Cohomology and Support Varieties

表示论研究线性空间的对称性。对于像二维平面这样容易处理的例子，人们可以通过它们产生的周期性镶嵌来研究对称性。美丽的例子，甚至完整的列表，这种镶嵌往往是通过艺术手段实现，如图所示，在著名的对称图纸或墙纸图案在阿尔汉布拉在西班牙。事实上，已知平面上正好有十七种本质上不同的镶嵌。枚举更复杂和更高维的物体的对称性通常是一项令人生畏的任务，如果不是完全不可能的话。然而，掌握如此复杂的对称性被证明不仅在表示论中而且在许多其他领域都很重要，无论是纯粹的还是应用的，包括几何，拓扑，代数，以及物理和化学。在这个项目中，PI将研究以有趣的几何形状形式出现的对称性的某些不变量。她的目标是研究这些几何对象包含多少关于原始对称性的数据，使它们成为研究对称性本身的有用工具。 成立于一百多年前，表示论现在是一个蓬勃发展的领域，与其他数学领域有着许多深刻的联系。模表示理论从许多其他领域，包括代数几何和拓扑学中汲取了方法和动机。在1971年，奎伦奠定了基础，代数几何应用群上同调，开辟了新的篇章，研究模块化表示理论，正在积极探索这一天。这个研究项目有其根源在奎伦的工作，寻求发展理论的支持品种在几个不同的，但相互关联的情况下。这涉及到解决几个基本问题的结构表示和上同调的有限维代数。这些应用将提供与有限超群方案、Schur代数、Nichols代数、还原群的Frobenius核和李超代数的表示相关的各种三角分类的全球结构的新信息。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Representation theory studies symmetries of linear spaces. For such tractable example as a two-dimensional plane one can study symmetries via periodic tessellations they generate. Beautiful examples and even complete lists of such tessellations are frequently implemented via artistic means, as illustrated in Escher's famous symmetry drawings or wallpaper patterns in the Alhambra in Spain. In fact, it is known that there are exactly seventeen essentially different tessellations of the plane. Enumerating symmetries of more complicated and higher dimensional objects is often a daunting if not outright impossible task. Yet, having a grasp on such complicated symmetries proves to be important not only in representation theory but in many other areas, both pure and applied, including geometry, topology, algebra, as well as physics and chemistry. In this project, the PI will investigate certain invariants of symmetries that come in the form of interesting geometric shapes. Her goal is to investigate how much data about the original symmetries these geometric objects contain, shaping them to be a useful tool in the study of symmetries themselves. Founded more than a hundred years ago, representation theory is now a thriving field that exhibits many deep connections with other areas of mathematics. Modular representation theory draws methods and motivation from a host of other areas, including algebraic geometry and topology. In 1971, Quillen laid the foundations for algebraic geometry applications to group cohomology, opening a new chapter in the study of modular representation theory that is being actively explored to this day. This research project has its roots in Quillen's work, seeking to develop the theory of support varieties in several different, but interrelated contexts. This involves solving several fundamental problems concerning the structure of representations and the cohomology of finite dimensional algebras. The applications will provide new information on the global structure of various triangulated categories associated to representations of finite supergroup schemes, Schur algebras, Nichols algebras, Frobenius kernels of reductive groups, and Lie superalgebras.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.