School of Mathematics

We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension n >=3, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension n >=3.

We will discuss the notion of loops in linguistic structures, mainly in dictionaries. In a simplified view, a dictionary is a graph that links every word (vertex) to a set of alternative words (the definition) which in turn point to further descendants. Iterating through definitions, one may loop back to the original word. We will examine possible links between such definitional loops and the emergence of new concepts during the evolution of languages. Potential relation to living systems will be briefly discussed.

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic applications. One can formulate a Whittaker analogue of the Ichino-Ikeda conjecture. We use the descent method of Ginzburg-Rallis-Soudry to reduce the Whittaker version to a purely local identity which we prove in the p-adic case under some mild hypotheses. Joint work with Zhengyu Mao

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Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space LM of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of LM and the “dual” loop cohomology product. If X is a homology class on LM, the "minimax" critical level Cr(X) is a critical value of the length function.