Mathematical Sciences: Elliptic Genera and Elliptic Cohomology

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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.