## EVENT DETAILS AND ABSTRACT

**Geometry/Physics Seminar**
**Title:** Equivariant Hodge theory and noncommutative algebraic geometry (pre-talk)

**Speaker:** Daniel Halpern-Leistner

**Speaker Info:** Columbia University

**Brief Description:**

**Special Note**:

**Abstract:**

Recent results have revealed a mysterious foundational phenomenon: some quotient stacks X/G behave as if they are proper schemes, even when G is a positive dimensional algebraic group and X itself is not proper. I will discuss this phenomenon from the perspective of Hodge theory. One can consider the noncommutative Hodge-de Rham sequence for the dg-category D^{b}Coh(X/G), whose first page is Hochschild homology and which converges to periodic cyclic homology. This spectral sequence agrees with the usual Hodge-de Rham sequence for smooth and proper schemes and thus degenerates on the first page. It turns out that this degeneration also occurs for many "cohomologically proper" quotient stacks. With a little work, this leads to a canonical pure Hodge structure on the Atiyah-Segal equivariant topological K-theory of the complex analytification of X. The associated graded of the Hodge filtration is the space of functions on the "derived loop space" of the stack. A similar degeneration property holds for many equivariant Landau-Ginzburg models whose critical locus is cohomologically proper.

**Date:** Tuesday, October 18, 2016

**Time:** 1:00pm

**Where:** Lunt 107

**Contact Person:** Ezra Getzler

**Contact email:**

**Contact Phone:**

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