The Mathematical Challenge of Large Networks

László Lovász
Eotvos Lorand University, Budapest; Institute for Advanced Study
October 24, 2011

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.

On the Rigidity of Black Holes

Sergiu Klainerman
Princeton University
October 25, 2011

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.

Knots and Quantum Theory

Edward Witten, Charles Simonyi Professor
Institute for Advanced Study
October 28, 2011

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.

C^0 Limits of Hamiltonian Paths and Spectral Invariants

Sobhan Seyfaddini
University of California at Berkeley
October 28, 2011

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.

Chow Rings, Decomposition of the Diagonal and the Topology of Families

Claire Voisin
CNRS, Institut de Mathematiques de Jussieu, Paris
November 1, 2011

Summary: These lectures are devoted to the interplay between cohomology and Chow groups of a complex algebraic variety, and also to the consequences, on the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general fiber. A crucial notion is that of coniveau of the cohomology and its conjectural relation with the shape of Chow groups of small dimension. A common theme will be that of decomposition of the diagonal, which will appear in various contexts.