Modular Representation Theory and Algebraic K-theory

PI建议进一步追求他对模块化表示理论的研究，突出代数几何的见解和对p-幂零算子的作用的关注，使用对模块支持多样性的新视角，这导致了更精细的“局部不变量”，新的计算工具和有趣的表示类。由PI和他的合著者开发的技术导致了“全球不变量”的形式的代数向量束和拟议的研究是深入研究这些不变量的意义有限群计划，并将其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。这包括调查空间之间的关系，经常地图和空间的连续映射，从投影光滑品种齐次品种，重温早期的“半拓扑”建设，以提供洞察到一些最具挑战性的问题，经典代数几何。所采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”来研究真实的代数几何问题。所提出的研究目标是为经典的、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和计算，有时抽象和理论。 PI建议增加他最近的“基本”建设的代表性理论和他的“半拓扑”方法代数几何与新的建设，基础的结果，明确的例子，和一般的结果。 在表示论中，PI提议研究类群结构在向量空间上的作用，这些向量空间不属于经典框架的一部分，但对数学的许多方面都非常重要。 从这种方法得到的结果模块表示理论可能会提供启发性的例子，在代数几何中的困难的代数。 这种方法的动机部分来自于明确的性质的行动，使建设的例子，重要的，但非常困难的几何结构。 在代数几何，PI将继续寻找技术，这可能会照亮一些最根本的挑战，出现在几何研究的解决方案多项式方程。

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.