Georgia Institute of Technology College of Sciences

A2 conjecture asked to have a linear estimate for simplest weighted singular operators in terms of the measure of goodness of the weight in question.We will show how the paradigm of non-homogeneous Harmonic Analysis (and especially its brainchild, the randomized BCR) was used to eventually solve this conjecture.

Understanding the computations performed by neuronal circuits requires characterizing the strength and dynamics of the connections between individual neurons. This characterization is typically achieved by measuring the correlation in the activity of two neurons through the computation of a cross-correlogram or one its variants. We have developed a new measure for studying connectivity in neuronal circuits based on information theory, the incremental mutual information (IMI). IMI improves on correlation in several important ways: 1) IMI removes any requirement or assumption that the interactions between neurons is linear, 2) IMI enables interactions that reflect the connection between neurons to be differentiated from statistical dependencies caused by other sources (e.g. shared inputs or intrinsic cellular or network mechanisms), and 3) for the study of early sen- sory systems, IMI does not require that the external stimulus have any specific properties, nor does it require responses to repeated trials of identical stimulation. We describe the theory of IMI and demonstrate its utility on simulated data and experimental recordings from the visual system.

In this presentation, we show a significant role that symmetry, a fundamental concept in convex geometry, plays in determining the power of robust and finitely adaptable solutions in multi-stage stochastic and adaptive optimization problems. We consider a fairly general class of multi-stage mixed integer stochastic and adaptive optimization problems and propose a good approximate solution policy with performance guarantees that depend on the geometric properties such as symmetry of the uncertainty sets. In particular, we show that a class of finitely adaptable solutions is a good approximation for both the multi-stage stochastic as well as the adaptive optimization problem. A finitely adaptable solution specifies a small set of solutions for each stage and the solution policy implements the best solution from the given set depending on the realization of the uncertain parameters in the past stages. To the best of our knowledge, these are the first approximation results for the multi-stage problem in such generality. (Joint work with Vineet Goyal, Columbia University and Andy Sun, MIT.)

Knot contact homology (KCH) is a combinatorially defined topological invariant of smooth knots introduced by Ng. Work of Ekholm, Etnyre, Ng and Sullivan shows that KCH is the contact homology of the unit conormal lift of the knot. In this talk we describe a monodromy result for knot contact homology,namely that associated to a path of knots there is a connecting homomorphism which is invariant under homotopy. The proof of this result suggests a conjectural interpretation for KCH via open strings, which we will describe.