See  Fermat's Last Tango  for more details.

Abstract: Black holes are among the more striking predictions of  Einstein's theory of General Relativity. 
Not only do they lead to new classical physics but they also seem to hold the key to the unknown 
physics that lies in the intersection between Quantum physics and General Relativity. I will describe 
the aspects of black hole physics 
that lead to this conclusion and will give a particular 
Ansatz that sheds some light into this new region of physics.

Abstract: If we apply a generalized cohomology 
theory to the classifying space of a 
group, we obtain a ring whose structure reveals
something about the group.  As a first approximation we can consider the variety defined by this ring. 
The first result in this direction was the work of Quillen
in 1971, describing the
variety defined by the mod p cohomology in terms of the
elementary abelian subgroups.  A number of such results for other cohomology theories 
have now been proven.  We shall review
these and discuss in detail the case of connective K-theory, which provides a deformation from 
representation theory to cohomology (at the level of varieties) and speculate on the
theories which interpolate between them.

Abstract:  It is often a complicated matter to estimate the 
the C*-norm (the usual Hilbert-space operator-norm) 
of the sum of two complex  matrices.   Nevertheless, 
an ultimate answer (without hard computation) can be sought 
for the best bound of the norm of T = A + B where 
A and B are (non-commuting ) normal matrices with 
known eigenvalues. Moreover, the main result can be 
extended to cover  the case of the sum of two non-normal
matrices.

Abstract: In this talk I will describe a new
family of invariants for graphs, which are similar
to the classical homotopy invariants of a topological
space.  These invariants can also be applied to
simplicial complexes.  I will describe
applications of combinatorial homotopy theory
of graphs to the study of
hyperplane arrangements, and to the study of
social, biological, and communications networks.
The talk is accessible to students,
who are encouraged to attend.

Abstract: Already known in ancient Greece 
as "Dido's Problem", this is the following 
task: Among all simple closed planar curves 
of fixed length 
 find the one enclosing the largest area! 
That the solution is the circle was rigorously 
shown to be true only at the end of the 19th
century. Even today there are a number of 
open questions related to this famous problem in the geometric calculus of variations.

Abstract: In this talk, I will describe an unusual 
approach to analyzing the ways in which artists use 
color, contrast and luminosity to create beautiful art. 
 Using this approach,  I'll show you the evidence 
for a hidden mathematical aesthetic held in 
common by great artists across many centuries. 
We'll look at many striking pieces of art by artists 
ranging from Tintoretto to Picasso and see that the 
"better" artists stay very close to what is called 
a "lognormal" curve. The lognormal family of models 
depends on two parameters and we'll see that the 
values of these parameters are fairly constant
for the paintings of a particular artist.  This 
observation will allow us to construct a 
two-dimensional "map" of fine art on which 
we can clearly see each artist's special niche. 
At the end of the talk,  I'll show you some 
of my own artwork created using the mathematics 
of dynamical systems.  If you have a piece of art that you would like to have analyzed, 
please bring it to the talk on a disc.

Abstract: Black holes are among the more striking predictions of  Einstein's theory of General Relativity. 
Not only do they lead to new classical physics but they also seem to hold the key to the unknown 
physics that lies in the intersection between Quantum physics and General Relativity. I will describe 
the aspects of black hole physics 
that lead to this conclusion and will give a particular 
Ansatz that sheds some light into this new region of physics.

Abstract: If we apply a generalized cohomology 
theory to the classifying space of a 
group, we obtain a ring whose structure reveals
something about the group.  As a first approximation we can consider the variety defined by this ring. 
The first result in this direction was the work of Quillen
in 1971, describing the
variety defined by the mod p cohomology in terms of the
elementary abelian subgroups.  A number of such results for other cohomology theories 
have now been proven.  We shall review
these and discuss in detail the case of connective K-theory, which provides a deformation from 
representation theory to cohomology (at the level of varieties) and speculate on the
theories which interpolate between them.

Abstract: Let n be a positive integer, w a group word. Consider the class of all groups G satisfying the identity w^n = 1 and having the verbal subgroup w(G) locally finite. We show that in many cases this is a variety.

Abstract: In analogy with forming a subgroup by closing a subset under the group operation, logicians form an 
elementary submodel of a model of set theory (a 
collection of sets satisfying the axioms of set theory) by 
closing under witnesses for existential quantifiers. Such an elementary submodel reflects the properties of the model 
that can be expressed by quantifying over elements of it. 
Given such an elementary submodel M containing a 
topological space (X , T), one forms the topological space 
X_M by endowing X \cap M with the topology generated 
by {U \cap M : U \in  (T \cap M)}.  Surprisingly, if X_M is homeomorphic to the real line, then X = X_M. One can 
think of this as saying that the logic has captured the
topology.  For the rationals, the corresponding assertion 
is false.The corresponding problem for the irrationals is unsolved, but a positive answer is consistent with the usual axioms of set theory. Except for the consistency proof (the details of which will be omitted) the methods are reasonably accessible: classical topology of the real line and elementary consequences of the definition of ''elementary submodel''.
 

