# Algebra and Algebraic Geometry Seminar

## @ McMaster University

This is the homepage of the Algebra and Algebraic Geometry Seminar at McMaster University. During Fall 2023, talks take place Thursday 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: 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.

# 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.