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; 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

作为田间研究所主题计划的一部分，将于2013年春季在加拿大多伦多举行三场研讨会和会议。 关于谎言理论的几何方法的讲习班将于2013年3月18日至29日举行，春季学校和关于扭转，动机和共同体学的讲习班将于2013年5月6日至17日举行，关于Torsors，Torsors，非相关代数和共同体的无效计划将于6月10日至14年10月14日，TORMER INSTORS THE TRINERS THE TRENORS和TRINERS THE TOBLER THE TRENORS THE BRENBRENT和TRINESORS结束。最近看到了两个基本进步。首先是基于Norm Quadric的动机共同体学的计算，V。Voevodsky（Fields Medal，2002）证明了Milnor的猜想。除其他外，这启发了对二次形式的深入研究，例如Karpenko的ICM 2010讲座调查了正交群体，其动机和共同体不变的扭转。第二个发现是由于Z. Reichstein造成的，并涉及线性代数群体基本和规范维度的概念（Reichstein的ICM 2010讲座）。粗略地说，这些数字不变性表征了托索尔的复杂性（分裂特性）。代数几何形状中有几种经典的开放猜想，它们与Torsors密切相关（Grothendieck-Serre，Serre II）。这是春季学校的中心主题，也是关于Torsors，Motive和Somomological不变性的研讨会。非缔合性（Lie，Jordan等）代数的理论在表示理论，组合学和理论物理学中有许多应用。当将其视为质心上的代数时，可以认为许多有趣的无限尺寸谎言代数被认为是有限的尺寸，而是在给定基地上的代数。从这个角度来看，所讨论的代数看起来像更简单对象的扭曲形式。这种类型的行为的典型例子由著名的仿射kac-moody Lie代数给出，这些代数在理论物理学中特别重要，例如保形场理论以及确切的可解决模型的理论。该地区最近的大部分活动都致力于扩展仿射代数，大致说明了仿射Kac-Moody Lie代数的高维类似物。代数几何“形式”观点对无限维谎言理论代数的影响将是关于谎言理论的几何方法研讨会的中心主题之一。 Torsors和非社交代数之间的桥梁是最终关于Torsors，非缔合代数和共同体不变式的核心主题，由各种共同体不变的人提供，例如。 de Rham和Galois共同体，动机，Chow群，K理论，代数COBORDISM。这在非缔合代数和托索尔的理论之间提供了牢固的联系。例如，著名的约旦代数的著名Rost-Serre不变性在Milnor K理论中提供了共同体的不变性，并且与Bloch-Kato猜想的（3,3） - 案例有关。第一个涉及非缔合代数结构的研究（Lie，Jordan，替代代数）。第二项研究并分类了所谓的代数对象的扭曲形式，例如组，代数，代数品种。两者在工程，计算机科学和数学物理学中都有许多应用。例如，在粒子物理学中使用谎言基和谎言代数的表示理论来描述基本颗粒的不同量子状态。转型群体的理论在描述2D和3D动作中起着重要作用。 E_8型谎言组的紧凑形式出现在磁相互作用的ISING模型中。描述和分类非缔合代数和Torsors，人们使用了共同体理论和共同体不变的语言。几十年来，后者一直是代数几何形状的中心主题，例如霍奇猜想的证明是克莱数学研究所建立的千年奖问题之一，涉及代数品种的同种学环的结构。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汇集在一起​​，讨论最新的发展和结果，以概述当前的研究和应用程序，并刺激新的进步。会议的URL是：http：//www.fields.utoronto.ca/programs/scientific/12-13/torsors/index.html