Georgia Institute of Technology College of Sciences

Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.

Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls, including two infinite families. This is joint work with Selman Akbulut.

Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).