Schedule, Notes/Handouts & Videos

At its current state of the art, ∞-category theory is challenging to explain even to specialists in closely related mathematical areas. Nevertheless, historical experience suggests that in, say, a century's time, we will routinely teach this material to undergraduates. This talk describes one dream about how this might come about --- under the assumption that 22nd century undergraduates have absorbed the background intuitions of homotopy type theory/univalent foundations. To illustrate the utility of this alternate foundational system, we'll share a new computer formalized proof of the ∞-categorical Yoneda lemma that reveals how close it is to the classical proof of the 1-categorical Yoneda lemma.

In the first part of the talk, we give an overview of various ways in which deformation theory shapes conceptions of noncommutative spaces. In the second part of the talk, we focus on prestacks which occur as deformations of schemes. We introduce the framework of box operads in order to establish the relevant higher structure on the Gerstenhaber-Schack complex of a prestack, allowing for a self-contained treatment of the higher order deformation theory of prestacks. This is joint work with Hoang Dinh Van and Lander Hermans.

b-Structures and other generalizations (such as E-symplectic structures) are ubiquitous and sometimes hidden, unexpectedly, in a number of problems including the space of pseudo-Riemannian geodesics and regularization transformations of the three-body problem. E-symplectic manifolds include symplectic manifolds with boundary, manifolds with corners, compactified cotangent bundles and regular symplectic foliations. Their deformation quantization was studied à la Fedosov by Nest and Tsygan. How general can such structures be? In this talk, I first explain how to associate an E-symplectic structure to a Poisson structure with transverse structure of semisimple type (joint work with Ryszard Nest) and I will connect this to a result by Cahen, Gutt and Rawnsley on tangential star products. This result illustrates how E-symplectic manifolds serve as a trampoline to the investigation of the geometry of Poisson manifolds and the different facets of their quantization. This should let us address a number of open questions in Poisson Geometry and the study of its quantization from a brand-new perspective.

Let X be a complex manifold. By homological mirror symmetry one expects an action of the fundamental group of the "moduli space of Kähler structures" of X on the derived category of X. If X is a crepant resolution of a Gorenstein affine toric variety we obtain an action on the derived category of the toric boundary divisor of X which leads to an action on the Grothendieck group of X. This is a joint work with Michel Van den Bergh.

Moderator: Sarah WItherspoon, Texas A & M University