Colloquia, AY 2007-2008

Colloquium, AY 2007-2008

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

Tea at 3:30 pm in 1425 Stevenson Center

September 6, 2007  

Franco Montagna, University of Siena, Italy

Advances in normal GBL-algebras



Contact person: Constantine Tsinakis

September 13, 2007  

John Lott, University of Michigan

Long-time behavior of Ricci flow

Perelman proved Thurston’s geometrization conjecture
using Hamilton’s Ricci flow. The first part of the
talk will be an introduction to this work.
Perelman gave enough
information about the long-time behavior of a
3-dimensional Ricci flow to prove the validity of
Thurston’s geometric decomposition. However, it
is not known whether Ricci flow performs the decomposition
for you, i.e. whether as time passes one sees the
various geometries appearing. I will give some
results in this direction.

Contact person: Guoliang Yu

September 20, 2007  
Vladimir Manuilov, Moscow State University

Almost representations and asymptotic representations of discrete groups

There are now several versions of weakening the notion of a group
representation and keeping some of its properties. One of them,
due to A. Connes, M. Gromov and H. Moscovici, is to consider, for a finitely
presented group, maps from the set of generators to the (infinite) unitary
group topologized by the operator norm, such that the relations are
satisfied up to some $\varepsilon>0$. A continuous family of such
$\varepsilon_t$-almost representations is an asymptotic representation of
the group if $\lim_{t\to\infty}\varepsilon_t=0$. We shall discuss
the relation between almost representations and asymptotic ones.
Typically, asymptotic representations do not converge to any genuine
representation and there is a topological obstruction for that: a
construction by A. Mishchenko shows that almost representations can be
used to define vector bundles over classifying spaces of groups similarly
to genuine representations and that these vector bundles may represent
non-trivial elements of the $K^0$-group of the classifying space.

Contact person: Gennadi Kasparov

September 27, 2007  
Carl Sundberg, University of Tennessee, Knoxville

Von Neumann’s inequality and model theory for several commuting operators

Let T be a contraction on a Hilbert space H, i.e. a bounded linear
operator on H whose norm is less than or equal to 1. Von Neumann’s
inequality says that if p is an analytic polynomial then the norm of p(T)
is bounded by the supremum of the values of |p(z)| for z in the unit disk.
Following Sz. Nagy-Foias this can be proved by developing a “model-theory”
for contractions, i.e. one shows that every contraction can be modelled as
the restriction to an invariant subspace of a particular nice kind of
operator for which von Neumann’s inequality can be easily proved. An
abstract approach to model theory was developed by Agler and this approach
suggests far reaching applications to the study of a variety of classes of

There are many ways to generalize von Neumann’s inequality to several
commuting operators, some of which work and some of which don’t. We will
discuss these generalizations, focusing particularly on one due to Drury
in 1978. This generalization is to “row-contractions” of commuting
operators. Drury’s result was rediscovered and further developed by
Arveson, who produced a model theory for row contractions. We discuss
Drury and Arveson’s work along with recent joint work with Stefan Richter,
in which we show how Arveson’s model theory can be produced using Agler’s

Contact person: Dechao Zheng, Guoliang Yu

October 4, 2007  
Uffe Haagerup, University of Southern Denmark

Random matrices and the Ext-invariant for C*-algebras

In this talk I will discuss a surprising connection between two quite
different areas of mathematics, namely “random matrices” and “operator
algebras” (i.e. C*-algebras and von Neumann algebras), a connection that
has been developed over the last 16 years. In 1991 Voiculescu introduced
a random matrix model for a free semicircular system, which has led to
the solution of a number of classical problems in von Neumann algebra theory.
More recently, Steen Thorbjoernesn and I have developed methods which
allowed us to apply random matrices to problems in C*-algebra theory as
well. In particular, we proved (Annals of Math. 2005) that the
Brown-Douglas-Fillmore Ext-invariant for the reduced C*-algebra C*r(F2)
for the free group on two generators is not a group, but only a
semi-group, a problem which had been open since 1978. Further results in
this direction have been obtained in collaboration with Hanne Schultz
and Steen Thorbjoernsen (Advances in Math. 2006).

Contact person: Dietmar Bisch

October 11, 2007  
Zhilan Feng, Purdue University

Influence of anti-viral drug treatments on evolution of HIV-1 pathogen

Mathematical models are used to study possible impact of
drug treatment of infections with the human immunodeficiency
virus type 1 (HIV-1) on the evolution of the pathogen.
Treating HIV-infected patients with a combination of several
antiretroviral drugs usually contributes to a substantial
decline in viral load and an increase in CD4+ T cells.
However, continuing viral replication in the presence
of drug therapy can lead to the emergence of drug-resistant
virus variants, which subsequently results in incomplete
viral suppression and a greater risk of disease progression.
As different types of drugs (e.g., reverse transcriptase
inhibitors, protease inhibitors and entry inhibitors) help to
reduce the HIV replication at different stages of the cell
infection, infection-age-structured models are useful
to more realistically model the effect of these drugs.
The model analysis will be presented and the results are
linked to the biological questions under investigation.
By demonstrating how drug therapy may influence the within
host viral fitness we show that while a higher treatment
efficacy reduces the fitness of the drug-sensitive virus,
it may provide a stronger selection force for drug-resistant
viruses which allows for a wider range of resistant strains
to invade.

