Difference between revisions of "Algebraic Geometry Seminar Spring 2018"

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(Tasos Moulinos)
(Tasos Moulinos)
Line 51: Line 51:
  
 
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math>
 
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math>
taking values in the infinity category of  <math> KU </math>-modules. In this talk I describe a relative version
+
taking values in the <math> \infty </math>-category of  <math> KU </math>-modules. In this talk I describe a relative version
of this construction; namely for X a quasi-compact, quasi-separated C-scheme I construct a
+
of this construction; namely for <math>X</math> a quasi-compact, quasi-separated <math> \mathbb{C} </math>-scheme I construct a
functor valued in the infinity category of sheaves of spectra on X(C), the complex points of X. For inputs
+
functor valued in the <math> \infty </math>-category of sheaves of spectra on <math> X(\mathbb{C}) </math>, the complex points of <math>X</math>. For inputs
of the form Perf(X, A) where A is an Azumaya algebra over X, I characterize the values
+
of the form <math>\operatorname{Perf}(X, A)</math> where <math>A</math> is an Azumaya algebra over <math>X</math>, I characterize the values
of this functor in terms of the twisted topological K-theory of X(C). From this I deduce
+
of this functor in terms of the twisted topological K-theory of <math> X(\mathbb{C}) </math>. From this I deduce
a certain decomposition, for X a finite CW-complex equipped with a bundle P of projective
+
a certain decomposition, for <math> X </math> a finite CW-complex equipped with a bundle <math> P </math> of projective
spaces over X, of KU(P) in terms of the twisted topological K-theory of X ; this is
+
spaces over <math> X </math>, of <math> KU(P) </math> in terms of the twisted topological K-theory of <math> X </math> ; this is
 
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer
 
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer
 
schemes.
 
schemes.

Revision as of 06:56, 17 January 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) TBA Michael
February 23 Aron Heleodoro (Northwestern) TBA Dima
March 9 Phil Tosteson (Michigan) TBA Steven
April 20 Alena Pirutka (NYU) TBA Jordan
April 27 Alexander Yom Din (Caltech) TBA Dima

Abstracts

Tasos Moulinos

Derived Azumaya Algebrais and Twisted K-theory

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