Algebraic Geometry Seminar Spring 2018: Difference between revisions
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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 | 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 | 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 12:56, 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]\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