Colloquia, AY 2006-2007
Colloquium, AY 2006-2007
Tea at 3:30 pm in 1425 Stevenson Center
|September 7, 2006||
|September 14, 2006||
Quantum bluffing in entangled poker – An introduction to quantum game theory
Computers and networks that exploit the bizarre properties of quantum mechanics
will have capabilities far exceeding those of the conventional computing
environment. The encryption of data, the searching of databases, and even the
play of simple games such as on-line poker will undergo profound changes when
implemented in the quantum environment.
This is because players who communicate their strategic choices via
|September 21, 2006||
Lattices in products of trees and simple groups
The study of groups of tree automorphisms viewed as analogs of
semisimple Lie groups led to various interesting results. Studying
lattices in products of trees one encounters both properties familiar
from the case of Lie groups as well as new phenomena. The interplay
between these led to a construction, jointly with Marc Burger, of a
finitely presented torsion free simple group, this turned out also to be
the first example of a simple group which is an amalgam of two free
groups. More recently using a similar strategy (and many other ideas)
P.-E. Caprace and B. Remy have shown the simplicity of a large family
of Kac-Moody groups.
|September 28, 2006||
Compactifications of symmetric and locally symmetric spaces
Symmetric and locally symmetric spaces are special Riemannian
manifolds closely related to Lie groups. They arise naturally in many
different subjects, for example,
as the moduli spaces of elliptic curves and abelian varieties,
and the moduli spaces of positive definite quadratic forms and
their equivalence classes.
Many such natural spaces are non-compact, and an important problem is to
compactify them. In fact, a number of different compactifications
have been constructed, motivated by various applications.
In this talk I will give a survey of compactifications of both symmetric
symmetric spaces, using simple examples such as the upper half-plane
and its arithmetic quotients.
Compactifications of symmetric spaces have often been studied by methods
quite different from those those used for locally symmetric spaces.
Here we will emphasize a uniform approach to both problems.
|October 5, 2006||
Richard Thompson’s group F: around the amenability problem
We well discuss the famous problem about amenability of Richard
Thompson’s group F. This will include descriptions of the most
popular criteria of amenability and some basic properties of the
group F. Also we are going to give a survey of recent progress in
this area and discuss some approaches to the problem.
|October 12, 2006||
Alexander Volberg, Michigan State University
Scattering for Jacobi matrices and discrete Schroedinger operators,
non-linear Fourier transforms and related questions of weighted Hilbert
We introduce a certain notion of scattering, show its relations
with non-linear Fourier transform on one side and with weighted estimates
(with matrix weights) for the Hilbert transform on the other side.
|October 19, 2006||
Formal proofs in geometry
Traditional mathematical proofs are written in a way to make them easily
understood by mathematicians. Routine logical steps are omitted.
An enormous amount of context is assumed on the part of the reader.
Proofs, especially in topology and geometry, rely on intuitive arguments
in situations where a trained mathematician would be capable of
translating those intuitive arguments into a more rigorous argument.
In a formal proof, all the intermediate logical steps are supplied.
No appeal is made to intuition, even if the translation from intuition
to logic is routine. Thus, a formal proof is less intuitive, and yet
less susceptible to logical errors. It is generally considered a
major undertaking to transcribe a traditional proof into a formal proof.
In recent years, a number of fundamental theorems in mathematics have
been formally verified by computer, including the Prime Number
Theorem, the Four Color Theorem, and the Jordan Curve Theorem.
|October 26, 2006||
Algebraic statistics for computational biology
This lecture gives an introduction to a book with this title
which was published recently by Lior Pachter and myself. It concerns
interactions between algebra and statistics and their emerging applications
to computational biology. Statistical models of independence and sequence
alignment will be illustrated by means of a fictional character, DiaNA,
who rolls tetrahedral dice with face labels “A”, “C”, “G” and “T”.
|November 2, 2006||
4:10-5:00pm at SC 4327, Co-hosted by the Department of Physics and Astronomy
Hidden noncommutative geometric structure of space-time
The geometry of space-time is reconstructed from the low-energy spectrum
defined by the quarks and leptons. I show that there is a hidden
noncommutative structure and that the dynamics of the unified geometrical
theory is governed by the “Spectral Action Principle”.
