# Nonlinear Diffusions and an Application to Image Processing.

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An introduction to questions and ideas of nonlinear partial

differential equations will be given. Nonlinear diffusions and an

application to image processing will be emphasized.

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# Classifying Ergodic Measure Preserving Transformations

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Many concrete deterministic dynamical systems exhibit apparently random

behaviour. This puzzle is studied my finding a time invariant probability

measure and discussing the statistical phenomenon using this measure. In

this way various systems (e.g. given by PDE's) can be said to be

"completely random" or "completely deterministic".

This leads to the project of classifying invariant probability measures.

The ergodic decomposition theorem shows that the basic building blocks of

these measures are the ergodic measures, which form a dense G_\delta set.

The equivalence relation of isomorphism is given by a Polish group action.

Thus the tools of descriptive set theory directly apply and one can show

that the action is "turbulent" and complete analytic. This precludes any

kind of recognizable classification.

This is joint work with Dan Rudolph and Benjy Weiss.

# Mean value theorems and local regularity theorem for Ricci flow

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We shall show a general mean value theorem on Riemannian manifold and how it leads to new monotonicity formulae for evloving metrics. As an application we show a local regularity theorem for Ricci flow.

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# NO MEETING THIS WEEK DUE TO VETERAN"S DAY

# Ranks of elliptic curves

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Abstract: In this lecture we will introduce elliptic curves

and some of the fundamental questions about them. The rank

of an elliptic curve is a measure of the number of solutions

of the equation which defines the curve. In recent years

there has been spectacular progress in the theory of elliptic

curves, but the rank remains very mysterious. Even basic

questions such as how to compute the rank, or whether the rank

can be arbitrarily large, are not settled. In this talk we

will survey what is known, as well as what is conjectured but

not known, about ranks of elliptic curves.

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In recent years, I have been thinking about

Mathematical Visualization, and developing a

program that does high quality, customized

visualizations of mathematical objects and

processes. I will demonstrate this program

and discuss some of the interesting and

unexpected ways that certain mathematical

theorems turned out to be just what was

needed to find fast and efficient algorithms

that solve a number of difficult rendering

problems. (Some of these rendering problems

involve seeing 3D objects in stereo, and I

will bring along the red/green glasses needed

for the demonstation.)