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射影变异上的向量束联系起来，从而在表示理论和代数几何之间建立一种新的联系。提出的研究需要它的根在PI和合作者在有限维代数的表示和上同调以前的工作。所提出的发展的独特视角在代数几何和表示理论中都有重要应用的潜力，从向量束的新示例的构建到分类结果。其他具体的研究领域包括支持变量的重要例子的计算，小量子群的新不变量的构造，以及常数约旦型模块理论的进展。此外，Pevtsova建议对“三角形几何”做出贡献，“三角形几何”是一种新的几何理论，它编码了三角分类的结构。Pevtsova将有限群方案和三角形几何的项目结合在一起，试图通过几何来比较与不同代数对象相关的衍生类别。表示理论研究群和其他代数结构在向量空间上的作用。它起源于对对称的研究，并在大约一百年前Frobenius和Schur的工作中成为一门独立的学科。在目前的发展阶段，表征理论已经被发现与许多数学分支，如几何学、拓扑学和组合学，以及物理学紧密地交织在一起。佩夫佐夫对几何学的联系特别感兴趣。她一直着迷于代数和几何之间美丽的相互作用，这种相互作用超越了许多数学研究领域。在她的CAREER项目中，她试图建立一种新的代数-几何联系，在可以用矩阵表示的代数描述和可以被认为是空间的某些几何不变量之间建立一座桥梁。在这一努力中，她希望开发出一种技术，这种技术将有可能阐明表征理论和几何中一些长期存在的问题。PI还有一个广泛的教育项目。她打算监督本科生和研究生的研究，并在华盛顿大学普特南团队的培训中开设一门关于解决问题的课程。基于她现有的拓展经验，PI打算在西雅图公立学校为有天赋的中小学生创建一个课后“数学挑战”项目网络。她还提议在2012年夏季为表征理论及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从事数学科学领域的职业。

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.