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Support theories: axiomatics, realizations and calculations

支持理论：公理、实现和计算

基本信息

批准号：
2200832
负责人：
Julia Pevtsova
金额：
$ 23万
依托单位：
University of Washington
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200832&HistoricalAwards=false
关键词：
Support theories axiomatics realizations calculations

项目摘要

Representation theory is a study of symmetries of linear spaces. Going back to its founders, Frobenius, Schur, Burnside, and Brauer, who first developed the subject over a century ago, the standard approach is to decompose a linear space with its symmetry into a sum of "simple ones," and then classify the simple ones into coherent and (relatively) comprehensible lists and families. The "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups" is a classical and fundamental example of this strategy in action. In this project, the PI turns to the study of symmetries which do not subject themselves to such nice decompositions as in the classical case, which live in the world of "wild" representation theories. Such theories are ubiquitous in mathematics. They arise in algebra, topology, math physics, combinatorics, and, of course, representation theory itself. The PI will employ the new and rapidly developing subject of tensor triangular geometry, which combines topological, homological, and categorical techniques, to induce some structure in this wild representation territory, thus advancing our general understanding of this complicated world of symmetries. This project will provide research and training opportnities for undergraduate and graduate students.In more detail, this projects will advance knowledge in several new directions within the realm of tensor triangular geometry. It builds on PI's expertise in support theories in modular settings and branches out to different areas where the representation theories to be studied differ from the setting of finite groups in one or more significant aspects. The most interesting is when the theory is monoidal but not symmetric, such as for small quantum groups and their Borel subalgebras. Besides quantum groups, the PI will study and develop further the tensor triangular picture for complex Lie superalgebras, Nichols algebras of diagonal type, Schur algebras, Frobenius kernels, and finite supergroup schemes. The PI will also study local properties of some representation categories associated to Gorenstein algebras and the cohomology of finite dimensional Hopf and Nichols algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

表示论是一门研究线性空间对称性的学科。追溯到它的创始人Frobenius、Schur、Burnside和Brauer，他们在一个多世纪前首次提出了这一主题，标准方法是将具有对称性的线性空间分解成几个“简单的”，然后将简单的分解成连贯的和(相对)可理解的列表和族。《有限群图集：单群的极大子群和普通特征标》就是这一策略在实践中的一个经典和基本的例子。在这个项目中，PI转向对对称性的研究，这些对称性不会像经典情况下那样受到很好的分解，因为经典情况下生活在“狂野的”表示理论的世界中。这样的理论在数学中无处不在。它们产生于代数、拓扑学、数学物理、组合学，当然还有表示论本身。PI将使用迅速发展的张量三角几何这一新的学科，它结合了拓扑学、同调和分类技术，在这个狂野的表示领域中归纳出一些结构，从而促进我们对这个复杂的对称世界的总体理解。这个项目将为本科生和研究生提供研究和培训机会。更详细地说，这个项目将在张量三角几何领域内的几个新方向上推进知识。它以PI在模集的支持理论方面的专业知识为基础，并扩展到不同的领域，在这些领域中，要研究的表示理论在一个或多个重要方面不同于有限群的设置。最有趣的是当理论是么一但不对称的时候，例如对于小量子群和它们的Borel子代数。除了量子群，PI还将进一步研究和发展复李超代数、对角型Nichols代数、Schur代数、Frobenius核和有限超群方案的张量三角图。PI还将研究与Gorenstein代数相关的一些表示范畴的局部性质以及有限维Hopf和Nichols代数的上同调。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。