Ramification in Algebra and Geometry at Emory

s printed May 21, 2011 Sanghoon Baek (UNIVERSITY OF OTTAWA) Essential dimension of simple algebras and its application to algebraic groups of type An In this talk, we introduce the notion of essential dimension of an algebraic structure and discuss some recent results on the essential dimension of certain classes of central simple algebras. We also relate these results to the essential dimension of split simple groups of type An. Eva Bayer-Fluckiger (ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE) Isometries of quadratic spaces Let k be a field of characteristic not 2, and let V be an inner product space over k. In 1969, Milnor raised the following question: which monic, irreducible polynomials can be minimal polynomials of isometries of V ? The aim of this talk is to give a partial answer to this question, in particular when k is an algebraic number field. Jodi Black (EMORY UNIVERSITY) Zero cycles on principal homogeneous spaces The following is an open question of Jean-Pierre Serre. “Let k be a field, let G be a connected linear algebraic group over k, and let X be a principal homogeneous space under G over k. If X admits a zero cycle of degree one, does X have a k-rational point?” We give a positive answer to the question in certain settings. Baptiste Calmès (UNIVERSITÉ D’ARTOIS) Cohomology theories on projective homogeneous varieties I’ll explain how to compute cohomology groups of projective homogeneous varieties for various types of cohomology theories: cobordism, Witt groups, etc. Vladimir Chernousov (UNIVERSITY OF ALBERTA) On conjugacy of Cartan subalgebras in extended affine Lie algebras One of the central theorems of classical Lie theory is that all split Cartan subalgebras of a finite dimensional simple Lie algebra over an algebraically closed field are conjugate, a theorem of Chevalley. This result yields the most elegant proof that the type of the root system of a simple Lie algebra is its invariant. In infinite dimensional Lie theory maximal abelian diagonalizable subalgebras (MADs) plays the role which Cartan subalgebras plays in the classical theory. In the talk we address to the problem of conjugacy of MADs in a big class of Lie algebras which are called in the literature by extended affine Lie algebras (EALA). To attack this problem we develop a bridge which connects the world of MADs in infinite dimensional Lie algebras and world of torsors over the Laurent polynomial rings. Jean-Louis Colliot-Thélène (UNIVERSITÉ PARIS-SUD XI, CNRS) Unramified cohomology This is a survey talk on unramified cohomology groups of function fields. A classical example of such a group is the Brauer group of a smooth projective variety. Motivation for the study of these groups comes from at least two directions: 1. These groups are birational invariants and are essentially trivial when computed on a retract rational variety. They may thus detect nonrationality. 2. These groups may yield information on the Chow groups of algebraic cycles. Charles De Clercq (UNIVERSITÉ PIERRE ET MARIE CURIE PARIS VI) Motivic indecomposable summands of projective PGL1(A)-homogeneous varieties Motivic decompositions of projective homogeneous varieties and their relations with classical discrete invariants have been intensively studied recently. We will consider the case of adjoint absolutely simple algebraic groups of inner type An, that is to say PGL1(A)-homogeneous varieties for some central simple algebra A. We give a purely algebraic classification of indecomposable direct summands arising in such varieties in the category of Grothendieck Chow motives with coefficients in Fp, and discuss their behaviour when extending