Georgia Institute of Technology College of Sciences

The past 10 years have seen a confluence of research in sparse approximation amongst computer science, mathematics, and electrical engineering. Sparse approximation encompasses a large number of mathematical, algorithmic, and signal processing problems which all attempt to balance the size of a (linear) representation of data and the fidelity of that representation. I will discuss several of the basic algorithmic problems and their solutions, including connections to streaming algorithms and compressive sensing.

Given points $z_1,\ldots,z_n$ on a finite open Riemann surface $R$ and complex scalars $w_1,\ldots,w_n$, the Nevanlinna-Pick problem is to determine conditions for the existence of a holomorphic map $f:R\to \mathbb{D}$ such that $f(z_i) = w_i$. In this talk I will provide some background on the problem, and then discuss the extremal case. We will try to discuss how a method of McCullough can be used to provide more qualitative information about the solution. In particular, we will show that extremal cases are precisely the ones for which the solution is unique.

Many vertebrate motor and sensory systems "decussate," or cross the midline to the opposite side of the body. The successful crossing of millions of axons during development requires a complex of tightly controlled regulatory processes. Since these processes have evolved in many distinct systems and organisms, it seems reasonable to presume that decussation confers a significant functional advantage - yet if this is so, the nature of this advantage is not understood. In this talk, we examine constraints imposed by topology on the ways that a three dimensional processor and environment can be wired together in a continuous, somatotopic, way. We show that as the number of wiring connections grows, decussated arrangements become overwhelmingly more robust against wiring errors than seemingly simpler same-sided wiring schemes. These results provide a predictive approach for understanding how 3D networks must be wired if they are to be robust, and therefore have implications both regenerative strategies following spinal injury and for future large scale computational networks.

Here we will introduce the basic definitions of bordered Floer homology. We will discuss bordered Heegaard diagrams as well as the algebraic objects, like A_\infinity algebras and modules, involved in the theory. We will also discuss the pairing theorem which states that if Y = Y_1 U_\phi Y_2 is obtained by identifying the (connected) boundaries of Y_1 and Y_2, then the closed Heegaard Floer theory of Y can be obtained as a suitable tensor product of the bordered theories of Y_1 and Y_2.Note the different time and place!This is a 1.5 hour talk.