Modular Representation Theory and Algebraic K-theory

PI建议进一步深入他对模表示理论的研究，强调从代数几何的见解和对p-幂零算子的作用的关注，使用关于模的支持种类的新视角，这导致了更好的“局部不变量”，新的计算工具，和有趣的表示类。PI和他的合著者发展的技术导致了代数向量丛形式的“全局不变量”，所提出的研究是更深入地挖掘这些不变量对有限群方案的意义，并将它们的适用性扩展到其他有限维代数以及代数群的有理表示。PI还建议进一步研究代数簇上的代数圈和代数向量丛。这包括研究从射影光滑簇到齐次簇的正则映射空间和连续映射空间之间的关系，重温早期的“半拓扑”结构，以便对经典代数几何中一些最具挑战性的问题提供见解。所使用的技术将来自抽象代数几何、代数K理论和同伦理论。PI还建议使用形态上同调和半拓扑K-理论的稳定方法来研究实代数几何问题，其目的是为经典的、熟悉的和基本的数学对象找到新的结构和新的关系。他的研究有时是具体的和计算的，有时是抽象的和理论的。他建议用新的结构、基本的结果、显式的例子和一般的结果来加强他最近在表示理论中的“初等”结构和他对代数几何的“半拓扑”方法。在表示理论中，PI建议研究群结构在向量空间上的作用，这些向量空间不是经典框架的一部分，但对数学的许多方面都非常重要。从模表示理论的这种方法得到的结果可以为代数几何中的困难猜想提供有启发性的例子。这种方法的动机部分来自于动作的显性性质，这使得能够为重要但非常困难的几何结构构造示例。在代数几何中，PI将继续寻找可能阐明在多项式方程的几何研究中出现的一些最基本的挑战的技术。

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.