Algebra and 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 9	Juliette Bruce (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
May 4	John Lesieutre (UIC)	TBA	Daniel

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