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# Upcoming Seminars

In the talk I will tell about a theory of cyclic elements in semisimple Lie algebras. This notion was introduced by Kostant, who associated a cyclic element with the principal nilpotent and proved that it is regular semisimple. In particular, I will tell about the classification of all nilpotents giving rise to semisimple and regular semisimple cyclic elements. The results are from my joint work with V. Kac and E. Vinberg.

There is a movement in design of experiments away from the classic randomization put forward by Fisher, Cochran and others to one based on optimization. In fixed-sample trials comparing two groups, measurements of subjects are known in advance and subjects can be divided optimally into two groups based on a criterion of homogeneity or "imbalance" between the two groups. These designs are far from random. This talk seeks to understand the benefits and the costs over classic randomization in the context of different performance criterions such as Efron's worst-case analysis. In the criterion that we motivate, randomization beats optimization. However, the optimal design is shown to lie between these two extremes. Much-needed further work will provide a procedure to find this optimal designs in different scenarios in practice. Until then, it is best to randomize.

In the vector balancing problem, we are given N vectors v_1,..., v_N in an n-dimensional normed space, and our goal is to assign signs to them, so that the norm of their signed sum is as small as possible. The balancing constant of the vectors is the smallest number beta, such that any subset of the vectors can be balanced so that their signed sum has norm at most beta.

The vector balancing constant generalizes combinatorial discrepancy, and is related to rounding problems in combinatorial optimization, and to the approximate Caratheodory theorem. We study the question of efficiently approximating the vector balancing constant of any set of vectors, with respect to an arbitrary norm. We show that the vector balancing constant can be approximated in polynomial time to within factors logarithmic in the dimension, and is characterized by (an appropriately optimized version of) a known volumetric lower bound. Our techniques draw on results from geometric functional analysis and the theory of Gaussian processes. Our results also imply an improved approximation algorithm for hereditary discrepancy.

Joint work with Aleksandar Nikolov, Nicole Tomczak-Jaegermann and Kunal Talwar.