Colloquia, AY 2005-2006

Colloquium, AY 2005-2006

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted

Tea at 3:30 pm in 1425 Stevenson Center


September 22, 2005 

Paula Cohen, Texas A&M, College Station

Subvarieties of Shimura varieties, special values of classical
functions, and monodromy

Inspired by Hilbert’s 7th problem, Siegel (1932) and Schneider
(1937) obtained the first significant results about the transcendence of
periods of doubly periodic functions and special values of modular
functions. In particular, they found a relation to the class fields
occurring in algebraic number theory. Siegel formulated similar problems
for G-functions, a special case of which is the classical hypergeometric
function. The modern development of this circle of ideas was made possible
by the work of Alan Baker and its outgrowth. In our lecture we
focus on recent results, for example, the characterization of points at
which hypergeometric functions take algebraic values. We describe the
surprising role, first noticed by Wolfart, played by non-arithmetic
monodromy groups acting on the complex ball. We also show how such
problems are related to questions on subvarieties of Shimura varieties.
The lecture will be self-contained and accessible to a general audience.

Contact person: Dietmar Bisch

October 6, 2005 

Ian H. Sloan, University of New South Wales, Sidney

Numerical integration in high dimensions – lifting the curse of

Numerical integration in high dimensions confronts us with the curse of
dimensionality — the number of function values needed to obtain an
acceptable approximation can grow exponentially in the number of dimensions
d. The exponential increase is clearly inevitable with any form of product
integration rule, and for many theoretical settings is now known to be
unavoidable no matter how the integration rule is chosen.

It has been known since 1998 that the curse of dimensionality can in
principle be overcome within the “weighted Sobolev space” setting introduced
by Sloan and W\’ozniakowki, if the “weights” that describe the behaviour
with respect to different variables satisfy a certain (necessary and
sufficient) condition. In that work it was show that, under the appropriate
condition on the weights, there exist integration rules for which the
“worst-case error” is bounded independently of d. That 1998 result was
non-constructive, giving no clue as to how we might construct “good”
integration rules. More recently it has been shown that “good” rules can be
found within the much smaller class of (shifted) lattice rules, and even more
recently that good rules can be constructed one component at a time.

This talk will review these developments, from early existence proofs and
non-constructive methods to recent fast constructions of good integration
rules in hundreds of dimensions, that may use hundreds of thousands of sample

Contact person: Ed Saff

October 13, 2005 

William B. Johnson, Texas A&M, College Station

A survey of non linear Banach space theory

There are three reasonable notions of geometric equivalence for
metric spaces: Lipschitz equivalence, uniform equivalence, and
Gromov’s notion of coarse equivalence (which has recently
attracted interest from geometric analysts because of its relation
to the Novikov and Baum-Connes conjectures). I’ll survey what is
known about these types of equivalences when at least one of the
metric spaces is a Banach space. When both spaces are Banach spaces,
a fundamental question is: When does the existence of one of these
non linear equivalences between the spaces imply the existence of a
linear equivalance (i.e., an isomorphism)?
If there is time I’ll also discuss the recently introduced concept of
Lipschitz quotient maps, which are closely related to the non collapsing maps
studied by David and Semmes.
The talk is suitable for graduate students as well as faculty.

Contact person: Guoliang Yu

October 27, 2005 

Shmuel Weinberger, University of Chicago

Homology manifolds and their resolution

Homology manifolds originally were considered as an abstraction (the
least you need for Poincare duality) and as a source of counterexamples.
However, they are now important objects in their own right and fill in
gaps in the theory of manifolds. This talk is mainly about the
connections between manifold theory and homology manifolds, a conjectural
picture and its heuristic evidence, and, finally some of the evidence
that the hard-headed like: partial results.

Contact person: Bruce Hughes

November 3, 2005 

Michael Lacey, Georgia Tech, Atlanta

Convergence of Fourier Series: Past, Present, Future

Carleson’s celebrated Theorem on the pointwise convergence of Fourier
series is the topic of this talk. We will review the statement,
and explain why the proof is hard, as well as why
one might want to know some elements of the proof. We conclude
with a brief description of related results, and some difficult
questions that remain unanswered.

