Georgia Institute of Technology College of Sciences

The study of random tilings of planar lattice regions goes back to the solution of the dimer model in the 1960's by Kasteleyn, Temperley and Fisher, but received new impetus in the early 1990's, and has since branched out in several directions in the work of Cohn, Kenyon, Okounkov, Sheffield, and others. In this talk, we focus on the interaction of holes in random tilings, a subject inspired by Fisher and Stephenson's 1963 conjecture on the rotational invariance of the monomer-monomer correlation on the square lattice. In earlier work, we showed that the correlation of a finite number of holes on the triangular lattice is given asymptotically by a superposition principle closely paralleling the superposition principle for electrostatic energy. We now take this analogy one step further, by showing that the discrete field determined by considering at each unit triangle the average orientation of the lozenge covering it converges, in the scaling limit, to the electrostatic field. Our proof involves a variety of ingredients, including Laplace's method for the asymptotics of integrals, Newton's divided difference operator, and hypergeometric function identities.

Consider the 2-d ideal incompressible fluid moving inside a bounded domain (say 2-d torus). It is described by 2-d Euler equations which have unique global solution; thus, we have a dynamical system in the space of sufficiently regular incompressible vector fields. The global properties of this system are poorly studied, and, as much as we know, paradoxical. It turns out that there exists a global attractor (in the energy norm), i.e. a set in the phase space attracting all trajectories (in spite the fact that the system is conservative). This apparent contradiction leads to some deep questions of non-equilibrium statistical mechanics.

Due to Alexander, it is well known that every closed oriented 3-manifold has an open book decomposition. In this talk, we will define open book decompositions of 3-manifolds. We will discuss various examples and sketch the proof of Alexander's theorem. Further, we will discuss the importance of the open books in manifold theory, in particular in contact geometry.

We will give an overview of the company as it relates to the work of a quant. We will discuss projects of interest, typical lifecycle of a project, and involved areas.