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
|January 26||Tasos Moulinos (UIC)||Derived Azumaya Algebras and Twisted K-theory||Michael|
|February 2||Daniel Erman (Wisconsin)||TBA||Local|
|February 8 (unusual date! 2:30-3:30 in VV B113)||Roman Fedorov (University of Pittsburgh)||A conjecture of Grothendieck and Serre on principal bundles in mixed
|February 9||Juliette Bruce (Wisconsin)||TBA||Local|
|February 16||Andrei Caldararu (Wisconsin)||Computing a categorical Gromov-Witten invariant||Local|
|February 23||Aron Heleodoro (Northwestern)||TBA||Dima|
|March 2||Moisés Herradón Cueto (Wisconsin)||TBA||Local|
|April 6||Phil Tosteson (Michigan)||TBA||Steven|
|April 20||Alena Pirutka (NYU)||TBA||Jordan|
|April 27||Alexander Yom Din (Caltech)||TBA||Dima|
|May 4||John Lesieutre (UIC)||TBA||Daniel|
Derived Azumaya Algebras and Twisted K-theory
Topological K-theory of dg-categories is a localizing invariant of dg-categories over taking values in the -category of -modules. In this talk I describe a relative version of this construction; namely for a quasi-compact, quasi-separated -scheme I construct a functor valued in the -category of sheaves of spectra on , the complex points of . For inputs of the form where is an Azumaya algebra over , I characterize the values of this functor in terms of the twisted topological K-theory of . From this I deduce a certain decomposition, for a finite CW-complex equipped with a bundle of projective spaces over , of in terms of the twisted topological K-theory of ; this is a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer schemes.
A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic
Let G be a reductive group scheme over a regular local ring R. An old conjecture of Grothendieck and Serre predicts that such a principal bundle is trivial, if it is trivial over the fraction field of R. The conjecture has recently been proved in the "geometric" case, that is, when R contains a field. In the remaining case, the difficulty comes from the fact, that the situation is more rigid, so that a certain general position argument does not go through. I will discuss this difficulty and a way to circumvent it to obtain some partial results.
Computing a categorical Gromov-Witten invariant
In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.
In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.
Alexander Yom Din