School of Mathematics
This is from joint works with D. Knopf and I. M. Sigal. In this talk I will present a new strategy in studying neckpinching of mean curvature flow. Different from previous results, we do not use backward heat kernel, entropy estimates or subsequent convergence, instead we apply almost precise estimates, invented in the past few years, to obtain the first result on asymmetric surface.
Does the world embody beautiful ideas? This is a question that people have thought about for a long time. Pythagoras and Plato intuited that the world should embody beautiful ideas; Newton and Maxwell demonstrated how the world could embody beautiful ideas, in specific impressive cases. Finally in the twentieth century in modern physics, and especially in quantum physics, we find a definitive answer: Yes! The world does embody beautiful ideas.
The PCP theorem (Arora et. al., J. ACM 45(1,3)) asserts the existence of proofs that can be verified by reading a very small part of the proof. Since the discovery of the theorem, there has been a considerable work on improving the theorem in terms of the length of the proofs, culminating in the construction of PCPs of quasi-linear length, by Ben-Sasson and Sudan (SICOMP 38(2)) and Dinur (J. ACM 54(3)).
Often mathematicians refer to a "beautiful" result or a "beautiful" proof. In this special lecture, Enrico Bombieri, Professor Emeritus in the School of Mathematics, addresses the question, "What is beauty in mathematics?"
For all practical purposes, the Micali-Vazirani algorithm, discovered in 1980, is still the most efficient known maximum matching algorithm (for very dense graphs, slight asymptotic improvement can be obtained using fast matrix multiplication). However, this has remained a ``black box" result for the last 32 years. We hope to change this with the help of a recent paper giving a simpler proof and exposition of the algorithm:
We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.
We study the nonlinear Klein-Gordon equation, in one dimension, with a qudratic term and variable coefficient qubic term. This equation arises from the asymptotic stability theory of the kink solution.Our main result is the global existence and decay estimates for this equation. We discovered a striking new phenomena in this problem: a resonant interaction between the spacial frequencies of the nonlinear coefficient and the temporal oscillations of the solution.