Date/Time Date(s) - 29/09/202610:30 am - 11:20 am
Location: HH 410
Talk #1: Title: Pointed Pairing Data in Rank 2 and Morphisms of Polyptych Lattices Speaker: Emma Naguit (McMaster)
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.
Talk #2: Title: The Lefschetz Properties of Artinian Rings Associated to the van der Waerden Simplicial Complex Speaker: Naveena Ragunathan (McMaster)
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.