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

一系列三个研讨会和会议将于2013年春天在加拿大多伦多举行，作为菲尔德研究所关于扭量、非结合代数和上同调不变量的主题计划的一部分。李理论中的几何方法研讨会将于2013年3月18日至29日举行，扭量、动机和上同调不变量的春季学校和研讨会将于2013年5月6-17日举行，扭量、非结合代数和上同调不变量的会议将于2013年6月10-14日结束。最近，扭量理论和相关的线性代数群有了两个基本的进展。第一种是V.Voevodsky(Fields Medal，2002)基于二次范数的基元上同调的计算证明了Milnor猜想。其中，这启发了对二次型的深入研究，例如，正交群的挠量、它们的动机和上同调不变量，由Karpenko的ICM 2010演讲调查)。第二个发现归功于Z.Reichstein，它涉及到线性代数群的本质维度和标准维度的概念(Reichstein的ICM 2010演讲)。粗略地说，这些数值不变量描述了扭矩的复杂性(分裂性质)。在代数几何中有几个与扭量密切相关的经典开放猜想(Grothendieck-Serre，Serre II)。这是关于扭矩、动机和上同调不变量的弹簧学派和研讨会的中心主题。非结合(Lie，Jordan等)代数的理论在表示论、组合学和理论物理中有着广泛的应用。许多有趣的无限维李代数，当被看作它们的质心上的代数时，可以被认为是有限维的，而不是被看作给定基域上的代数。从这个角度来看，所讨论的代数看起来像是更简单物体的扭曲形式。这类行为的典型例子是著名的仿射Kac-Moody李代数，它在理论物理中具有特别重要的意义，例如保角场论和精确可解模型理论。最近该领域的许多活动都致力于扩展仿射李代数，粗略地说，是仿射Kac-Moody李代数的高维类似。代数-几何“形式”观点对无限维李代数理论的影响将是李理论几何方法研讨会的中心主题之一。扭量和非结合代数之间的桥梁是关于扭量、非结合代数和上同调不变量的最后一次会议的中心主题，各种上同调不变量，如De Rham和Galois上同调、Motives、Chow群、K-理论、代数余边等都提供了这一桥梁。这提供了非结合代数理论和扭量之间的紧密联系。例如，著名的例外Jordan代数的Rost-Serre不变量给出了Milnor K-理论中的上同调不变量，它与Bloch-Kato猜想的(3，3)-情形有关。非结合代数理论和挠率理论是现代数学中公认的领域。第一部分是关于非结合代数结构(李代数、乔丹代数、交错代数)的研究。第二种是研究和分类所谓的扭曲形式的代数对象，例如群、代数、代数簇。两者在工程、计算机科学和数学物理中都有许多应用。例如，李群和李代数的表示理论在粒子物理中被用来描述基本粒子的不同量子态；变换群理论在描述二维和三维运动中起着重要作用；E8型李群的紧致形式出现在伊辛的磁相互作用模型中。用上同调理论和上同调不变量来描述和分类非结合代数和上同调不变量。后者几十年来一直是代数几何的中心主题，例如Hodge猜想，它的证明是由Clay数学研究所建立的千禧年奖问题之一，它涉及到代数簇的上同调环的结构。该计划的目的是将在这些领域工作的专家和年轻研究人员聚集在一起，讨论最新的发展和结果，概述当前的研究和应用，并鼓励新的进展。会议网址为：http://www.fields.utoronto.ca/programs/scientific/12-13/torsors/index.html

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