Georgia Institute of Technology College of Sciences

Man has grappled with the meaning and utility of randomness for centuries. Research in the Theory of Computation in the last thirty years has enriched this study considerably. I'll describe two main aspects of this research on randomness, demonstrating respectively its power and weakness for making algorithms faster. I will address the role of randomness in other computational settings, such as space bounded computation and probabilistic and zero-knowledge proofs.

Optimization problems involving sparse vectors or low-rank matrices are of great importance in applied mathematics and engineering. They provide a rich and fruitful interaction between algebraic-geometric concepts and convex optimization, with strong synergies with popular techniques like L1 and nuclear norm minimization. In this lecture we will provide a gentle introduction to this exciting research area, highlighting key algebraic-geometric ideas as well as a survey of recent developments, including extensions to very general families of parsimonious models such as sums of a few permutations matrices, low-rank tensors, orthogonal matrices, and atomic measures, as well as the corresponding structure-inducing norms.Based on joint work with Venkat Chandrasekaran, Maryam Fazel, Ben Recht, Sujay Sanghavi, and Alan Willsky.

Problem: describe the asymptotic behavior of the coefficients a_{ij} of the Taylor series for 1/Q(x,y) where Q is a polynomial. This problem is the simplest of a number of such problems arising in analytic combinatorics whose answer was not until recently known. In joint work with J. van der Hoeven and T. DeVries, we give a solution that is completely effective and requires only assumptions that are met in the generic case. Symbolic algebraic computation and homotopy continuation tools are required for implementation.

Motivated by a problem in the synthesis of nanowires, a sequential space filling design, called Sequential Minimum Energy Design (SMED), is proposed for exploring and searching for the optimal conditions in complex black-box functions. The SMED is a novel approach to generate designs that are model independent, can quickly carve out regions with no observable nanostructure morphology, allow for the exploration of complex response surfaces, and can be used for sequential experimentation. It can be viewed as a sequential design procedure for stochastic functions and a global optimization procedure for deterministic functions. The basic idea has been developed into an implementable algorithm, and guidelines for choosing the parameters of SMED have been proposed. Convergence of the algorithm has been established under certain regularity conditions. Performance of the algorithm has been studied using experimental data on nanowire synthesis as well as standard test functions.(Joint work with V. R. Joseph, Georgia Tech and T. Dasgupta, Harvard U.)