地点：海纳苑2幢204室

时间：6月1日，16:15 - 17:35

报告人：徐平 Ping Xu（宾夕法尼亚州立大学）

摘要:  

Pretalk: Introduction to DG Manifolds 

Abstract:I will give a brief introduction to dg manifolds, a useful framework for studying geometric objects that may have singularities. Roughly speaking, dg manifolds extend the notion of smooth manifolds by incorporating homological (chain complex) data into their structure.

DG manifolds of amplitude [−n,−1] are equivalent to Lie n-algebroids, which can be viewed as infinitesimal models of higher groupoids. On the other hand, DG manifolds of amplitude [1,n] are closely related to derived manifolds, which arise in derived geometry and allow one to systematically handle derived or hidden intersections.

More generally, DG manifolds with amplitude [−m,n] provide a unified language that can encode both stacky and derived types of singularities, making them a flexible tool in modern differential geometry.

Research talk: Duflo--Kontsevich Type Theorem for DG Manifolds

Abstract: It is a classical result that for any dg algebra A, the pair of its Hochschild (co)homologies (H•(A, A), H•(A, A)) carries rich algebraic structures resembling the usual Cartan calculus, often referred to as the Tamarkin--Tsygan calculus.

DG manifolds provide a useful geometric framework for describing spaces with singularities. In this talk, I will discuss the Tamarkin--Tsygan calculus associated with the dg algebra of a dg manifold and present a Duflo--Kontsevich type theorem in this setting.

As applications to several important examples, we recover the Duflo theorem on the center of the universal enveloping algebra of a Lie algebra and Kontsevich's theorem on the Hochschild cohomology of complex manifolds, placing them within a unified framework.

This is joint work with Hsuan-Yi Liao and Mathieu Stiénon.