Georgia Institute of Technology College of Sciences

We study an online algorithm for making a well—equidistributed random set of points in an interval, in the spirit of "power of choice" methods. Suppose finitely many distinct points are placed on an interval in any arbitrary configuration. This configuration of points subdivides the circle into a finite number of intervals. At each time step, two points are sampled uniformly from the interval. Each of these points lands within some pair of intervals formed by the previous configuration. Add the point that falls in the larger interval to the existing configuration of points, discard the other, and then repeat this process. We then study this point configuration in the sense of its largest interval, and discuss other "power of choice" type modifications. Joint work with Pascal Maillard.

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2. We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1) and {b1, b2}⊆V(G2). In this talk, we will prove the existence of 5-edge configurations in (G, a0, a1, a2, b1, b2). Joint work with Changong Li, Robin Thomas, and Xingxing Yu.

In this talk I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on L^p(R^n) by means of testing functions as general as possible. In the classical theory for boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend. As a by-product, the results also describe those Calderón-Zygmund operators whose boundedness can be checked with non-accretive testing functions.

Almost all biological activities involve transport and distribution of ions and charged particles. The complicated coupling and competition between different ionic solutions in various biological environments give the intricate specificity and selectivity in these systems. In this talk, I will introduce several extended general diffusion systems motivated by the study of ion channels and ionic solutions in biological cells. In particular, I will focus on the interactions between different species, the boundary effects and in many cases, the thermal effects.