RTG: Algebraic Geometry and Representation Theory

代数几何和表示理论是两个非常活跃的数学领域，与数学的其他领域（例如，数论、拓扑和辛几何）和理论物理（例如，弦理论、共形场论和统计力学）有着深刻的联系。代数几何研究由多项式方程及其高维类似物定义的曲线和曲面，而表示理论研究对称。这两门学科相互作用：几何物体具有对称性，人们可以用几何来研究对称性。这个研究培训小组项目旨在招募和培训年轻人，让他们参与这些令人兴奋的数学领域的研究。该项目涉及各级教育，从本科到博士后研究。小组活动包括：(a)通过参加“微积分之桥”高中项目和“女孩之角”数学俱乐部，在大学预科阶段进行拓展；(b)开发新的本科课程，为本科生提供系列讲座，本科生研究经验，并由本科生小组成员指导“微积分之桥”学生；(c)专题研究生讨论会和小型课程、开发新的研究生课程、研究生暑期学校和研究生指导本科生；(d)博士后参与小组研究研讨会和研究会议，由博士后指导本科生和研究生；(e)由博士后科研人员的教师积极培训，帮助他们进一步发展自己的独立研究项目和专业技能。小组活动集中在代数几何和表示理论的研究主题，包括派生范畴和同调代数技术，hyper-Kähler流形，量子上同调和计数不变量，霍奇理论，以及与物理和镜像对称的联系。本科生的研究经验活动包括有限群的表示理论（包括模表示理论）、晶体、颤振、交换代数和代数几何的计算方面、初等交理论。该项目包括在第五年组织一次大型国际研究会议。

Algebraic geometry and representation theory are two extremely active fields of mathematics, with deep connections to other areas of mathematics (for example, number theory, topology, and symplectic geometry) and theoretical physics (for example, string theory, conformal field theory, and statistical mechanics). Algebraic geometry studies curves and surfaces defined by polynomial equations and their higher dimensional analogs, while representation theory studies symmetry. The two subjects interact: geometric objects have symmetries and one can use geometry to study symmetry. This Research Training Group project aims to recruit and train young people by engaging them in research in these exciting areas of mathematics. The project involves all levels of education, from undergraduate to postdoctoral studies. The group activities include: (a) outreach at the pre-college level though participation in the Bridge to Calculus high school program and in the Girls' Angle math club; (b) development of new undergraduate courses, a lecture series for undergraduates, undergraduate research experiences, and mentoring of Bridge to Calculus students by undergraduate group members; (c) topical graduate seminars and mini-courses, development of new graduate courses, summer schools for graduate students, and graduate student mentoring of undergraduates; (d) participation of postdoctoral associates in group research seminars and a research conference, and mentoring by postdoctoral associates of undergraduate and graduate students; and (e) active training by the faculty of postdoctoral researchers to help them further develop their independent research programs and professional skills.The group activities are centered on research themes in algebraic geometry and representation theory that include derived categories and homological algebra techniques, hyper-Kähler manifolds, quantum cohomology and counting invariants, Hodge theory, and connections with physics and mirror symmetry. The undergraduate research experience activities include representation theory of finite groups (including modular representation theory), crystals, quivers, computational aspects of commutative algebra and algebraic geometry, and elementary intersection theory. The project includes organization of a large international research conference in the fifth year.