November 8, 2006
4:10-5:00pm at SC 1308, Special Lecture
Zero-one laws in discrete mathematics
The fraction of n-vertex finite graphs that are connected grows to 1 as n
grows to infinity. In that sense almost all finite graphs are connected.
There are numerous results like that. Almost all graphs are Hamiltonian,
not 3-colorable, rigid, etc. Each of these results required a separate
proof. Is there a general phenomenon behind results of that sort? It
turns out that much depends on the logical form of the property in
question. In particular, every claim expressible in predicate logic is
almost surely true or almost surely false on finite structures. This
zero-one law was generalized in various directions. We will explain some
of the results.
Contact person: Mark Sapir
|November 9, 2006||
Higher-dimensional analogs of stable curves
Stable curves were introduced in the 60s by Deligne-Mumford (and
versions by Mayer, Knudsen, Grothendieck…). Stable maps is a more
recent invention of Kontsevich. They have a myriad of applications:
most notably to Gromov-Witten invariants and quantum cohomology, but
also to such diverse topics as resolutions of singularities in
positive characteristic and universal bounds for the number of
solutions of diophantine equations.
What happens if one replaces ‘curve’ in ‘stable curve or map’ by
‘higher-dimensional variety’? I will explain the current state of the
art in the theory that results.
|November 16, 2006||
Homological Mirror Symmetry and Applications
In this talk we will give an elementary introduction to Homological Mirror
Symmetry. We will discuss applications to classical problems in Algebraic
|November 23, 2006||
|November 30, 2006||
On the superrigidity of malleable actions
Please klick for the
|December 7, 2006||
A Taste of Compressed Sensing
The usual paradigm for encoding signals is based on the Shannon
sampling theorem. If the signal is broad-banded then this requires a
high sampling rate even though the information content in the signal may
be small. Compressed Sensing is an attempt to get out of this dilemma
and sample at close to the information rate.
The fact that this may be possible is embedded
in some old mathematical results in functional analysis,
geometry and approximation.
This talk will be an excursion into these topics which will focus on
the relation between the number of samples we take of a signal and
how well we can approximate the signal. It will take place in the
discrete setting for vectors in Euclidean space.
The talk should be understandable to graduate students
and non specialists.
|January 11, 2007||
Increasing and decreasing subsequences
We survey the theory of increasing and decreasing subsequences of
permutations. Enumeration problems in this area are closely related to
the RSK algorithm. The asymptotic behavior of the expected value of
the length is(w) of the longest increasing subsequence of a
permutation w of 1,2,…,n was obtained by Vershik-Kerov and (almost)
by Logan-Shepp. The entire limiting distribution of is(w) was then
determined by Baik, Deift, and Johansson. These techniques can be
applied to other classes of permutations, such as involutions, and are
related to the distribution of eigenvalues of elements of the
classical groups. We will also briefly discuss two variations of
increasing and decreasing subsequences, viz., alternating
subsequences and crossings and nestings of matchings.
|January 18, 2007||
|January 25, 2007||
On the Langlands functoriality for automorphic forms
The Langlands functorialty is a central conjecture in
the modern theory of automorphic forms, the Langlands program.
To motivate my lecture, I recall first some basic problems in number theory,
including the famous Artin Conjecture for Galois representations.
Then I will discuss the in some details the recent work on the Langlands
functoriality and applications to number theory, including for example,
the Inverse Galois Problem.
