattice model. Although ASMs have a robust combinatorial underpinning and although their irreducible components are matrix Schubert varieties, they are nevertheless much more difficult to get a handle on than matrix Schubert varieties themselves. In this talk, we will define ASMs, compare and contrast with matrix Schubert varieties, and state some open problems.

Title: Oops, all Hilbert! Hilbert functions, h-polynomials, and the Hilbertian property

Abstract: Many fundamental properties of projective varieties are encoded by their Hilbert functions, which record dimensions of the graded pieces of their coordinate rings. A variety is called Hilbertian if its Hilbert function is a polynomial. We will introduce these notions from the ground up before explaining the ubiquity of Hilbertian varieties in combinatorial commutative algebra via Stanley-Reisner theory. If time permits we will also discuss possible generalizations of these ideas to the multigraded setting.

Title: Asymptotic Invariant of Graded Families

Abstract: In this talk we shall present a duality for sequences of numbers which interchanges superadditive and subadditive sequences, and inverts their asymptotic growths. We shall discuss at least two algebro-geometric contexts where this duality shows up: how it interchanges the sequence of initial degrees of symbolic powers of an ideal of points with the sequence of regularities of a family of ideals generated by powers of linear forms; and how it underpins the reciprocity between the Seshadri constant and the asymptotic regularity of a finite set of points. This is joint work with Michael DiPasquale and Alexandra Seceleanu.

Title: Barile-Macchia resolutions

Abstract: In this talk I will introduce Barile-Macchia resolutions, which is a special type of resolutions for monomial ideals constructed via discrete Morse theory. These resolutions are minimal for many classes of monomial ideals, including edge ideals of weighted oriented forests and (most) cycles. I will also discuss recent follow-up work on this topic.

### Oct 12: No Seminar

### Oct 19: No Seminar

Title: The Kruskal-Katona Theorem

Abstract: We can associate to a simplicial complex of dimension d a tuple f =(f_0,f_1,...,f_d),where f_i counts the number of faces of the simplicial complex of dimension i. The tuple f is called the f-vector of the simplicial complex. In this expository talk, I will describe the Kruskal-Katona theorem, which characterizes what vectors can be the f-vector of a simplicial complex. If there is time, I will explain some connections to commutative algebra.

### Nov 2: No Seminar

### Nov 9: Kieran Bhaskara & Aniketh Sivakumar (McMaster)

Title of Kieran's talk: Regularity and projective dimension of toric ideals of bipartite graphs

Abstract: The regularity and projective dimension of combinatorially-defined ideals are frequently studied invariants in combinatorial commutative algebra. In particular, much work has been done towards understanding the values these invariants can achieve for toric ideals I_G associated to a graph G. In this talk, we fully describe the possible values of these invariants for I_G as G ranges over all bipartite graphs on a fixed number of vertices. As a corollary, we show that any pair of positive integers can be realized as the regularity and projective dimension of a toric ideal of a bipartite graph. Finally, we demonstrate how our main result allows us to determine the values all five major invariants studied in the literature for this family of graphs.

Title of Aniketh's talk: An overview of binomial edge ideals

Abstract: In this talk we will introduce an ideal associated to a graph, called a binomial edge ideals. These ideals were first introduced in 2010, with an application to conditional independence statements. Since then a lot of work has been done studying various homological invariants of these ideals. Here, we will define these ideals and give some results on their minimal primes and grobner basis. If time permits, we will talk about some results on the Betti numbers and regularity of these ideals.

Title: An improved upper bound on a class of exponential sums

Abstract: In Yitang Zhang’s proof of bounded gaps between primes (2014), a certain 3-variable Kloosterman sum played a crucial role and was one of the deepest parts of his proof. Further improvements on bounded gaps will likely require deeper understanding of this sum. This particular exponential sum was originally studied by J. Friedlander and H. Iwaniec (1985) with upper bound first obtained by B. Birch and E. Bombieri (1985) by special arguments. Further improvements were made by N. Katz (1986) using ℓ-adic method, and very recently by C. Chen and X. Lin (2022) by p-adic method. In an attempt to understand a remark made by Katz at the end of his 1986 paper, M. Roth and I were able to find a way that leads to a slight improvement on the upper bound of this sum, as well as on a family of similar exponential sums. Our method is based on ℓ-adic cohomology, consisting of finer studying of the local monodromies at zero via representations of the inertia groups there. In this talk, I will survey the above-mentioned results and give a high-level overview of our procedure that gives rise to this new improvement. This is joint work in progress with M. Roth.

Title: On the parity conjecture for Hilbert schemes of points on threefolds

Abstract: The Hilbert scheme of points in affine n-dimensional space, which parametrizes zero-dimensional subschemes with a fixed degree, is a fundamental parameter space in algebraic geometry. Quot schemes are a generalization of Hilbert schemes, parametrizing finite length quotients of a locally free sheaf. We will explore some interesting phenomena and problems about these spaces that are specific to the three-dimensional case, focusing on the tangent space and recent progress on the parity conjecture of Okounkov and Pandharipande. This is joint work with Alessio Sammartano.