Georgia Institute of Technology College of Sciences

While this could be a lecture about our US Congress, it instead deals with problems that are asymptotically related to best-packing and best-covering. In particular, we discuss how to efficiently generate N points on a d-dimensional manifold that have the desirable qualities of well-separation and optimal order covering radius, while asymptotically having a prescribed distribution. Even for certain small numbers of points like N=5, optimal arrangements with regard to energy and polarization can be a challenging problem.

We study small perturbations of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant on a spatially periodic background. These solutions model a quiet fluid in a spacetime undergoing accelerated expansion. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. Our result extends the stability results of Rodnianski and Speck for the Euler-Einstein system with positive cosmological constant to the case of dust (i.e. a pressureless fluid). The main difficulty that we overcome is the degenerate nature of the dust model that loses one degree of differentiability with respect to the Euler case. To resolve it, we commute the equations with a well-chosen differential operator and develop a new family of elliptic estimates that complement the energy estimates. This is joint work with J. Speck.

Let p be a prime, let C/F_p be an absolutely irreducible curve inside the affine plane. Identify the plane with D=[0,p-1]^2. In this talk, we consider the problem of how often a box B in D will contain the expected number of points. In particular, we give a lower bound on the volume of B that guarantees almost all translations of B contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense. This is joint work with Alexandru Zaharescu.