The classical isoperimetric property, discovered by Queen Dido shortly after her arrival at the coast of Africa in 900 B.C., states that amongst all figures of equal perimeter the circle encloses the largest area. In his 1954, Mathematics and Plausible Reasoning (Vol.I, p.181), George Pólya writes: "The isoperimetric theorem, deeply rooted in our experience and intuition, so easy to conjecture, but not so easy to prove, is an inexhaustible source of inspiration." This assertion, made fifty years ago, remains true today as we shall illustrate in this talk.
In 1973, J.M. Luttinger proved that certian "multiple" convolutions are nondecreasing if each function is replaced by its symmetric decreasing rearrangement. These inequalities and subsequent extensions by Brascamp, Lieb, and Luttinger, provide a powerful and elegant method for proving not only the classical isoperimetric inequality but also many of its physical relatives such as the Rayleigh-Faber-Krahn isoperimetric inequality for the lowest eigenvalue of the Laplacian and Pólya's isoperimetric inequalities for torsional rigidity and electrostatic capacity. From the point of view of probability, Luttinger's inequalities are inequalities about the finite dimensional distributions of Brownian motion and raise questions concerning their validity for other stochastic processes. After discussing the Brownian motion case, we will explore versions of the Brascamp-Lieb-Luttinger inequalities which lead to isoperimetric inequalities for other Lévy processes such as the "symmetric stable processes" and the "relativistic Brownian motion." We will explain how these "generalized" isoperimetric inequalities give new information even in the Brownian motion case. This leads to new results on some open questions in geometry and PDE's including, for example, a proof of a special case of van den Berg's conjecture (problem #44 in Yau's 1990 "open problems in geometry") on the size of the spectral gap of Schrödinger operators and new information on geometric properties of eigenfunctions and heat kernels for certain non-elliptic and non-local operators.
This talk is designed for a general audience. We will show pictures, discuss some results and keep technicalities to a minimum.
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