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簇上的向量丛联系起来，从而在表示理论和代数几何之间建立一种新的联系。所提出的研究源于PI和合作者在有限维代数的表示和上同调方面的先前工作。所提出的发展的独特视角在代数几何和表示理论中都具有重要的应用潜力，从构造向量丛的新示例到分类结果。其他具体的研究领域包括支持簇的重要例子的计算，小量子群的新不变量的构造，以及对常量Jordan模理论的发展。此外，Pevtsova还建议对“三角几何”做出贡献，“三角几何”是一种编码三角范畴结构的新几何理论。Pevtsova把关于有限群方案和三角几何的项目带到一个交汇点，试图通过它们的几何来比较与不同代数对象相关联的派生范畴。表示论研究群和其他代数结构在向量空间上的行为。它起源于对对称性的研究，并在大约一百年前弗罗贝尼乌斯和舒尔的著作中独立成为一门学科。在目前的发展阶段，表示理论已被发现与许多数学分支紧密交织在一起，如几何、拓扑学、组合学以及物理学。佩夫索娃对几何学的联系特别感兴趣。她一直被代数和几何之间的美丽相互作用所吸引，这种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中，她试图发展一种新的代数-几何联系，在可以用矩阵表示的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，她希望开发出一些技术，这些技术将有可能阐明表示理论和几何中的一些长期存在的问题。国际和平协会还有一个广泛的教育项目。她打算指导本科生和研究生的研究，并结合华盛顿大学普特南团队的培训开发一门关于解决问题的课程。在她现有外展经验的基础上，她打算为西雅图公立学校的资优中小学生创建一个“数学挑战”课外项目网络。她还建议在2012年夏天为代表理论和相关领域的年轻研究人员组织一次暑期班。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投身数学科学领域。

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.