Mathematical Sciences: Elliptic Genera and Elliptic Cohomology

兰德韦伯教授将继续他与R.E.Stong(弗吉尼亚大学)和其他同事的合作项目，研究椭圆亏格和椭圆上同调。1986年，在Stong和D.C.Ravenel的共同工作下，一族新的周期上同调理论被证明存在；它们的系数环是2阶模函数环。现在被广泛认识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定义了水平N个椭圆亏格(N至少为2)，当N=2时，它成为导致椭圆上同调的椭圆亏格，本文对这些椭圆亏格进行了详细的研究。Tamanoi教授将研究具有不同专业度的Kaehler流形上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，并研究它们所蕴含的精细刚性性质。他通过仿射李代数的表示理论来研究这些问题。两位研究人员都在运用高度复杂的代数理论来解决流形几何中的问题。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的一种很有前途的方法。

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.