Georgia Institute of Technology College of Sciences

In recent work, an idea of Adam Jakubowski was used to prove infinite stable limit theory and precise large deviation results for sums of strictly stationary regularly varying sequences. The idea of Jakubowski consists of approximating tail probabilities of distributions for such sums with increasing index by the corresponding quantities for sums with fixed index. This idea can also be made to work for Laplace functionals of point processes, the distribution function of maxima and the characteristic functions of partial sums of stationary sequences. In each of these situations, extremal dependence manifests in the appearance of suitable cluster indices (extremal index for maxima, cluster index for sums,...). The proposed method can be easily understood and has the potential to function as heuristics for proving limit results for weakly dependent heavy-tailed sequences.

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.