Georgia Institute of Technology College of Sciences

In 1963 Fisher and Stephenson conjectured that the correlation function of two oppositely colored monomers in a sea of dimers on the square lattice is rotationally invariant in the scaling limit. More precisely, the conjecture states that if one of the monomers is fixed and the other recedes to infinity along a fixed ray, the correlation function is asymptotically $C d^(-1/2)$, where $d$ is the Euclidean distance between the monomers and $C$ is a constant independent of the slope of the ray. Shortly afterward Hartwig rigorously determined $C$ when the ray is in a diagonal direction, and this remains the only direction settled in the literature. We generalize Hartwig's result to any finite collection of monomers along a diagonal direction. This can be regarded as a counterpart of a result of Zuber and Itzykson on n-spin correlations in the Ising model. A special case proves that two same-color monomers interact the way physicists predicted.

We consider various dependence concepts for random fields. Special attention is paid to Gaussian and shot-noise fields. The multivariate central limit theorems (CLT) are proved for the volumes of excursion sets of stationary quasi-associated random fields on $\mathbb{R}^d$. Formulae for the covariance matrix of the limiting distribution are provided. Statistical versions of the CLT are established as well. They employ three different estimators of the asymptotic covariance matrix. Some numerical results are also discussed.

Individual chemical reactions between molecules are inherently stochastic, although for a large collection of molecules, the overall system behavior may appear to be deterministic. When deterministic chemical reaction models are sufficient to describe the behavior of interest, they are a compact way to describe chemical reactions. However, in other cases, these mass-action kinetics models are not applicable, such as when the number of molecules of a particular type is small, or when no closed-form expressions exist to describe the dynamic evolution of overall system properties. The former case is common in biological systems, such as intracellular reactions. The latter case may occur in either small or large systems, due to a lack of smoothness in the reaction rates. In both cases, kinetic Monte Carlo simulations are a useful tool to predict the evolution of overall system properties of interest. In this talk, an approach will be presented for generating approximate low-order dynamic models from kinetic Monte Carlo simulations. The low-order model describes the dynamic evolution of several expected properties of the system, and thus is not a stochastic model. The method is demonstrated using a kinetic Monte Carlo simulation of atomic cluster formation on a crystalline surface. The extremely high dimension of the molecular state is reduced using linear and nonlinear principal component analysis, and the state space is discretized using clustering, via a self-organizing map. The transitions between the discrete states are then computed using short simulations of the kinetic Monte Carlo simulations. These transitions may depend on external control inputs―in this application, we use dynamic programming to compute the optimal trajectory of gallium flux to achieve a desired surface structure.

One of the basic problems arising in many pure and applied areas of mathematics is to solve a system of polynomial equations. Numerical Algebraic Geometry starts with addressing this fundamental problem and develops machinery to describe higher-dimensional solution sets (varieties) with approximate data. I will introduce numerical polynomial homotopy continuation, a technique that is radically different from the classical symbolic approaches as it is powered by (inexact) numerical methods.