Contact person: Glenn Webb

October 18, 2007  
Vaughan Jones, UC Berkeley

Random matrices and planar algebras

I will discuss a connection between random matrices
and subfactors discovered through the planar algebra approach
to subfactors and the fact that there is a genus expansion
for certain integrals over large matrices with the leading
term being of genus zero. This is joint work with Alice
Guionnet and Dimitri Shlyakhtenko.

Contact person: Dietmar Bisch
October 25, 2007  
Konstantina Trivisa, University of Maryland

Multicomponent reactive flows

Multicomponent reactive flows arise in many practical applications
such as combustion, atmospheric modelling, astrophysics, chemical
reactions, mathematical biology etc. The objective of this work is
to develop a rigorous mathematical theory based on the principles of
continuum mechanics.

Contact person: Gieri Simonett

November 1, 2007  
Constantine Dafermos, Brown University

Hyperbolic conservation laws with involutions and contingent entropies

The lecture will discuss paradigms from the theory of hyperbolic conservation laws
in which physics imposes special challenges to the analyst,
but then also provides the means of overcoming these challenges.

Contact person: Gieri Simonett, Mikhail Perepelitsa

November 8, 2007  
Catherine Yan, Texas A&M

Crossings and nestings in combinatorics

We present results on the crossings and nestings for
various combinatorial strucutures, in particular, for
matchings and set partitions. By a variant
of the RSK algorithm from algebraic combinatorics,
matchings and set partitions are in one-to-one
correspondence with oscillating/vacillating tableaux,
which are certain random walks in the Hasse diagram of
the lattice of integer partitions. We prove that the
crossing numbers and the nesting numbers are distributed
symmetrically over all partitions of [n], as well as
over all matchings of [2n]. Many enumerative results
about the noncrossing/nonnesting partitions will be
discussed. At the end, we will extend the method to
filling of Ferrer shapes, filling of Moon polyominoes, and
pattern avoidence in general multi-graphs.

Contact person: Mark Ellingham


Friday, January 11, 2008  

Special Colloquium, 4:10-5:00 pm, SC 5211

Monday, January 21, 2008 

Special Colloquium, 4:10-5:00 pm, SC 1307

Tuesday, January 22, 2008  

Special Colloquium, 4:10-5:00 pm, SC 1308

Thursday, January 24, 2008  

Special Colloquium, 4:10-5:00 pm, SC 5211

Friday, January 25, 2008  

Special Colloquium, 4:10-5:00 pm, SC 5211

Tuesday, January 29, 2008  

Special Colloquium, 4:10-5:00 pm, SC 1308

Thursday, January 31, 2008  

Special Colloquium, 4:10-5:00 pm, SC 5211

Monday, February 4, 2008  

Special Colloquium, 4:10-5:00 pm, SC 1307

February 14, 2008  

Carlos Cabrelli, University of Buenos Aires

Optimal shift invariant spaces

Shift invariant spaces are spaces of functions that are invariant
under integer translates. They are important in many areas, in particular
in approximation theory, numerical analysis and wavelet theory.
They also serve as models for signal and image processing applications
as well as sampling theory. Their very rich and interesting mathematical
structure can be successfully exploited in many problems.

In this talk we will try to describe part of this structure and give
and application to the problem of finding the “best fitting” shift
invariant space for a given set of data.

Contact person: Akram Aldroubi

February 21, 2008  
Denis Osin, CUNY

Group theoretic Dehn surgery and its applications

In my talk I will introduce a group theoretic version of
Dehn surgery in 3-manifolds. It turns out that many basic facts about
ordinary Dehn surgery can be translated to algebraic language. The
main result in this direction is a group theoretic analogue of the
Thurston Hyperbolic Dehn Surgery Theorem. Although this result is
strongly motivated by geometry, it also has some unexpected algebraic
applications. For example, most results obtained by means of small
cancellation theory over hyperbolic and relatively hyperbolic groups
can be recovered using Dehn surgery of a very special type.

Contact person: Alexander Ol’shanskiy

February 28, 2008  

Ian Agol, University of California, Berkeley

Finiteness of arithmetic reflection groups

We show that there are only finitely many
maximal reflection groups in hyperbolic spaces. This
talk will define the terms in this theorem, and explain some
of the ingredients of the proof.

Contact person: John Ratcliffe

March 6, 2008  
Spring Break

March 13, 2008  
Serguei P. Novikov, University of Maryland and
Landau Institute

New discretization of complex analysis

A new discretization of complex analysis and geometric GL_n connections was found
(in collaboration with I. Dynnikov) a few years ago as a by-product of the theory
of integrable systems. We are using an equilateral triangle lattice instead
of the standard square lattice for complex analysis, and simplicial
complexes for GL_n connections.

