# Difference between revisions of "Algebraic Geometry Seminar Spring 2018"

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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 07:56, 17 January 2018

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

Here is the schedule for the previous semester.

## Contents

## Algebraic Geometry Mailing List

- Please join the AGS Mailing List to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## 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**