Georgia Institute of Technology College of Sciences

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).

This lecture will conclude the series. In a climactic finish the speaker will prove the Unique Linkage Theorem, thereby completing the proof of correctness of the Disjoint Paths Algorithm.

They may flow like fluids but under constraints of mechanical energies from their crystal aspects. As a result, they exhibit very rich phenomena that grant them tremendous applications in modern technology. Based on works of Oseen, Z\"ocher, Frank and others, a continuum theory (not most general but satisfactory to a great extent) for liquid-crystals was formulated by Ericksen and Leslie in 1960s. We will first give a brief introduction to this classical theory and then focus on various important special settings in both static and dynamic cases. These special flows are rather simple for classical fluids but are quite nonlinear for liquid-crystals. We are able to apply abstract theory of nonlinear dynamical systems upon revealing specific structures of the problems at hands.