CAREER: From Modular Representation Theory to Geometry: connections and interactions

Pevtsova打算发展一种理论，将有限群方案的表示与投射簇上的向量丛联系起来，从而在表示论和代数几何之间建立一种新的联系。拟议的研究需要在PI和合作者在有限维代数的表示和上同调以前的工作的根源。所提出的发展的独特视角具有潜在的显着的应用在代数几何和表示理论，从建设新的例子向量束的分类结果。 其他具体领域的拟议研究包括计算的重要例子的支持品种，建设新的不变量的小量子群，并取得进展的理论模块的恒定约旦型。 此外，Pevtsova提出对“三角几何”做出贡献，这是一种新的几何理论，编码三角范畴的结构。 Pevtsova将有限群方案和三角几何的项目带到了一个交汇点，他试图通过几何比较与不同代数对象相关的派生类别。表示论研究向量空间上的群和其他代数结构的作用。它起源于对称性的研究，并在大约一百年前的弗罗贝纽斯和舒尔的著作中作为一个独立的主题出现。在其目前的发展阶段，表征理论已被发现与数学的许多分支密切相关，如几何学，拓扑学和组合学，以及物理学。Pevtsova特别感兴趣的是与几何的连接。她一直着迷于代数和几何之间的美丽的相互作用，超越了数学研究的许多领域。在她的职业生涯项目，她寻求开发一种新的代数几何连接，建立一个桥梁之间的代数描述的表示，可以提出的矩阵和某些几何不变量，可以被认为是空间。 在这一奋进中，她希望开发出的技术，将有可能阐明一些长期存在的问题，在这两个表示理论和几何。 PI也有一个广泛的教育计划。她打算指导本科生和研究生的研究，并为华盛顿大学的普特南小组的培训开设一门解决问题的课程。在她现有的推广经验的基础上，PI打算为西雅图公立学校的天才小学生和中学生创建一个"数学挑战”课后项目网络。她还建议在2012年夏季为表征理论和相关领域的年轻研究人员组织一个暑期学校。通过为有才华的K-12学生，本科生和研究生创造这些机会，Pevtsova打算吸引更多合格的候选人在数学科学的职业生涯。

Pevtsova intends to develop a theory that relates representations of finite group schemes to vector bundles on projective varieties, thereby establishing a novel connection between representation theory and algebraic geometry. The proposed research takes its roots in previous work of the PI and collaborators in representations and cohomology of finite dimensional algebras. The unique perspective of the proposed development has a potential for significant applications in both algebraic geometry and representation theory, ranging from construction of new examples of vector bundles to classification results. Other specific areas of the proposed research include computations of important examples of support varieties, constructions of new invariants for small quantum groups, and advances to the theory of modules of constant Jordan type. In addition, Pevtsova proposes to make contributions to the ``triangular geometry", a new geometric theory that encodes the structure of triangulated categories. Bringing the projects on finite group schemes and triangular geometry to a meeting point, Pevtsova is seeking to compare derived categories associated to different algebraic objects via their geometry.Representation theory studies actions of groups and other algebraic structures on vector spaces. It takes its origins in the study of symmetries and has emerged as a subject on its own about hundred years ago in the work of Frobenius and Schur. In its current stage of development, representation theory has been discovered to be intimately intertwined with numerous brunches of mathematics, such as geometry, topology and combinatorics, as well as physics. Pevtsova is particularly interested in connections with geometry. She has always been fascinated by the beautiful interplay between algebra and geometry that transcends many areas of mathematical research. In her CAREER project, she seeks to develop a new algebra-geometric connection, building a bridge between algebraic descriptions of representations which can be presented in terms of matrices and certain geometric invariants which can be thought of as spaces. In that endeavor she hopes to develop techniques that will hold a potential to shed light on some longstanding problems in both representation theory and geometry. The PI also has an extensive educational program. She intends to supervise both undergraduate and graduate research and to develop a course on problem solving in connection with the training of the Putnam team at the University of Washington. Building on her existing outreach experience the PI intends to create a network of ``Math challenge" afterschool programs for gifted elementary and middle school students in Seattle Public schools. She also proposes to organize a summer school for young researchers in Representation Theory and related areas in the Summer of 2012. By creating these opportunities for talented K-12 students, undergraduates and graduate students, Pevtsova intends to attract more qualified candidates to careers in Mathematical Sciences.