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 2026, talks take place Tuesdays at 10:30-11:20am (Eastern) at HH 312.
Abstract: The Hilbert series of a homogeneous algebra reflects various algebraic properties of the algebra through the h-polynomial appearing in its numerator. In this talk, I will give an overview of the relationship between Hilbert series and ring-theoretic properties, e.g., the Cohen–Macaulay and Gorenstein properties, and explain some fundamental results on h-polynomials, including Macaulay’s criterion. I will also discuss Stanley–Reisner rings, which play an important role in the study of h-polynomials of homogeneous algebras. Finally, I will present some recent results from joint work with Kenta Ueyama and Takumi Yanagida on homogeneous algebras whose h-polynomials are products of cyclotomic polynomials.
Abstract: Hessenberg varieties are subvarieties of the flag variety Flags(\C^n) and have a rich geometry which intersects with the theory of combinatorial commutative algebra, Schubert geometry and Schubert calculus, and geometric representation theory, among other areas. The cohomology rings of Hessenberg varieties have been studied extensively due to their connections with the theory of quasisymmetric functions. I will give a curated introduction to this circle of ideas, with a focus on a selection of recent developments.
Abstract: Edge ideals of graphs provide a fertile ground for uncovering combinatorial expressions for algebraic invariants of squarefree monomial ideals. In fact, many algebraic invariants of these edge ideals directly reflect combinatorial properties of their underlying graphs. In this talk, I will discuss how the independence polynomial P_G(x), the generating function for independent sets of a graph G, encodes surprisingly rich information about the Hilbert series of the corresponding edge ideal I(G). In particular, I will explain how P_G(x) determines both the top coefficient and the degree of the h-polynomial. I will then demonstrate the results through familiar examples and a simple suspension construction that lets us track these invariants cleanly.
Abstract: Toric geometry provides a well-known dictionary between combinatorics and algebraic geometry. In 2024, Escobar, Harada and Manon introduced a combinatorial object called a polyptych lattice and accompanying theory which aims to generalize this dictionary. Recent developments in this theory include a method of storing the data of related polyptych lattices as a triple of data called pointed pairing data, as well as notions of maps between polyptych lattices. In this talk, we will explore these recent developments. Specifically, we will discuss a new family of examples of rank 2 pointed pairing data, general results about maps between polyptych lattices, and one of the first known computations of a hom-space between two polyptych lattices.
Abstract: The Lefschetz Properties are a well-studied characteristic of Artinian rings, with its history tracing back to Stanley’s study of the Lefschetz Properties on complete intersections. In this talk, I will discuss the occurrence of the Lefschetz Properties on Artinian rings created from the Stanley-Reisner ring of the van der Waerden simplicial complex, further quotiented out by the squares of all variables. These simplicial complexes, defined on positive integers n>k, are generated by all arithmetic progressions of size k on n variables. We prove that for the smallest and largest possible values of k, namely k=1,2, and n-1, we always have the occurrence of the Weak Lefschetz Property. We also examine the first occurrence of failure at k=3, and prove that this failure propagates for all higher n values. I will also give a complete characterization of when our simplicial complex is a pseudo-manifold, and the implications this has on the occurrence of the Weak Lefschetz Property in certain degrees. Finally, I will present some recent ideas from joint work with Dr. Brett Nasserden, in which we generalize the failure of the k=3 case to all odd values of k.
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We will share with our Dean some narratives of our activities and our research.
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During Winter 2027, talks take place Tuesdays at 10:30-11:20am (Eastern) at HH ??? (to be confirmed).
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