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

特别会议：Torsors、非结合代数和上同调不变量 多伦多菲尔兹研究所主题项目 2013 年 1 月至 6 月

• 批准号: 1222637
• 负责人: Alexander Merkurjev
• 金额: $ 8.03万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-11-01 至 2013-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1222637&HistoricalAwards=false
• 关键词: Special; Meeting; Torsors; Nonassociative; algebras

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

作为菲尔兹研究所关于环量、非结合代数和上同不变量主题计划的一部分，一系列的三个研讨会和会议将于2013年春季在加拿大多伦多举行。李论几何方法研讨会将于2013年3月18日至29日举行，环量、动机和上同调不变量春季研讨会将于2013年5月6日至17日举行，环量、非结合代数和上同调不变量会议将于2013年6月10日至14日结束。torsor和相关的线性代数群的理论最近有两个基本的进展。第一个是V. Voevodsky （Fields Medal, 2002）基于范数二次元的动机上同调的计算对Milnor猜想的证明。除此之外，这激发了二次型的深入研究，例如正交群的tortor，它们的动机和上同调不变量（Karpenko的ICM 2010讲座调查了这一点）。第二个发现是由于Z. Reichstein，并处理线性代数群的基本维和规范维的概念（Reichstein的ICM 2010讲座）。粗略地说，这些数值不变量表征了一个变形量的复杂性（分裂性质）。代数几何中有几个经典的开放猜想与旋量密切相关（Grothendieck-Serre, Serre II）。这是春季学校和工作坊的中心主题torsor，动机和上同不变量。非结合代数（Lie、Jordan等）理论在表示理论、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可以被认为是有限维的，当它们被看作是质心上的代数，而不是给定基域上的代数。从这个角度来看，所讨论的代数看起来像是简单物体的扭曲形式。这种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，例如共形场论和精确可解模型理论。该领域最近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李代数的高维类似物。代数-几何“形式”观点对无穷维李代数理论的影响将是李论几何方法研讨会的中心主题之一。各种上同调不变量，如de Rham和Galois上同调，动机，Chow群，k理论，代数共乘，提供了环量和非结合代数之间的桥梁，这是环量，非结合代数和上同调不变量最终会议的中心主题。这在非结合代数理论和环体理论之间提供了一个强有力的联系。例如，著名的例外Jordan代数的Rost-Serre不变量给出了Milnor k理论中的上同不变量，并且与Bloch-Kato猜想的(3,3)-情况有关。非结合代数理论和环量理论是现代数学中公认的领域。第一部分涉及非结合代数结构（Lie, Jordan，替代代数）的研究。第二部分研究和分类所谓的代数对象的扭曲形式，例如群、代数、代数变。两者在工程、计算机科学和数学物理中都有很多应用。例如，李群和李代数的表示理论在粒子物理学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；E_8型李群的紧化形式出现在磁相互作用的Ising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环量。后者几十年来一直是代数几何的中心主题，例如霍奇猜想，其证明是克莱数学研究所建立的千禧年奖问题之一，涉及代数变体的上同环的结构。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，讨论最近的发展和结果，提供当前研究和应用的概述，并促进新的进展。会议网址为：http://www.fields.utoronto.ca/programs/scientific/12-13/torsors/index.html