Contact person: Dechao Zheng

November 10, 2005 

Pawel Idziak, University of Cracow, Poland

On-line algorithms for graphs and posets

Traditional design and analysis of algorithms assumes that the entire
information of the input data is available for us before the algorithm
is executed. This assumption is not met in many situations,
where the complete knowledge of the entire input is not known in
Examples come from problems like:
real-time systems,
routing in communication networks,
scheduling tasks on servers,
paging in a virtual memory,
dynamic storage allocation, etc.

In these situations algorithms have to make their decision based only
on an accessible piece of information.
Data are delivered during the performance of the algorithm,
while the algorithm cannot change its earlier decisions.
Thus it often happens that a decision that seems to be correct at the
moment leads to a solution that is far from being optimal in the future.

On-line algorithms are supposed to deal with this kind of problems
in a way that is as close to an optimal solution as possible.
Thus, from a mathematical point of view they may be considered
as two persons games: between Spoiler and an Algorithm.
Spoiler presents data in the worst possible (from the Algorithm’s point
view) way, while Algorithms try to find a solution that is not far away
from an optimal off-line solution.

In this talk several on-line problems
concerning graph coloring and poset chain covering are discussed.

Contact person: Ralph McKenzie

November 24, 2005 

Thanksgiving Break

December 1, 2005 

Guido Weiss, Washington University, St. Louis

A general
description of Wavelets for those not in this field

I plan to to give a talk that describes the field
of Wavelets as a pure mathematical subject and its properties
that make it a good source for applications in many fields in
science and engineering. I strongly believe that this subject
is a very beautiful mathematical field. I hope to convince a
rather wide audience of this. Students, Engineers, Scientists
and Mathematicians having little background in Analysis should
be able to understand the message I will try to give.

Contact person: Akram Aldroubi


January 12, 2006 

Yanqin Fan, Department of Economics, Vanderbilt University

Sharp Correlation Bounds and Their Applications

In this paper, we establish asymptotic properties, including the
consistency and asymptotic
normality, of nonparametric estimators of the sharp bounds on the
correlation between two random variables.
We demonstrate both theoretically and numerically that the
sharp bounds may
differ from the traditionally used bounds [-1,1] and the nonparametric
estimators of the sharp
bounds shed light on the strength of the type of dependence, linear or
nonlinear, between two
random variables. To facilitate inference on the true sharp bounds, we
provide easy-to-compute
estimators of the asymptotic variances of the nonparametric estimators of
the sharp bounds.
Using the sharp correlation bounds on the unobserved covariates, we derive
sharp bounds on
the correlation of durations in bivariate hazard rate models with
unobserved heterogeneity and
the correlation of dependent variables in bivariate log-linear regression
models with unobserved
covariates. These results provide insight on the selection of distributions
of the unobserved
heterogeneity in bivariate hazard rate models and unobserved covariates in
log-linear regression

Contact person: Gieri Simonett

January 19, 2006 

Frank Morgan, Williams College

Double Bubbles and Gauss Spaces

The Ancient Greeks claimed and Schwarz (1884) proved that a round
spherical “soap bubble” provides the least-perimeter way to enclose a
given volume of air. Our Double Bubble Theorem (Annals of Math, 2002)
says that the familiar double soap bubble provides the
least-perimeter way to enclose and separate two given volumes of air.
I’ll also discuss extensions to other spaces, many by students, and
open questions.

Featured today will be Gauss space, Euclidean space with
Gaussian density. Despite the rotational symmetry of Gauss space, the
best single bubble is not round, but another familiar shape. Gauss
space is the model example of the important “new” category of
manifolds with density.
No special prerequisites: undergraduate math majors welcome.

