Spring 2024
January 31: François Greer (Michigan State University)
Location: STON 215
Title: Hodge structures and period domains
Abstract: A smooth projective variety (over C) is a special topological space, which enjoys several additional structures on its cohomology groups. In many cases, these structures are fine enough to recover the variety. Since cohomology is a linear algebraic object, this allows us to solve difficult nonlinear problems using linear algebra.
February 7: Morgan Opie (UCLA)
Note: This will be a special joint Algebraic Geometry/Topology Seminar.
Location: STON 215
Title: Enumerating stably trivial topological vector bundles with higher real K-theories
Abstract: The goal of the pre-talk will be to set up some basic notions and ideas for the talk "Enumerating stably trivial topological vector bundles with higher real K-theories", to make the main talk more accessible. I will discuss vector bundles (and how I view them as a homotopy theorist); will talk about some historical methods for enumerating vector bundles; and will also talk a bit about "higher real K-theories" and why these generalized cohomology theories come into play. I aim for this can be an interactive talk, so if you have questions about specific things in the abstract please feel free to ask!
February 14: Xiaojiang Cheng (Washington University in St. Louis)
Location: STON 215
Title: How to find Hodge classes algebraically?
Abstract: Let S be an arithmetic quotient of a Hermitian symmetric domain and X/S be a family of varieties over S. One interesting problem is to find the Hodge classes of X, and if possible, prove the Hodge conjecture for X. Automorphic forms are certain special functions generalizing the classical elliptic modular forms. In this talk, we will introduce this problem and explain the automorphic form approach to this problem.
March 6: Salim Tayou (Harvard University)
Location: STON 215
Title: Local systems on algebraic varieties.
Abstract: What does the fundamental group of a smooth projective algebraic variety look like? A useful way to approach this question is via representation theory of the fundamental group, which leads naturally to studying local systems/flat bundles on algebraic varieties. In his seminal work, Simpson introduced a new approach to studying local systems via Higgs bundles and stated several conjectures about local systems which should be "motivic". This pre-talk will be a gentle introduction to this circle of ideas.
March 20: Laure Flapan (Michigan State University)
Location: STON 215
Title: Why study moduli spaces
Abstract: This talk will give a gentle introduction to the idea of moduli spaces and their utility. I will describe different ways moduli spaces are useful and give examples from some of my own work.
March 27: Tong Zhou (UC Berkeley)
Location: STON 215
Title: Constructible Sheaves on the Affine Line
Abstract: The goal of this talk is to present a clear picture of constructible étale sheaves on the affine line in positive characteristic, as an invitation to the audience to the world of ℓ-adic sheaves. Along the way, we introduce tools and theorems for their study, and discuss generalisations to higher dimensions, and open problems.