Contact person: Gennadi Kasparov

March 20, 2008  
Bruce Kleiner, Yale University

A new proof of Gromov’s theorem on groups of polynomial growth

In 1981 Gromov showed that any finitely generated
group of polynomial growth contains a finite index nilpotent subgroup.
This was a landmark paper in several respects. The proof was based on
the idea that one
can take a sequence of rescalings of an infinite group G,
pass to a limiting metric space, and apply deep results
about the structure of locally compact groups to draw conclusions about
the original group G. In the process,
the paper introduced Gromov-Hausdorff convergence, initiated the
subject of geometric group theory, and gave
the first application of the Montgomery-Zippin
solution to Hilbert’s fifth problem (and subsequent
extensions due to Yamabe). The purpose of the lecture is to give a new,
much shorter, proof of Gromov’s theorem. The main step involves showing
that any
infinite group of polynomial growth admits a finite dimensional
linear representation with infinite image. We establish this
using harmonic maps, thereby avoiding the Montgomery-Zippin-Yamabe
theory of locally compact groups which was used in Gromov’s
original proof.

I will explain the proof in a manner accessible to a broad

Contact person: Mark Sapir

March 27, 2008  

Petar Markovic, University of Novi Sad, Serbia

Computational complexity of the constraint satisfaction problem

I will give an overview of the current state of knowledge on the
following conjecture: Given the class of all constraint
satisfaction problems with a fixed template, the complexity of
deciding if there is a solution of an instance of such a
constraint satisfaction problem is either polynomial-time
solvable, or NP-complete (depending on the template).

In the first part of the talk I will provide all definitions
needed for understanding the problem. In the second part I will
overview the best partial results known so far using the
approaches of logic, complexity theory, universal algebra and
graph theory. In the third part I will concentrate on a subcase
of the problem when the template is a directed graph, as the
dichotomy conjecture for computational complexity is equivalent to the
restricted conjecture for this subcase. I will present the most
recent results which prove the conjecture in the case that the
template has no sources and no sinks, and some extensions of this
obtained in collaboration with R. McKenzie this Fall.
This approach uses a combination of graph-theoretic and universal
algebraic methods, and we feel that it is the most promising one
of all that have been attempted so far.

Contact person: Ralph McKenzie

April 3, 2008  
Yair Minsky, Yale University

Geometry and rigidity of mapping class groups

The mapping class group of a compact surface S is the group of
homeomorphisms of S modulo isotopy. Via its Cayley graph it can be
viewed as an infinite-diameter metric space, whose large-scale
geometry is strongly connected with the algebraic properties of the
group.  In joint work with Behrstock, Kleiner and Mosher, we study
this large-scale geometry and prove in particular that its
quasi-isometries are bounded perturbations of the action of the group.
This implies (as was also independently shown by Hamenstadt) that the
group is quasi-isometrically rigid.

Contact person: Mark Sapir
April 10, 2008  
Sergei Ivanov,
University of Illinois at Urbana-Champaign

On Dehn functions of infinite presentations of groups

Abstract: We discuss two new types of Dehn functions of group
presentations which seem more suitable (than the standard Dehn function)
for infinite group presentations and indicate the fundamental equivalence
between the solvability of the word problem for a group presentation
defined by a decidable set of defining words and the property of being
computable for one of the newly introduced functions (this equivalence
fails for the standard Dehn function). Elaborating on this equivalence
and making use of this function, we present a characterization of
finitely generated groups for which the word problem can be solved in
nondeterministic polynomial time. We also give upper bounds for these
functions, as well as for the standard Dehn function, in the case of the
Grigorchuk 2-group of intermediate growth and free Burnside groups of
sufficiently large exponents, defined by some minimal systems of defining
words. This talk is based on joint with R.I. Grigorchuk work.

Contact person: Alexander Ol’shanskii

April 17, 2008  
Yasuyuki Kawahigashi, University of Tokyo

Moonshine and operator algebras

”Moonshine” is a name for mysterious relations between
elliptic modular functions and the largest finite simple group among
the 26 sporadic groups, the Monster. The Moonshine vertex operator
algebra is a mathematical object to understand these relations, and
I will present its operator algebraic counterpart in the framework
of conformal field theory.

Contact person: Pinhas Grossman,
Dietmar Bisch

April 24, 2008  
Anna Gilbert, University of Michigan

(Fast) algorithms for compressed sensing

“Compressed sensing” captures a new paradigm which connects sparse
representations, high-dimensional geometry, probability, and algorithms. It
suggests a new paradigm in information acquisition and processing of
compressible signals. These signals can be approximated using an amount of
information much smaller than the nominal dimension of the signal.
Traditional approaches acquire the entire signal and process it to extract
the information. The new approach acquires a small number of nonadaptive
linear measurements of the signal and uses sophisticated algorithms to
determine its information content. Emerging technologies can compute these
general linear measurements of a signal at unit cost per measurement.

I will discuss the connections to randomized algorithms and signal
processing. In particular, I will focus on extremely fast algorithms and
measurement designs for compressed sensing, including some prototype
hardware designs.

Contact person: Alex Powell


Colloquium Chair (2007-2008): Gieri Simonett

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