Mathematical Sciences: Elliptic Genera and Elliptic Cohomology

数学科学：椭圆属和椭圆上同调

• 批准号: 8903048
• 负责人: Peter Landweber
• 金额: $ 18.84万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1993-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8903048&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Elliptic; Genera; Cohomology

Professor Landweber will continue his study of elliptic genera and elliptic cohomology, a joint project with R. E. Stong (University of Virginia) and other colleagues. A new family of periodic cohomology theories was shown to exist in 1986, in joint work with Stong and D. C. Ravenel; their coefficient rings are rings of modular functions of level 2. The main problem, now widely recognized, is to determine the intrinsic geometric nature of these cohomology theories. Recently, F. Hirzebruch has defined level N elliptic genera (N at least 2), which for N = 2 become the elliptic genera leading to elliptic cohomology; it is intended to study these genera in detail. Professor Tamanoi will study the internal infinite dimensional symmetries present in elliptic genera on Kaehler manifolds with varying degrees of speciality. He proposes to explore the topological applications of these symmetries, and to study refined rigidity properties implied by them. He approaches these problems via representation theory of affine Lie algebras. Both investigators are bringing highly sophisticated algebraic theories to bear upon problems in the geometry of manifolds. Applications include the theory of strings, which has been offered as a promising approach to quantum physics of systems of interacting particles.

兰德韦伯教授将继续他的椭圆属和椭圆上同的研究，这是他与R. E. Stong（弗吉尼亚大学）和其他同事的一个联合项目。1986年，一个新的周期上同理论家族被证明存在，这是与strong和d.c. Ravenel共同完成的工作；它们的系数环为二级模函数环。现在公认的主要问题是确定这些上同调理论的内在几何性质。最近，F. Hirzebruch定义了N阶椭圆属（N至少2），当N = 2时成为椭圆属，导致椭圆上同；本文拟对这些属进行详细的研究。Tamanoi教授将研究Kaehler流形上具有不同程度特殊性的椭圆属的内部无限维对称性。他建议探索这些对称性的拓扑应用，并研究它们隐含的精细刚性性质。他通过仿射李代数的表示理论来解决这些问题。两位研究者都将高度复杂的代数理论应用于流形的几何问题。它的应用包括弦理论，它被认为是研究相互作用粒子系统的量子物理学的一种很有前途的方法。