Georgia Institute of Technology College of Sciences

Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to obtain broad applicable correlation measures, we use a pair of r-squares of generalized regression to deal with asymmetries in explained variances, and linear or nonlinear relations between random variables. We call the pair of r-squares of generalized regression generalized measures of correlation (GMC). We present examples under which the paired measures are identical, and they become a symmetric correlation measure which is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and decision making. In statistical inferences, the joint asymptotics of the kernel based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. In real data analysis, this talk presents ideas of using GMCs as an indicator of suitability of asset pricing models, and hence new pricing models may be motivated from this indicator.

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this setup to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as "combinatorial shadows" of curves. This tropical relationship between graphs and algebraic curves has led to beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs.

An introduction for non-experts on real and finite Euler sums, also known as multiple zeta values.

(Alon-Tarsi Conjecture): For even n, the number of even nxn Latin squares differs from the number of odd nxn Latin squares. (Stones-Wanless, Kotlar Conjecture): For all n, the number of even nxn Latin squares with the identity permutation as first row and first column differs from the number of odd nxn Latin squares of this type. (Aharoni-Berger Conjecture): Let M and N be two matroids on the same vertex set, and let A1,...,An be sets of size n + 1 belonging to both M and N. Then there exists a set belonging to both M and N and meeting all Ai. We prove equivalence of the first two conjectures and a special case of the third one and use these results to show that Alon-Tarsi Conjecture implies Rota's bases conjecture for odd n and any system of n non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.Joint work with Ron Aharoni.