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.
In this talk I will discuss some of my recent work on Probability Theory and its applications. The topics will include the rate of dispersion of oil slicks under turbulent random flow, statistical properties of randomly growing surfaces and a model for the behavior of magnetic fields on stars.
In this talk, I will use the Korteweg-de
Vries equation, non-linear Schrodinger equation, and
the sine-Gordon equation as models to explain some
remarkable properties of a certain class of non-linear
wave equations, the so called "soliton equations".
Some relations to differential geometry will be
discussed. I will also use Richard Palais'
3D-XplorMath Visualization Computer Program to help us
"see" some of these properties.
I will give an overview of my work on some problems founds at the
interface between mathematical and life sciences. This includes
modeling of initiation and development of cancer (viewed as an
evolutionary problem), some problems in biophysics (studying the
dynamics of RNA transcription); learning theory, and the evolution of
natural languages.
Among the nicest spaces in topology and geometry are
manifolds, i.e. spaces which locally look like an open ball in R^n. If X
is such a manifold and G is a finite group acting on it, the usual
quotient X/G in general will not be a manifold anymore (if the G-action
has stabilizers). The theory of orbifolds is a different approach to
taking quotients, leading to objects which behave as if they were
manifolds, but also have some surprising properties defying our
intuition.
