Modular Representation Theory and Algebraic K-theory

模表示理论和代数K理论

• 批准号: 1067088
• 负责人: Eric Friedlander
• 金额: $ 15.4万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1067088&HistoricalAwards=false
• 关键词: Modular; Representation; Theory; Algebraic; theory

The PI proposes to further pursue his investigation of modular representation theory, highlighting insights from algebraic geometry and a focus on the role of p-nilpotent operators, using a new perspective on support varieties for modules which has led to finer "local invariants", new computational tools, and interesting classes of representations. The techniques developed by the PI and his coauthors have led to "global invariants" in the form of algebraic vector bundles and the proposed research is to delve deeper into the significance of these invariants for finite group schemes, and to extend their applicability to other finite dimensional algebras as well as to rational representations of algebraic groups. The PI also proposes to further investigate algebraic cycles and algebraic vector bundles on algebraic varieties. This includes investigating the relationship between spaces of regular maps and spaces of continuous maps from projective smooth varieties to homogeneous varieties, revisiting earlier "semi-topological" constructions in order to offer insights into some of the most challenging questions of classical algebraic geometry. Techniques to be employed will come from abstract algebraic geometry, algebraic K-theory, and homotopy theory. The PI also proposes to study questions of real algebraic geometry using "stable methods" of morphic cohomology and semi-topological K-theory.Goals of the proposed research are to find new structures and new relationships for mathematical objects which are classical, familiar, and fundamental. His research at times is concrete and calculational, at times abstract and theoretical. The PI proposes to augment his recent ``elementary" constructions in representation theory and his "semi-topological" approach to algebraic geometry with new constructions, foundational results, explicit examples, and general results. In representation theory, the PI proposes to investigate the actions of group-like structures on vector spaces which are not part of the classical framework but which are highly important to many aspects of mathematics. Results obtained from this approach to modular representation theory may provide enlightening examples related to difficult conjectures in algebraic geometry. Motivation for this approach arises in part from the explicit nature of the actions which enable the construction of examples for important, but very difficult geometric structures. In algebraic geometry, the PI will continue to search for techniques which might illuminate some of the most fundamental challenges which arise in the geometric study of solutions to polynomial equations.

PI提议进一步追求对模块化表示理论的调查，强调代数几何形状的见解，并着重于P-Nilpotent Operators的作用，利用对模块的支持品种的新观点，这些观点导致了“本地不变式”，新的计算工具，新的计算工具和有趣的表示。 PI及其合着者开发的技术以代数矢量捆绑的形式导致了“全球不变性”，拟议的研究是为了深入研究这些不变的有限组方案的重要性，并将它们的适用性扩展到其他有限的尺寸代数的代数层，并将其扩展到Algebality Algebrationals Algebrationals的其他有限尺寸。 PI还建议进一步研究代数品种上的代数循环和代数载体束。这包括研究常规地图的空间与从投射平滑品种到均匀品种的连续地图之间的关系，重新审视了早期的“半目标”结构，以便深入了解古典代数几何学的一些最具挑战性的问题。要使用的技术将来自抽象代数几何，代数K理论和同型理论。 PI还提议使用形态的共同体和半主体K理论的“稳定方法”来研究实际代数几何的问题。拟议的研究的目标是找到新的结构和新的关系，以了解具有古典，熟悉和基本性的数学对象的新结构和新的关系。 他的研究有时是具体的，有时是抽象和理论的。 PI建议在代表理论中增强他最近的``基本''结构及其“半音阶”方法对代数几何的方法，并通过新的结构，基础结果，明确的实例，明确的实例，明确的例子和一般结果。在代表理论中，PI提出，PI提出了在群体中不受群体框架的行动而不是在许多方面进行诸如群体框架的一部分，但这些框架的一部分是多个基本框架的一部分。模块化的代表理论可以提供与代数几何形状中的困难猜想有关的启发性例子，部分原因是该行动的明确性质，可以为重要的几何结构而建立示例，以实现重要的几何形状。方程式。