### Liu Qing: 1. "The Brauer group of surfaces" 2. "On the Grothendieck ring of varieties"

Prof. Liu Qing from Institut de Mathématiques de Bordeaux, Université Bordeaux 1

will visit us from Apr.20th to 29th. The following are two talks he will give.

Talk 1:

Time: April 23 (Wed.) 2:00-3:00 pm

Location: MCM 410

Title: "The Brauer group of surfaces"

Abstract:

This is a joint work with Dino Lorenzini and Michel Raynaud.

Let X be a projective smooth surface defined over a finite

field. Let Br(X) be its Brauer group. It is conjectured that

this group is finite. For a long time, people believed it is

order is not always a perfect square because of a computation

mistake by Y. Manin. We prove in this work that actually

the order of Br(X), if finite, is a perfect square. The proof

is based on Kato-Trihan's theorem (on Birch-Swinnerton-Dyer's

conjecture for abelian varieties over a finite field),

Poonen-Stoll's theorem on the order of the Tate-Shafarevich group

of Jacobians, Milnes's solution to Artin-Tate conjecture and

our previous working relating Brauer group and Tate-Shafarevich

group.

Talk 2:

Time:Apr. 25 (Friday) 2:00-3:00 pm

Location: MCM 410

Title: "On the Grothendieck ring of varieties"

Abstract:

This is a joint work with Julien Sebag. Let k be a field.

Let K_0(V_k) be the Grothendieck ring of algebraic varieties

over k (it is the ring of isomorphic classes of algebraic

varieties over k, quotient by the relations [X]=[Y]+[X\Y]

whenever X is an algebraic variety over k and Y is a

closed subvariety of X). It is a kind of universal

motives ring. Our aim is to give some properties of huge ring.

Most results are true only for characteristic zero field k

using desingularization and factorization of birational

morphisms, but in positive characteristics we can also

derive some basic properties.

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