EMSW21-RTG: Research Training Group in Interactions of Representation Theory, Geometry and Combinatorics

这个项目的目的是将加州大学伯克利分校在表示理论、几何和组合学领域的研究人员以及一批博士后助理和研究生聚集在一起，以促进合作研究和对年轻人未来工作的培训。这个项目涉及的数学的三个分支以极其重要的方式联系在一起。在这些领域中的每一个领域，当前许多最令人兴奋的研究进展都涉及到它们之间的相互联系，因此对其中任何一个领域的认真研究都需要对另外两个领域有扎实的了解。该项目的一个中心目标是提供一个环境，让博士后和研究生获得所需的时间、机会和合作氛围，以更好地掌握所有三个领域的全面技术。该项目将设立两个新的研讨会，在整个学年举行会议，一个集中于标准课程中通常不存在的专业主题的教学，另一个是关于当前研究的联合研讨会。该项目还将每年举办一次为期一周的强化暑期研讨会，由杰出的外部访客和加州大学教职员工就高级主题举行5-6场系列讲座，旨在为研究生和博士后水平的观众提供服务。我们将欢迎和促进来自其他机构的年轻研究人员的参与。列举最近的数学发展存在于表示论、几何和组合学的交叉点，未来还有更多的工作要做。组成研究小组核心的教职员工(教授)。Haiman、Reshetikhin、Borcherds、Frenkel、Givental、Knutson、Serganova和Wolf)有着广泛的重叠兴趣，每个人在自己的工作中都使用了所有这三个领域的技术。该小组多名成员共同的具体研究兴趣包括：量子场论；无限维李代数的表示理论；量子群和正则基；Gromov-Witten不变量和镜像对称性；以及Schubert簇和旗形的几何和组合学。该项目的活动旨在促进研究合作，最重要的是为年轻数学家提供培训，使他们掌握在上述主题的研究前沿所需的广泛智力工具。

Abstract for RTG award DMS-0354321 of HaimanThe purpose of this project is to bring together U.C. Berkeley'sresearch faculty in the areas of Representation Theory, Geometry andCombinatorics, along with a group of postdoctoral associates andgraduate students, in order to promote collaborative research and thetraining of young people for future work. The three branches ofmathematics addressed by this project are connected in profoundlyimportant ways. In every one of these areas, many of the mostexciting current research advances involve the interconnectionsbetween them, so that serious study of any of them requires strongknowledge of the other two. A central goal of the project is toprovide an environment in which postdocs and graduate students aregiven the time, opportunity and collaborative atmosphere needed tomaster a full range of techniques in all three areas. The projectwill establish two new seminars, to meet throughout the academic year,one concentrating on instruction in specialized topics not usuallyfound in standard courses, and the other a joint seminar on currentresearch. The project will also hold an annual week-long intensivesummer workshop, featuring series of 5--6 lectures on advanced topicsby distinguished outside visitors and U.C. faculty members, intendedfor a graduate student to postdoctoral level audience. We willwelcome and facilitate participation by young researchers from otherinstitutions.Exciting recent mathematical developments lie at the intersection ofrepresentation theory, geometry and combinatorics, with much more tobe done in the future. The faculty members forming the core of theresearch group (Profs. Haiman, Reshetikhin, Borcherds, Frenkel,Givental, Knutson, Serganova and Wolf) have extensive overlappinginterests, each using techniques from all three areas in his or herown work. Specific research interests common to multiple members ofthe group include topics in quantum field theory; representationtheory of infinite-dimensional Lie algebras; quantum groups andcanonical bases; Gromov-Witten invariants and mirror symmetry; andgeometry and combinatorics of Schubert varieties and flag manifolds.The activities of the project are designed to foster researchcollaboration and above all to equip young mathematicians in trainingwith the broad range of intellectual tools needed to reach thefrontiers of research on such topics such as those mentioned.