Special Meeting: Torsors, Nonassociative algebras and Cohomological invariants Thematic Program at the Fields Institute Toronto January - June 2013

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

A series of three workshops and conferences will be held in Toronto, Canada in the spring of 2013, as part of the Fields Institute Thematic Program on Torsors, Nonassociative Algebras, and Cohomological Invariants. The Workshop on Geometric Methods in Lie Theory will take place March 18-29, 2013, the Spring School and Workshop on Torsors, Motives, and Cohomological Invariants will be held May 6-17, 2013, and the Conference on Torsors, Nonassociative Algebras, and Cohomological Invariants will conclude the program on June 10-14, 2013. The theory of torsors and the associated linear algebraic groups has recently seen two fundamental advances. The first is the proof of Milnor's conjecture by V. Voevodsky (Fields Medal, 2002), based on the computation of the motivic cohomology of the norm quadric. Among other things, this inspired an intensive study of quadratic forms, e.g. torsors for orthogonal groups, their motives and cohomological invariants, surveyed by Karpenko's ICM 2010 lecture). The second discovery is due to Z. Reichstein and deals with the notions of essential and canonical dimensions of linear algebraic groups (Reichstein's ICM 2010 lecture). Roughly speaking, these numerical invariants characterize the complexity (splitting properties) of a torsor. There are several classical open conjectures in algebraic geometry which are closely related to torsors (Grothendieck-Serre, Serre II). This is the central theme of the Spring School and Workshop on Torsors, Motives and Cohomological Invariants. The theory of nonassociative (Lie, Jordan, etc) algebras have many applications in representation theory, combinatorics and theoretical physics. Many interesting infinite dimensional Lie algebras can be thought as being finite dimensional when viewed as algebras over their centroids, instead as algebras over the given base field. From this point of view, the algebras in question look like twisted forms of simpler objects. The quintessential example of this type of behavior is given by the celebrated affine Kac-Moody Lie algebras which have particular importance in theoretical physics, for example conformal field theory, and the theory of exactly solvable models. Much of the recent activity in the area has been devoted to extended affine Lie algebras, roughly speaking higher-dimensional analogues of the affine Kac-Moody Lie algebras. The impact of the algebra-geometric "forms" point of view on the theory of infinite-dimensional Lie algebras will be one of the central theme of the Workshop on Geometric Methods in Lie Theory. The bridge between torsors and nonassociative algebras, which is the central theme of the final conference on Torsors, Nonassociative Algebras and Cohomological Invariants, is provided by various cohomological invariants, e.g. de Rham and Galois cohomology, motives, Chow groups, K-theory, algebraic cobordism. This provides a strong connection between the theory of nonassociative algebras and torsors. For instance,the celebrated Rost-Serre invariant of exceptional Jordan algebras gives a cohomological invariant in Milnor K-theory and is related to the (3,3)-case of the Bloch-Kato conjecture. The theory of nonassociative algebras and the theory of torsors are well-established areas of modern mathematics. The first deals with the study of nonassociative algebraic structures (Lie, Jordan, alternative algebras). The second studies and classifies so-called twisted forms of algebraic objects, e.g. groups, algebras, algebraic varieties. Both have many applications in engineering, computer science and mathematical physics. For instance, the representation theory of Lie groups and Lie algebras is used in particle physics to describe the different quantum states of elementary particles; the theory of transformation groups plays an important role in describing the 2D and 3D-motions; the compact form of the Lie group of type E_8 appears in the Ising model for magnetic interactions. To describe and classify nonassociative algebras and torsors one uses the language of cohomology theories and cohomological invariants. The latter has been a central theme of algebraic geometry for decades, e.g. the Hodge Conjecture, whose proof is one of the Millennium Prize problems established by the Clay Mathematical Institute, concerns the structure of the cohomology ring of an algebraic variety. The purpose of the program is to bring together specialists and young researchers working in these areas to discuss recent developments and results, to provide an overview of the current research and applications, and to stimulate new advances. The URL of the conference is: http://www.fields.utoronto.ca/programs/scientific/12-13/torsors/index.html