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Friday, October 30, 2009 - 15:00 ,
Location: Skiles 269 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

In this talk I will discuss a generalizations and/oo applications of bordered Floer homology. After reviewing the basic definitions and constructions, I will focus either on an application to sutured Floer homology developed by Rumen Zarev, or on applications of the theory to the knot Floer homology. (While it would be good to have attended the other two talks this week, this talk shoudl be independent of them.) This is a 2 hour talk.

Series: SIAM Student Seminar

After a brief introduction of the theory of orthogonal polynomials, where we touch on some history and applications, we present results on Müntz orthogonal polynomials. Müntz polynomials arise from consideration of the Müntz Theorem, which is a beautiful generalization of the Weierstrass Theorem. We prove a new surprisingly simple representation for the Müntz orthogonal polynomials which holds on the interval of orthogonality, and in particular we get new formulas for some of the classical orthogonal polynomials (e.g. Legendre, Jacobi, Laguerre). This allows us to determine the strong asymptotics on the interval, and the zero spacing behavior follows. We also look at the asymptotic behavior outside the interval, where we apply the method of stationary phase.

Series: Joint ACO and ARC Colloquium

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.

Series: Geometry Topology Seminar

We recall the Schur Weyl duality from representation theory and show how this can be applied to express the colored Jones polynomial of torus knots in an elegant way. We'll then discuss some applications and further extensions of this method.

Series: Analysis Seminar

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.

Series: Research Horizons Seminar

The Soul Theorem, proved by Cheeger and Gromoll forty
year ago, reveals a beautiful structure of noncompact complete
manifolds of nonnegative curvature. In the talk I will sketch
a proof of the Soul Theorem, and relate it to my current work
on moduli spaces of nonnegatively curved metrics.

Series: Mathematical Biology Seminar

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.

Joint ACO/DOS - Approximability of Combinatorial Problems with Multi-agent Submodular Cost Functions

Series: Other Talks

Organizer: Daniel Dadush, ACO Student, ISyE

Applications in complex systems such as the Internet have spawned recent interest in studying situations involving multiple agents with their individual cost or utility functions. We introduce an algorithmic framework for studying combinatorial problems in the presence of multiple agents with submodular cost functions. We study several fundamental covering problems (Vertex Cover, Shortest Path, Perfect Matching, and Spanning Tree) in this setting and establish tight upper and lower bounds for the approximability of these problems. This talk is based on joint work with Gagan Goel, Chinmay Karande and Wang Lei. This is a joint ACO/DOS seminar, so please come a little early for pizza and refreshments sponsored by ACO.

Wednesday, October 28, 2009 - 10:00 ,
Location: Skiles 255 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

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.

Series: PDE Seminar

We first discuss blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations,
where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. In the second part of the talk we discuss some observations on the Euler equations with symmetries, which shows that the point-wise behavior of the pressure along the flows is closely related to the blow-up of of solutions.