Support theories: axiomatics, realizations and calculations

表征理论是对线性空间对称性的研究。回到它的创始人Frobenius， Schur， Burnside和Brauer，他们在一个多世纪前首次发展了这一学科，标准的方法是将具有对称性的线性空间分解为“简单空间”的总和，然后将简单空间分类为连贯且（相对）可理解的列表和家族。“有限群图集：单群的极大子群和普通字符”是这一策略的一个经典和基本的例子。在这个项目中，PI转向了对对称性的研究，这些对称性不像经典案例那样受到如此美妙的分解，它们生活在“野生”表示理论的世界中。这样的理论在数学中无处不在。它们出现在代数、拓扑学、数学物理、组合学，当然，还有表示理论本身。PI将采用新的和快速发展的张量三角形几何学科，它结合了拓扑、同调和分类技术，在这个野生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一般理解。该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形几何领域的几个新方向上推进知识。它建立在PI在模块化设置支持理论方面的专业知识的基础上，并扩展到不同的领域，其中要研究的表示理论在一个或多个重要方面与有限群的设置不同。最有趣的是当理论是一元而非对称的时候，比如小量子群和它们的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的张量三角形图、对角型Nichols代数、Schur代数、Frobenius核和有限超群方案。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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.