Georgia Institute of Technology College of Sciences

I will give 2 or 3 lectures on Legendrian contact homology. This invariant has played a big role in our understanding of Legendrian submanifolds of contact manifolds in all dimensions. We will discuss the general definition but focus on the 3-dimensional setting where it easiest to compute (and describe Legendrian knots). I will also discuss the A^\infty structure associated to the linearized co-chain groups of contact homology.

Our brains have evolved in a three-dimensional environment, and so we are very good at visualising two- and three-dimensional objects. But what about four-dimensional objects? The best we can really do is to look at threedimensional "shadows". Just as a shadow of a three-dimensional object squishes it into the two-dimensional plane, we can squish a four-dimensional shape into three-dimensional space, where we can then makea sculpture of it. If the four-dimensional object isn't too complicated and we choose a good way to squish it, then we can get a very good sense of what it is like. We will explore the sphere in four-dimensional space, thefour-dimensional polytopes (which are the four-dimensional versions of the three-dimensional polyhedra), and various 3D printed sculptures, puzzles, and virtual reality experiences that have come from thinking about thesethings.

The Integration by Parts Formula, which is equivalent withthe DivergenceTheorem, is one of the most basic tools in Analysis. Originating in theworks of Gauss, Ostrogradsky, and Stokes, the search for an optimalversion of this fundamental result continues through this day and theseefforts have been the driving force in shaping up entiresubbranches of mathematics, like Geometric Measure Theory.In this talk I will review some of these developments (starting from elementaryconsiderations to more sophisticated versions) and I will discuss recentsresult regarding a sharp divergence theorem with non-tangential traces.This is joint work withDorina Mitrea and Marius Mitrea from University of Missouri, Columbia.

This talk is about the Dirac equation. We consider an electron modeled by awave function and evolving in the Coulomb field generated by a nucleus. Ina very rough way, this should be an equation of the form$$i\partial_t u = -\Delta u + V( \cdot - q(t)) u$$where $u$ represents the electron while $q(t)$ is the position of thenucleus. When one considers relativitic corrections on the dynamics of anelectron, one should replace the Laplacian in the equation by the Diracoperator. Because of limiting processes in the chemistry model from whichthis is derived, there is also a cubic term in $u$ as a correction in theequation. What is more, the position of the nucleus is also influenced bythe dynamics of the electron. Therefore, this equation should be coupledwith an equation on $q$ depending on $u$.I will present this model and give the first properties of the equation.Then, I will explain why it is well-posed on $H^2$ with a time of existencedepending only on the $H^1$ norm of the initial datum for $u$ and on theinitial datum for $q$. The linear analysis, namely the properties of thepropagator of the equation $i\partial_t u = D u + V( \cdot - q(t))$ where$D$ is the Dirac operator is based on works by Kato, while the non linearanalysis is based on a work by Cancès and Lebris.It is possible to have more than one nucleus. I will explain why.(Joint work with F. Cacciafesta, D. Noja and E. Séré)