Contact person: Doug Hardin

January 26, 2006 

George Metcalfe,
Vienna University of Technology

How to make your algebras dense: a proof-theoretic approach

Proof theory provides an algorithmic way of representing and reasoning
about ordered algebraic structures. In this talk, I show how proof
can be used to solve an algebraic problem important in the wider context
of fuzzy logic; namely, that certain varieties are generated by their
dense chains. The strategy consists of two parts. First, validity in all
dense chains of the variety is shown to be equivalent to derivability in
a calculus for the variety extended with a special density rule. It is
then shown that applications of the density rule can be eliminated from
proofs in the calculus.

Contact person: Constantine Tsinakis

February 2, 2006 

Yehuda Shalom, Tel Aviv University (visiting Princeton)

Some analytic methods in group theory

We shall try to explain how analytic tools, involving spectral
analysis, ergodic theory and probability on groups, combine
together to yield purely algebraic properties of some interesting classes of
groups. We shall also use the main result, which is entirely elementary
in its statement, as a motivation to discuss some aspects of the fundamental
notions of amenability and property (T), assuming no prior familiarity. The talk
should be accessible to every graduate student.

Contact person:
Mark Sapir

February 9, 2006 

Dean Billheimer, Department of Biostatistics, Vanderbilt University

Compositional Data in Biomedical Research

Modern methods of compositional data analysis are not well known in
biomedical research. Moreover, there appear to be few mathematical and
statistical researchers working on compositional biomedical
problems. Like other areas of science, biomedicine has many problems
in which the relevant scientific information is encoded in the
relative abundance of key species or categories. These vectors of
relative amounts, or proportions, constitute compositional

I review standard approaches to compositional data analysis, and
describe several recent theoretical advances. These newer methods
exploit a vector space structure of the simplex that extends data
modeling, and eases interpretation of results. In addition, I
describe a problem in cancer research in which analysis of
compositions plays an important role. The problem involves
subcellular localization of the BRCA1 protein, and its role in breast
cancer patient prognosis. This talk contains a tutorial component,
and should be accessible to students with exposure to inner product
spaces and multivariate statistics.

Contact person: Gieri Simonett

February 16, 2006 

Leon Glass, Centre for Nonlinear Dynamics,
Department of Physiology,
McGill University

Dynamics of Paroxysmal Tachycardia

Reentrant tachycardias are abnormal cardiac arrhythmias in which the
period is set by the time it takes for the excitation to travel in a
circuitous path. Simple conceptual models of reentrant tachycardia
include waves circulating in rings or annuli, spiral waves in a
or scroll waves in three dimensions. In this talk I discuss various
aspects of reentrant tachycardia including: the stability of the
circulation as the path length of the reentrant circuit is
the resetting of tachycardia by single pulses and predicting the
effects of multiple pulses, control of instabilities during the
reentrant rhythm by adjusting the timing of stimuli delivered during
the course of the tachycardia, spontaneous breakup of rotating
waves and the spontaneous generation of bursting rhythms.
models for these types of systems can be generated by growing
embryonic chick heart cells in tissue culture and optically imaging
the activity using voltage and calcium sensitive dyes. Mathematical
models can be developed suitable for comparison with experimental
and clinical data.

Contact person: Daphne Manoussaki

February 23, 2006 

Benson Farb, University of Chicago

Problems and progress in understanding the
Torelli group

The Torelli group T(S) associated to a surface S is defined to be the group of
homotopy classes of homeomorphisms of S acting trivially on H_1(S,Z). The study
of T(S) connects to 3-manifold theory, symplectic representation theory,
combinatorial group theory, and algebraic geometry. In this talk I will explain
some of the main themes in this beautiful topic. I will then describe a new
thread, very recently discovered by Chris Leininger and Dan Margalit and me,
which might be summarized as “algebraic complexity implies dynamical
complexity”. Much of this talk should be accessible to beginning graduate

Contact person: Bruce Hughes

March 2, 2006 
Daniele Mundici, University of Florence, Italy

From MV-algebras to AF-algebras
via ordered groups: the first twenty years

Elliott’s theory yields a classification of AF-algebras in
terms of countable Riesz groups. In particular,
AF-algebras whose Murray von Neumann order
of projections is a lattice correspond to
(countable) lattice-ordered abelian groups
with order-unit. These groups are categorically
equivalent to (countable) MV-algebras,
the algebras of Lukasiewicz infinite-valued logic.
Propositions in this logic are states of knowledge
in a R{‘e}nyi-Ulam game of Twenty Questions with lies,
just as propositions in boolean logic are
states of knowledge in the Twenty Questions game.
We survey recent research on the
interplay between these structures, including
new representation theorems for lattice-ordered
abelian groups, and decidability/undecidability
results for the isomorphism problem of
finitely presented AF C*-algebras.

