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.
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.
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.
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.)
In Calculus and PDE, people study rather complicated functions, equations on
relatively simple spaces (real line or n-dim Euclidean spaces). On the other hand,
in topology, people study complicated spaces with relatively simple function theory
on them. We are going to introduce a kind of calculus that takes the underlying
topological space into account. Thus we can see how topology interacts with calculus
naturally. The kind of new Calculus is called differential geometry. From this point
of view, Calculus and topology are finally unitfied into differential geometry.
Number theory and algebraic geometry have numerous applications,
including to cryptography. Cryptography is concerned with encrypting
and decrypting secret messages. This talk will give an elementary
introduction to elliptic curve cryptography and pairing-based
cryptography, and will discuss some interesting open problems. Only
undergraduate algebra will be assumed.
