Georgia Institute of Technology College of Sciences

The Joint Athens-Atlanta Number Theory Seminar meets once a semester, usually on a Tuesday, with two talks back to back, at 4:00 and at 5:15. Participants then go to dinner together.

In graph/hypergraph theory, perfect matchings are fundamental objects of study. Unlike the graph case, perfect matchings in hypergraphs have not been well understood yet. It is quite natural and desirable to extend the classical theory on perfect matchings from graphs to hypergraphs, as many important problems can be phrased in this framework, such as Ryser's conjecture on transversals in Latin squares and the Existence Conjecture for block designs. I will focus on Dirac-type conditions (minimum degree conditions) in uniform hypergraphs and discuss some recent progresses. In particular, we determine the minimum codegree threshold for the existence of a near perfect matching in hypergraphs, which confirms a conjecture of Rodl, Rucinski and Szemeredi, and we show that there is a polynomial-time algorithm that can determine whether a k-uniform hypergraph with minimum codegree n/k has a perfect matching, which solves a problem of Karpinski, Rucinski and Szymanska completely.

We will continue our discussion of intuitive dyadic calculus. We will begin discussing multi-parameter Calderon-Zygmund operators and oscilation.

Observations of high energy density environments, from supernovae implosions/explosions to inertial confinement fusion, are determined by many different physical effects acting concurrently. For example, one set of equations will describe material motion, while another set will describe the spatial flow of energy. The relevant spatial and temporal scales can vary substantially. Since direct measurement is difficult if not impossible, and the relevant physics happen concurrently, computer simulation becomes an important tool to understand how emergent behavior depends on the constituent laws governing the evolution of the system. Further, computer simulation can provide a means to use observation to constrain underlying physical models. This talk shall examine the challenges associated with developing computational multiphysics simulation. In particular this talk will outline some of the physics, the relevant mathematical models, the associated algorithmic challenges, some of which are driven by emerging compute architectures. The problem as a whole can be formidable and an effective solution couples many disciplines together.