# Transition from the network of the thin wave guides to limiting graphs

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# Computational surface partial differential equations

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Partial differential equations on and for evolving surfaces occur in many applications.

For example, traditionally they arise naturally in fluid dynamics and materials

science and more recently in the mathematics of images.

In this talk we describe computational approaches to the formulation and

approximation of transport and diffusion of a material quantity on an

evolving surface.

We also have in mind a surface which not only evolves in the normal direction

so as to define the surface evolution but also has a tangential velocity

associated with the motion of material points in the surface which advects material

quantities such as heat or mass.This is joint work with G. Dziuk

# Integral Geometry and the X-ray transform

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n this talk I discuss some principal problems in

Reconstructive Integral Geometry with emphasis on the X-ray transform

with recent results on range problems and inversion formulas.

# Universality for mathematical and physical systems

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All physical systems in equilibrium obey the laws of

thermodynamics. In other words, whatever the precise nature of the

interaction between the atoms and molecules at the microscopic level,

at the macroscopic level, physical systems exhibit universal behavior in

the sense that they are all governed by the same laws and formulae of

thermodynamics.

The speaker will recount some recent history of universality ideas in

physics starting with Wigner's model for the scattering of neutrons

off large nuclei and show how these ideas have led mathematicians to

investigate universal behavior for a variety of mathematical systems.

This is true not only for systems which have a physical origin, but also

for systems which arise in a purely mathematical context such as the

Riemann hypothesis, and a version of the card game solitaire called

patience sorting.

# Gamma Rhythms of the Nervous System: From Biophysics to Cognition

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# The Distribution Functions of Random Matrix Theory

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It is now believed, but proved only in a few cases, that the distribution

functions

of random matrix theory are universal for a wide class of stochastic

problems in combinatorics,

growth processes, and statistics. These developments will be surveyed.

No prior knowledge

of random matrix theory will be assumed.

# Compressive Sampling

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Conventional wisdom and common practice in acquisition and

reconstruction of images from frequency data follows the basic

principle of the Nyquist density sampling theory. This principle

states that to reconstruct an image, the number of Fourier samples we

need to acquire must match the desired resolution of the image, i.e.

the number of pixels in the image.

This talk introduces a newly emerged sampling theory which shows that

this conventional wisdom is inaccurate. We show that perhaps

surprisingly, images or signals of scientific interest can be

recovered accurately and sometimes even exactly from a limited number

of nonadaptive random measurements. In effect, the talk introduces a

theory suggesting "the possibility of compressed data acquisition

protocols which perform as if it were possible to directly acquire

just the important information about the image of interest." In other

words, by collecting a comparably small number of measurements rather

than pixel values, one could in principle reconstruct an image with

essentially the same resolution as that one would obtain by measuring

all the pixels, a phenomenon with far reaching implications.

The reconstruction algorithms are very concrete, stable (in the sense

that they degrade smoothly as the noise level increases) and

practical; in fact, they only involve solving convenient convex

optimization programs. If time allows, I will discuss connections

with other fields such as statistics and coding theory.

# Estimation and Prediction with HIV Treatment Interruption Data

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We consider longitudinal clinical data for HIV patients undergoing treatment interrupt

ions. We use a nonlinear dynamical mathematical model in attempts to fit individual pa

tient data. A statistically-based censored data method is combined with inverse proble

m techniques to estimate dynamic parameters. The predictive capabilities of this appro

ach are demonstrated by comparing simulations based on estimation of parameters using

only half of the longitudinal observations to the full longitudinal data sets.

# New Periodic Orbit of the Classical N-Body Problem

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Since the discovery in 1993 of the figure-8 orbit by Cris Moore, a large number of periodic orbits for equal n masses have been found having beautiful symmetries and topologies. Most of these orbits are either planar or have been obtained from perturbation of planar orbits.

Recently Moore and I have found also a number of new three-dimensional periodic orbits of this kind which have cubic symmetries. We found these orbits by symmetry considerations, and by minimizing numerically the action integral directly as a function of the Fourier coefficients for the periodic orbit coordinates. I will review some of the early history of periodic orbits, discuss our method, and present video animations of recent results.