Wednesday, February 28
Poincare’s variational problem in potential theory
In a famous Acta Mathematica article, Poincare has stated
a variational principle as a heuristic basis for his work in potential
theory. Later on, Carleman has developed in 2D some of Poincare’s ideas,
but they were never put into the framework of modern mathematics. I will
show in my talk how this can be done, and what perspectives Poincare’s
programme opens today. In particular I will touch the theory of
symmetrizable linear operators and some classical aspects of function
theory related to the Beurling transform. Based on recent work with Dmitry
Khavinson and Harold S. Shapiro.
|March 8, 2007||
|March 15, 2007||
Free probability and free entropy dimension
More than 20 years ago, Voiculescu found a noncommutative
probabalistic notion called freeness that corresponds to the situation
of words in a free group. The theory based on this definition is called
free probability theory and in this theory free products play a role
analogous to the usual (Cartesian) product of spaces in the classical
probabalistic theory of independence. The parallels with classical
probability theory are far-reaching and surprising. After introducing
freeness, we will describe one of its fundamental examples, which is the
asymptotic behavior of random matrices as the matrix size grows without
bound. We’ll also describe the use of freeness to investigate von
Neumann algebras, including applications of the related quantity, free
|March 22, 2007||
Diffusion methods in image processing
One of the most famous approaches to the denoising and contrast enhancing of
blurred images is based on the famous Perona – Malik equations. Unfortunately
these equations are ill posed. Thus no mathematical justification for numerical
algorithms based on them is possible.
Since the Perona – Malik technique seems to produce excellent results,
been proposed many modifications of the underlying equations with the aim to
obtain, on the one hand side, a sound mathematical theory and, on the other
hand side, to preserve the desirable features of the Perona – Malik equations.
However, as shall be explained in this talk, these modifications result in an
unavoidable smearing of sharp edges and, consequently, in an undesirable loss
In our talk we shall present a new approach to the Perona – Malik equations
which does not have these shortcomings. In contrast to the widely used space
regularization we propose a time regularization technique. It allows for a sound
mathematical theory as well, but avoids blurring of sharp edges.
We shall addresses a general mathematical audience without any prior knowledge
of image processing, explain the diffusion theoretical approach in elementary
terms, and illustrate the basic ideas by numerical experiments.
|March 29, 2007||
The Kadison-Singer problem in Mathematics and Engineering
We will see that the 1959
Kadison-Singer Problem in C*-algebras is
equivalent to fundamental unsolved problems
in a dozen areas of research in pure mathematics,
applied mathematics and engineering. This
gives all these research areas common ground on which
to interact as well as explaining
why each of them has volumes of literature
on their respective problems without a satisfactory
resolution. We will
look at some of the equivalences of KS in
operator theory, Banach space theory, harmonic
analysis, and applied math/engineering.
|April 5, 2007||
|April 12, 2007||
Discrete versus continuous in analysis
Analysis might be defined as the study of functions, and function spaces.
Such a broad definition encompasses real-valued functions of a real
variable, maps between Banach spaces of operators, and much, much more.
While functions can be defined on discrete spaces (such as functions on the
natural numbers, otherwise known as sequences), most analysts seem to
regard the discrete aspects of the subject as being but stepping stones to
the ultimate truth, which surely lies in the continuous realm. One might
wonder, for instance, at the scant attention paid to difference equations
in the undergraduate curriculum, compared to differential equations.
In applied harmonic analysis, classical tools like the Fourier transform
that apply to analog signals must be discretized in order to apply to
digital signals. This has stimulated research on discrete versions of
classical function spaces. For example I recently obtained new results on
the “analysis” (continuous-to-discrete) and “synthesis”
(discrete-to-continuous) operators on Lebesgue and Hardy spaces. I’ll
describe this work near the end of the talk, after first exploring some
continuous/discrete dichotomies drawn from differential equations,
probability, mathematical physics and elementary geometry.
|April 19, 2007||
The Poincaré duality theorem and its converse
The Poincaré duality theorem established isomorphisms between the
homology and cohomology of a topological manifold. The duality
isomorphisms are the fundamental algebraic topology consequences of a
space being locally Euclidean. The surgery theory developed over the
last 40 years characterizes the homotopy types of manifolds of
dimension >4 in terms of the topological K-theory of vector bundles and
the algebraic L-theory of quadratic forms. The talk will describe a
simplicial version of surgery theory, using a combinatorial version of
sheaf theory to obtain a converse of the duality theorem in dimension
>4: a simplicial complex is homotopy equivalent to a topological manifold
if and only if it has sufficient Poincaré duality.
Colloquium Chair (2006-2007): Gieri Simonett
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