This is the homepage of the Algebra and Algebraic Geometry Seminar at McMaster University. 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, or Adam Van Tuyl,
During Fall 2025, talks take place Tuesdays at 11:30-12:20am (Eastern) at HH 312.
Let E be an elliptic curve over the rationals given by an integral Weierstrass model and let P be a rational point of infinite order. The multiple nP has the form (A_n/B_n^2,C_n/B_n^3) where A_n, B_n, C_n are integers with A_nC_n and B_n coprime, and B_n positive. The sequence (B_n) is called the elliptic divisibility sequence generated by P. In this talk we answer the question posed in 2007 by Everest, Reynolds and Stevens: does the sequence (B_n) contain only finitely many perfect powers?
We discuss (mostly) recent work from several authors on epsilon and Hilbert Kunz multiplicities.
The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds over algebraically closed fields is a notoriously difficult problem. In contrast to the situation in lower dimensions, rank 2 bundles over smooth affine fourfolds are no longer uniquely determined up to isomorphism by their Chern classes. In this talk, we survey classification results in low dimensions and analyze the cohomological obstructions for the set of isomorphism classes of rank 2 bundles with prescribed Chern classes over a fixed smooth affine fourfold to be finite (resp. a singleton). This enables us in many cases to enumerate isomorphism classes of rank 2 bundles with prescribed Chern classes; we can even completely classify rank 2 bundles over some concrete smooth affine fourfolds. The talk is based on joint work with Thomas Brazelton and Morgan Opie.
Title: An introduction to jets of graphs
The notion of jets comes from geometry, where it is used to study singularities. The space of jets of an affine variety can be constructed directly from the equations of the variety, and this procedure naturally extends to ideals in a polynomial ring. After reviewing this construction, we will examine how it applies to (squarefree) monomial ideals which leads to a new notion of jets for (hyper)graphs. Finally, we will explain how vertex covers of jet graphs are related to covers of the original graph.
Title: Mutations of Exceptional Collections on Projective Varieties
Utilizing recent joint work with Michael Brown, Souvik Dey, and Guanyu Li on computing extensions of bounded complexes of coherent sheaves on projective varieties, we present several applications involving computations in D^b(X), including manipulations of exceptional collections consisting of complexes and computing explicit monads based on them. We will also discuss related problems and conjectures from Helix theory in triangulated categories, particularly D^b(P^n).
Title: Gapfree graphs and powers of their edge ideals
Abstract: Powers of edge ideals often reflect the combinatorics of the underlying graph. Let $G$ be a graph and let $I(G)$ be its edge ideal. For gapfree graphs, a conjecture of Nevo–Peeva states that high enough powers of the edge ideal have linear resolutions. In this talk, I will describe our approach to this conjecture via a new conjecture involving linear quotients. Our conjecture is monotone: once one power $I(G)^q$ has linear quotients, then every higher power $I(G)^s$ (for $s\geq q$) should as well. I’ll present partial progress, including hypotheses under which the problem reduces to checking just the second power $I(G)^2$. It is known that forbidding a cricket, a diamond, and a 4-cycle forces $I(G)^q$ to have a linear resolution for all $q\geq 2$. So, one of the results we will discuss is about a construction of gapfree graphs that do contain those subgraphs (and a 5-cycle), yet still have linear quotients for every $q\geq 2$. The talk will be example-driven, with quick primers on linear resolutions and linear quotients, and open questions at the end. Joint work with Erey, Faridi, Hà, Hibi, and Morey.
Title: Square-free monomial ideals via open neighborhoods of graphs.
A key branch of commutative algebra, combinatorial commutative algebra, focuses on the study of square-free monomial ideals using combinatorial structures. Many techniques have been developed to analyze these ideals, particularly through the use of simplicial complexes and (hyper)graphs. This talk explores square-free monomial ideals through the lens of open neighborhood ideals of finite simple graphs. We will explore which types of tree graphs have a Cohen-Macaulay open neighborhood ideal, and how these ideals and their corresponding graphs arise in different algebraic or combinatorial settings. Furthermore, we will answer what types of square-free monomial ideals can be realized as the open neighborhood ideal of a finite simple graph.