Problems in Analysis at the Interface with Geometry and Physics

Columbia University, New York NY

Investigators

Abstract

PI: Duong H. Phong, Columbia University DMS-0245371 ABSTRACT It is proposed to address several problems at the interface of analysis with geometry and physics. A first problem is the development of stable analytic methods. Stable methods have emerged as essential for the study of singularities and oscillatory integrals, and for the basic problem of finding canonical metrics in differential geometry. Canonical metrics are expected to be equivalent to the notion of stability in the sense of geometric invariant theory, and the relation between this global notion and stable estimates is to be investigated. The second problem in the proposal is the development of Feynman rules for string scattering amplitudes. Feynman rules had been unavailable due to a geometric difficulty encountered for two-loops or higher, namely ``supermoduli". But the situation is now much more promising, thanks to the recent success of Eric D'Hoker and the PI in overcoming this difficulty in the simplest case of two-loops. The third problem in the proposal is the construction/identification of integrable structures in supersymmetric gauge and string theories. A basic tool is a new Hamiltonian approach to soliton equations developed by the PI in collaboration with Igor Krichever. These are core problems in theoretical physics and applied mathematics. The search of natural laws at their most fundamental level relies more and more on geometric principles and symmetry. The problems addressed in the present proposal - stability and canonical metrics, Feynman rules for string theory, soliton equations - are essential to the understanding of the two theories which have led research in both mathematics and theoretical physics for the last two decades, namely string theory and supersymmetry, together with their underlying geometric structures.