ABC Algebra Workshop

The ABC Algebra Workshop is a series of one or two day workshops, featuring invited algebraists, from Alberta and British Columbia, and others from outside of Canada.

Location:

University of Alberta, Central Academic Building, Room 657.  

Click here for the Workshop Schedule

Speaker Abstracts: 

Asher Auel (Yale University)

Stable rationality of conic bundle fourfolds

The stable rationality problem has seen major recent advances since the introduction of a degeneration method for 0-cycles by Voisin and Colliot-Thélène/Pirutka.  This method requires the construction of singular varieties having both nontrivial unramified cohomology and well-behaved resolutions of singularities.  I will explain a new geometric construction of conic bundles over projective 3-space satisfying these properties.  In general, I will outline a formula for the unramified Brauer group of such conic bundles and describe their resolutions.   In the absence of a sufficiently developed 

theory of maximal orders over 3-dimensional schemes, the difficult part turns out to be finding examples!  This is joint work with Christian Boehning, Hans-Christian Graf von Bothmer, and Alena Pirutka.

David Favero (University of Alberta)

Fractional Calabi-Yau Categories and Q-Gorenstein Cones

A relatively nice triangulated category has a so called "Serre functor" attached to it which behaves like the canonical bundle of an algebraic variety.  If this Serre functor is trivial up to shift, we say that the category is Calabi-Yau.  I will discuss how Gorenstein cones appear in the context of mirror symmetry, and how they naturally lead to comparisons between Calabi-Yau categories and the derived category of certain (non-Calabi-Yau) toric complete intersections. This is based on joint work with Tyler Kelly.

Max Lieblich (University of Washington)

Finiteness and Tate

I will discuss the following innocent question: "Are there only finitely many K3 surfaces over a given finite field?" It turns out that this is essentially equivalent to the Tate conjecture. I will discuss some of this equivalence (proven by myself, Davesh Maulik, and Andrew Snowden) and the general idea of a proof of the Tate conjecture coming from these ideas (due to Francois Charles).

Alexander Merkurjev (UCLA):

Degree 3 unramified cohomology of semisimple groups.

J-P. Serre computed the unramified Brauer group (degree 2 cohomology) of a semisimple group G and found groups G with nontrivial unramified Brauer group, thus providing examples of nonrational semisimple groups. We compute degree 3 unramified cohomology of adjoint semisimple groups and give new examples of nonrational groups.

Andrei Rapinchuk (University of Virginia).

On algebraic groups with the same tori. 

Let G be an absolutely almost simple algebraic group over a finitely generated field K. We will discuss the problem of characterizing the K-forms of G that have the same maximal K-tori as G itself - the set of K-isomorphism classes of such forms is called the genus of G. One expects the genus of G to be finite assuming that characteristic of  K is ``good" for G. We will present results that, on the one hand, confirm this conjecture in some situations, and, on the other hand, link the general case with other finiteness properties. This is joint work with V.  Chernousov and I. Rapinchuk.

Ben Williams (UBC)

Topological Azumaya Algebras with involutions of the second kind

This talk represents joint work with U. First. Let A be an Azumaya algebra over a commutative ring R. An involution of the second kind on A is an order-two antiautomorphism of A which restricts to a nontrivial involution of R. In earlier work with A. Auel and U. First, we compared the case of Azumaya algebras of the first kind over R with Azumaya algebras over topological spaces, but we said nothing about the second kind. In this talk, I will discuss a general approach to Azumaya algebras of the second kind, sufficient to handle both the classical case and the case of topological Azumaya algebras. I will then explain a topological obstruction to the existence of involutions of the second kind on Azumaya algebras, and use this obstruction to manufacture an example of a low-degree Azumaya algebra which is Brauer equivalent to an Azumaya algebra with an involution of the second kind, but only at the expense of increasing the degree.

Click here for a list of confirmed participants