Contact person: Constantine Tsinakis

March 9, 2006 

Spring Break

March 16, 2006 

Richard Kadison, University of Pennsylvania

Re-examining the Pythagorean Theorem – A Functional Analyst’s

In the colloquium talk we’ll study some surprising
twists and connections with operator algebras, symmetric
spaces, Schur’s work – on reviewing the basic theorem. We’ll
start at the beginning.

In the second lecture (on Friday),
we’ll study the Schur-Horn extension and Pythagoras
in II1 factors, as well as connections with the work of Kostant, Atiyah, and
Guillemin-Sternberg. This may be a bit more technical.

Contact person: Dietmar Bisch

March 23, 2006 

Jonathan Rosenberg, University of Maryland

An analogue of the Novikov Conjecture in complex algebraic geometry

We introduce an analogue of the Novikov Conjecture on higher signatures in the
context of the algebraic geometry of (nonsingular) complex projective
varieties. This conjecture asserts that certain “higher Todd genera” are
birational invariants, and implies birational invariance of certain extra
combinations of Chern classes (beyond just the classical Todd genus) in the
case of varieties with large fundamental group (in the topological sense).
The conjecture is, in a certain sense, best possible, and unlike
the usual Novikov Conjecture, it is already known to be true in all
cases, though some variants are still open. An interesting biproduct of
this work is a curious analogy between the homotopy
category of smooth manifolds and the birational category of smooth
projective varieties.

Contact person: Guoliang Yu

March 30, 2006 

Noga Alon, Tel Aviv University (visiting Princeton)

The structure of graphs and Grothendieck type inequalities

I will describe a connection between a classical inequality of
Grothendieck, a constructive version of a powerful lemma of Szemeredi
in Graph Theory, and integrality gaps of certain integer programming
problems. The investigation of this connection suggests the definition
of a new graph parameter, called the Grothendieck constant of a graph,
whose study is motivated by algorithmic applications, and leads to several
extensions of the inequality of Grothendieck, to an improvement of a recent
result of Kashin and Szarek, and to a solution of a problem of Megretski
and of Charikar and Wirth.

Contact person: Paul Edelman

April 6, 2006 

Yang Wang, Georgia Tech

Hilbert’s 3rd Problem, Lattice Tiling, and Weyl-Heisenberg Bases

Steinhaus proposed in the 1950’s the following “infamous” problem: Is there
a set T such that every rotation of T tiles the plane by the integer
Jackson and Mauldin recently showed that such a T exists if the
set is not required to be measurable. The question remains open for
measurable sets. A similar question one may ask is that given a
collection of lattices can one find a measurable set T that tiles
Rn by
each of the lattices in the collection. This question has a surprising
link to time-frequency analysis, and to the classic Hilbert’s third problem.

Contact person: Alex Powell

April 13, 2006 

Faculty Meeting

April 20, 2006 
Peter Constantin, University of Chicago

Particles and Fluids

I will discuss systems in which particles interact with a fluid
that carries them. The particles are agitated by thermal noise and
constrained by inter-particle potentials. The system is modeled by
a nonlinear Fokker-Planck equation, describing the particles, coupled with
the Navier-Stokes equations, describing the fluid. In the absence of
coupling to a fluid, the nonlinear Fokker-Planck equation has a
gradient structure with multiple steady states. I
will talk about some of the existing results and some of the
open questions arising in the presence of coupling.

Contact person: Gieri Simonett

Colloquium Chair: Gieri Simonett

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