In this talk, we will present some recent results in the study of the nonlinear wave equation with cubic defocusing nonlinearity, describing the completion of a program to establish global well-posedness and scattering in the energy-supercritical regime under an assumed a priori uniform-in-time control of the critical norm. In particular, we discuss a series of recent results in which the concentration-compactness approach of Kenig and Merle is combined with tools from harmonic analysis to yield insight in this class of problems.
A knot is more or less what you think it is—a tangled mess of string in ordinary three-dimensional space. In the twentieth century, mathematicians developed a rich and deep theory of knots. And surprisingly, as Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, explains in this lecture, it turned out that many of the most interesting ideas about knots have their roots in quantum physics.
I discuss two examples of random symplectic maps in this talk. As the first example consider a stochastic twist map that is defined to be a stationary ergodic twist map on a planar strip. As a natural question, I discuss the fixed point of such maps and address a Poincare-Birkhoff type theorem. As the second example I consider stochastic flows associated with diffusions and discuss those diffusions which produce symplectic maps only in average sense. Using stochastic diffusions, it is possible to derive Iyer-Constantin Circulation Theorem for Navier-Stokes Equation.
After reviewing spectral invariants, I will write down an estimate, which under certain assumptions, relates the spectral invariants of a Hamiltonian to the C0-distance of its flow from the identity. I will also show that, unlike the Hofer norm, the spectral norm is C0-continuous on surfaces. Time permitting, I will present an application to the study of area preserving disk maps.
In this talk, I will present a formulation of the Gross-Zagier formula over Shimura curves using automorphic representations with algebraic coefficients. It is a joint work with Shou-wu Zhang and Wei Zhang.
The classical result on the uniqueness of black holes in GR, due to Hawking, which asserts that regular, stationary solutions of the Einstein vacuum equations must be isometric to an admissible black hole Kerr solution, has at its core a a highly unrealistic analyticity assumption for the metric. The goal of the talk is to survey recent results, obtained in collaboration with Alexakis and Ionescu, on the general rigidity problem, without analyticity.
It is becoming more and more clear that many of the most exciting structures of our world can be described as large networks. The internet is perhaps the foremost example, modeled by different networks (the physical internet, a network of devices; the world wide web, a network of webpages and hyperlinks). Various social networks, several of them created by the internet, are studied by sociologist, historians, epidemiologists, and economists. Huge networks arise in biology (from ecological networks to the brain), physics, and engineering.