Georgia Institute of Technology College of Sciences

I will describe the arithmetic of polynomials over idylls and various division algorithms and rules. For instance, that arithmetic might capture a total order/sign or an absolute value. These division algorithms will relate, for instance, the number of positive roots of a polynomial to the signs of the coefficients (Descartes's Rule of Signs).

Independent component analysis is a useful and general data analysis tool. It has found great successes in many applications. But in recent years, it has been observed that many popular approaches to ICA do not scale well with the number of components. This debacle has inspired a growing number of new proposals. But it remains unclear what the exact role of the number of components is on the information theoretical limits and computational complexity for ICA. Here I will describe our recent work to specifically address these questions and introduce a refined method of moments that is both computationally tractable and statistically optimal.

Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r \geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not between 0 and 1/2. In this talk, we will discuss the fillability of the contact structures M for 0

The chromatic index $\chi'(G)$ of a graph $G$ is the least number of colors assigned to the edges of $G$ such that no two adjacent edges receive the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The chromatic index and the total chromatic number are two of few fundamental graph parameters, and their correlation has always been a subject of intensive study in graph theory.

By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$. In 1984,  Goldberg conjectured that for any multigraph $G$, if $\chi'(G) \ge \Delta(G) +3$ then $\chi''(G) = \chi'(G)$. We show that Goldberg's conjecture is asymptotically true. More specifically,  we prove that for a multigraph $G$ with maximum degree $\Delta$ sufficiently large, $\chi''(G) = \chi'(G)$ provided $\chi'(G) \ge \Delta + 10\Delta^{35/36}$.  When $\chi'(G) \ge \Delta(G) +2$, the chromatic index $\chi'(G)$ is completely determined by the fractional chromatic index. Consequently,   the total chromatic number $\chi''(G)$ can be computed in polynomial-time in this case.