Algebraic Geometry Seminar Spring 2018

The seminar meets on Fridays at 2:25 pm in room B113.

Here is the schedule for the previous semester.

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Spring 2018 Schedule

date	speaker	title	host(s)
January 26	Tasos Moulinos (UIC)	Derived Azumaya Algebras and Twisted K-theory	Michael
February 2	Daniel Erman (Wisconsin)	TBA	Local
February 23	Aron Heleodoro (Northwestern)	TBA	Dima
April 6	Phil Tosteson (Michigan)	TBA	Steven
April 13	Reserved		Daniel
April 20	Alena Pirutka (NYU)	TBA	Jordan
April 27	Alexander Yom Din (Caltech)	TBA	Dima

Abstracts

Tasos Moulinos

Derived Azumaya Algebras and Twisted K-theory

Topological K-theory of dg-categories is a localizing invariant of dg-categories over [math]\displaystyle{ \mathbb{C} }[/math] taking values in the [math]\displaystyle{ \infty }[/math]-category of [math]\displaystyle{ KU }[/math]-modules. In this talk I describe a relative version of this construction; namely for [math]\displaystyle{ X }[/math] a quasi-compact, quasi-separated [math]\displaystyle{ \mathbb{C} }[/math]-scheme I construct a functor valued in the [math]\displaystyle{ \infty }[/math]-category of sheaves of spectra on [math]\displaystyle{ X(\mathbb{C}) }[/math], the complex points of [math]\displaystyle{ X }[/math]. For inputs of the form [math]\displaystyle{ \operatorname{Perf}(X, A) }[/math] where [math]\displaystyle{ A }[/math] is an Azumaya algebra over [math]\displaystyle{ X }[/math], I characterize the values of this functor in terms of the twisted topological K-theory of [math]\displaystyle{ X(\mathbb{C}) }[/math]. From this I deduce a certain decomposition, for [math]\displaystyle{ X }[/math] a finite CW-complex equipped with a bundle [math]\displaystyle{ P }[/math] of projective spaces over [math]\displaystyle{ X }[/math], of [math]\displaystyle{ KU(P) }[/math] in terms of the twisted topological K-theory of [math]\displaystyle{ X }[/math] ; this is a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer schemes.

Aron Heleodoro

TBA

Alexander Yom Din

TBA