Abstract: One of the typical tools in the classification of complex manifolds is the 
study of mapping discs of a given radius into 
a complex manifold, while specifying 
the initial position and velocity. The fact
that this radius is possibly bounded is purely complex analytic in nature,
and is based on the classical Schwarz Lemma, a.k.a. the Maximum Principle.
It is expected that for most complex manifolds, there is an absolute bound
on the aforementioned radius, depending only 
on the initial position and velocity. 
However, there are many examples where 
there is no bound, such as on complex Lie groups 
and Homogeneous spaces, to name a few.
We will review a couple (of the many) classical motivations behind consideration of maps 
of discs, and then pass to a limit and consider
maps of Euclidean spaces into complex manifolds. 
Most of the rest of our talk will be about the presence
of such maps in the theory of complex dynamics,
through the theorems of S. Sternberg and J.-P. Rosay-W. Rudin (and generalizations of those 
theorems due to the speaker), and recent results
of M. Jonsson and the speaker.  We will also discuss the new theory of random iteration, which is being 
currently explored by J.-E. Fornaess and the 
speaker.  One of the consequences of this theory 
is a new phenomenon in the classification problem 
for complex manifolds, the so-called Short C^k.
The talk will conclude with a discussion of the open problem of compactifications of Euclidean spaces. 
We will talk about K. Kodaira's
results in 2 dimensions, and about Y.-T. Siu's 
program and a new example
of J.-M. Hwang and the speaker which shows why the first step is so hard.

Abstract: Eilenberg and MacLane defined categories
in order to define functors, which compare categories. They defined functors in 
order to define natural transformations, which 
compare functors. What if one has two natural
transformations and wants to compare them? 
And then how does one compare two comparisons? These questions inevitably lead to the 
idea of an n-category, which has objects, maps, 
maps between maps, maps between maps
between maps, and so on. There are many compelling motivations for studying such things, and definitions have been proposed by people working 
not only in category theory but also in computer 
science, logic, algebraic topology, algebraic geometry, and mathematical physics. This looks 
like it will become an important area of mathematics. 
People proposing definitions are scattered, some 
of the main centers being Sydney, Nice, Cambridge, and Montreal. There are at least a dozen 
definitions and very few comparisons among
them. I will give an utterly superficial 
overview of this emerging field. In particular, I will 
describe a unification program aimed at avoiding the fate of the tower of Babel. Time permitting, I will 
also motivate and explain my own recent definition. 

Abstract: YoungBin Choe, James Oxley, Alan Sokal and I have been 
collaborating on reasearch generalizing some results in the 
theory of (linear,time-invariant) electrical networks.  As 
background, I will review brief proofs of Kirchhoff's 
formula for the effective admittance of a network, 
and of the physically sensible fact that if every branch
dissipates energy then the whole network dissipates energy. 
These results motivate the investigation of polynomials with the "half-plane property" -- if each variable is given a value with
positive real part, then the polynomial does not vanish. 
Such polynomials have a surprising combinatorial structure 
which generalizes the graph theory underlying classical
network theory.

Abstract: Information is always stored in a physical medium and manipulated by physical processes. Therefore the laws of 
physics dictate the capabilities and limitations of any information
processing device.  Quantum Information Processing, or Quantum Computation, treats information processing in a quantum 
mechanical framework.
The quantum features of nature lead to qualitatively different and
apparently more powerful models of computation and 
communication. For example, quantum computers could factor 
large numbers efficiently (this would compromise most of 
public-key cryptography in use today). No known classical 
algorithm can do this. As another example, quantum
information is intrinsically sensitive to eavesdropping, a feature 
that makes it very suitable for information security protocols.
In this talk I will introduce quantum computation as a natural
generalization of classical computation, and survey some 
interesting examples of how to exploit the quantum features
of information.

Abstract: We describe current proposals for the structure of spacetime at Planck
scale. These are based on the predictions for the Planck scale geometry
that resulted from the quantization of general relativity, and incorporate
features such as discreteness, causality, locality and local unitarity. 
These quantum spacetimes, or spin foams, are modeled as 2-complexes
labeled by group representations and so they are a departure from
classical manifold spacetimes.  It is then an open question how the
physical content of general relativity translates in this new language. 
We show how the requirement that the theory is cosmological (that is, it
consistently describes the entire universe) means that classical
observables should be varying sets and obey a Heyting rather than a
Boolean algebra.  We then discuss the same requirement, of the theory
being cosmological, for the full quantum spacetimes.