Georgia Institute of Technology College of Sciences

We will discuss a proof that finite energy solutions to the defocusing cubicKlein Gordon equation scatter, and will discuss a related result in thefocusing case. (Don't worry, we will also explain what it means for asolution to a PDE to scatter.) This is joint work with Rowan Killip andMonica Visan.

( This will be a continuation of last week's talk. )An n-dimensional topological quantum field theory is a functor from the category of closed, oriented (n-1)-manifolds and n-dimensional cobordisms to the category of vector spaces and linear maps. Three and four dimensional TQFTs can be difficult to describe, but provide interesting invariants of n-manifolds and are the subjects of ongoing research. This talk focuses on the simpler case n=2, where TQFTs turn out to be equivalent, as categories, to Frobenius algebras. I'll introduce the two structures -- one topological, one algebraic -- explicitly describe the correspondence, and give some examples.

Why did kamikaze pilots wear helmets? Why does glue not stick to the inside of the bottle? Why is lemonade made with artificial flavor but dishwashing liquid made with real lemons? How can I avoid traffic jams and be paid for it? While the first three are some of life's enduring questions, the fourth is the subject of a traffic decongestion research project at Stanford University. In this talk, I will briefly describe this project and, more generally, discuss incentive mechanisms for Societal Networks--- networks which are vital for a society's functioning; for example, transportation, energy, healthcare and waste management. I will talk about incentive mechanisms and experiments for reducing road congestion, pollution and energy use, and for improving "wellness" and good driving habits. Some salient themes are: using low-cost sensing technology to make societal networks much more efficient, using price as a signal to co-ordinate individual behavior, and intelligently "throwing money at problems".

In classical holomorphic dynamics, rational self-maps of the Riemann sphere whose critical points all have finite forward orbit under iteration are known as post-critically finite (PCF) maps. A deep result of Thurston shows that if one excludes examples arising from endomorphisms of elliptic curves, then PCF maps are in some sense sparse, living in a countable union of zero-dimensional subvarieties of the appropriate moduli space (a result offering dubious comfort to number theorists, who tend to work over countable fields). We show that if one restricts attention to polynomials, then the set of PCF points in moduli space is actually a set of algebraic points of bounded height. This allows us to give an elementary proof of the appropriate part of Thurston's result, but it also provides an effective means of listing all PCF polynomials of a given degree, with coefficients of bounded algebraic degree (up to the appropriate sense of equivalence).