combinatorics is rich with objects counted by the catalan numbers. one such set of objects is the set of noncrossing partitions of the numbers 1 through n. there is a natural generalization in which one considers the set of noncrossing partitions of kn with block sizes each divisible by k. in this talk, we'll consider a rational generalization of noncrossing partitions and discuss current research in this subject.

solving an unconstrained quadratic program means solving a linear equation where the matrix is symmetric and positive definite. this is a fundamental subproblem in nonlinear optimization. we discuss the behavior of the method of conjugate gradients and quasi-newton methods on a quadratic problem. we show that by interpreting the method of conjugate gradients as a particular exact line search quasi-newton method, necessary and sufficient conditions can be given for an exact line search quasi-newton method to generate a search direction which is parallel to that of the method of conjugate gradients. the analysis gives a condition on the quasi-newton matrix at a particular iterate, the projection is inherited from the method of conjugate gradients. we also analyze update matrices and show that there is a family of symmetric rank-one update matrices that preserve positive definiteness of the quasi-newton matrix. this is in contrast to the classical symmetric-rank-one update where there is no freedom in choosing the matrix, and positive definiteness cannot be preserved.

optimization is an indispensable tool in planning of intensity-modulated radiotherapy therapy (imrt) planning. conventional planning objectives are designed to minimize the violation of so-called dose-volume histogram (dvh) thresholds using penalty functions. in this study, we abandon the usual penalty-function framework and propose planning objectives that more explicitly relate to dvh statistics. the proposed planning objectives are based on mean-tail-dose, resulting in optimization problems of linear programming type. we investigate the potential of the proposed planning objectives as tools for optimizing dvh statistics through juxtaposition with the conventional planning objectives on two patient cases. an obstacle when changing from quadratic penalty functions to optimization of mean-tail-dose is that the dimension of the optimization problem is increased by several orders of magnitude. we demonstrate how to adapt a higher-order interior method to make this problem size manageable.

energy minimization is the process by which nature selects the configuration of a system and balances the relevant forces. but not all of the current standard energy functionals in electrostatics are bounded below, leading to a contradiction. by recognizing legendre transforms at the stationary points (or maximums) of non-convex (or concave) functionals, one can rewrite the functionals in terms of the transformed variable such that the new functionals are convex, hence better suited to standard optimization techniques. in this talk i will describe how legendre transforms can be applied to reformulate the electrostatic free energy functional and present a proof of the equivalence of the new formulation with the classical one. i will also discuss how the legendre transformed functional can be applied to the dielectric boundary problem in molecular solvation.

we will present a quantitative equidistribution result for adelic homogeneous subsets whose stabilizer is maximal and semisimple. some number theoretic applications will also be discussed. this is based on a joint work with einsiedler, margulis and venkatesh.

after a brief introduction to general relativity, i will discuss penrose's celebrated strong cosmic censorship conjecture. this conjecture can be viewed as a global uniqueness conjecture, which states that for generic initial data, the solution is uniquely determined. this is in contrast to the phenomenon present in some special explicit black hole solutions such that determinism breaks down. i will then discuss some recent mathematical progress regarding this conjecture.