Departmental Colloquium 2009-2010
Thursdays, 4:15 PM, JWB 335
October 8: No Colloquium -- Special Lecture sponsored by the Center for Science and Mathematics Education
NOTE: Different Room and time: Aline Skaggs Auditorium, 5:00 p.m.
Speaker:
Deborah Ball,
University of Michigan
Title: Improving Outcomes in Mathematics and Science: Making Change Work
Abstract:
Deborah Ball has been involved in mathematics education at many levels, from elementary school teacher,
through teacher educator, to Dean of a School of Education. She is among the very few who deserve and
enjoy genuine and deep respect from both research communities, in mathematics and education.
Her research focuses on the nature of mathematical knowledge, and the development of measures
that make possible the relationship among mathematical knowledge, quality of teaching and student performance.
She has long been involved in the study of interventions designed to improve quality and effectiveness of mathematical
instruction. Dr. Ball has served on many international commissions, and is at present on the National Mathematics Advisory Panel.
In this talk, Dr. Ball will discuss what we know about interventions that work, why they work, and ways in which serious
change can be effected.
October 29:
Speaker:
Oscar Garcia Prada,
Consejo Superior de Investigaciones Cientificas
Title:
Geometry of surface group representations
Abstract:
Given a compact real surface S and a semisimple
Lie group G, we consider the moduli space R(S,G) of representations
of the fundamental group of S in G (sometimes called the character
variety). This moduli space plays a central role in many problems
in geometry, topology and physics. By considering a complex
structure on the surface S (thus making it a Riemann surface),
the moduli space of representations is in bijection with a moduli
space of holomorphic objects, known as Higgs bundles. We explain
this correspondence and show how to use it to study the topology
of R(S,G). We give special attention to the case where G is the
isometry group of a non-compact Hermitian symmetric space.
In this situation the moduli space has special components that
can be regarded in some sense as generalizations of the
Teichmueller space of S (which can be identified with
a component of the character variety when G=PSL(2,R)).
November 5: SPECIAL COLLOQUIUM
Speaker:
Sam Payne,
Stanford University
Title:
Nonarchimedean geometry
Abstract:
The usual archimedean norm on the complex numbers and its associated
analytic geometry (holomorphic functions and differential forms) have
been fundamental tools for understanding the geometry and topology of
complex algebraic varieties, for instance through Hodge theory and
Lefschetz hyperplane theorems, since the beginnings of the subject.
Nonarchimedean norms, such as the p-adic norm on the rational numbers,
also have an associated analytic geometry which is well-known to
number theorists, but this theory is just beginning to be applied in
other areas of mathematics, such as algebraic geometry and dynamical
systems. This talk will be an introduction to nonarchimedean
geometry, through the basic notions of tropicalization and
analytification.
November 12:
Speaker:
Anthony Henderson,
University of Sydney
Title:
Enhancing the Jordan canonical form
Abstract:
In undergraduate linear algebra, we learned the Jordan canonical form theorem:
that every n x n complex matrix A is similar to an essentially unique matrix B
which is block-diagonal with each block having a very simple form (a single eigenvalue repeated down the diagonal,
ones on the super-diagonal, and zeroes elsewhere). This is of course very
useful for matrix calculations.
From the Lie-theoretic viewpoint, this theorem is about classifying the orbits of
the general linear group GL_n(C) in its adjoint representation. This suggests natural
follow-up questions about the closures of the orbits. It also raises the hope of finding
analogous orbit classifications for other representations of algebraic groups. After explaining
some of the general context, I will focus on a case which, despite its close proximity to the
Jordan canonical form theorem, has only recently been worked out: the direct
sum of the vector and adjoint representations of GL_n(C). This is joint work with Pramod Achar (Louisiana State University).
November 19:
Speaker:
George Papanicolaou,
Stanford University
Title:
Imaging with noise
Abstract:
It is somewhat surprising at first that it
is possible to locate a network of sensors from
cross correlations of noise signals that they
record. This is assuming that the speed of
propagation in the ambient environment is known and that the
noise sources are sufficiently diverse. If the
sensor locations are known and the propagation
speed is not known then it can be estimated from
cross correlation information. Although a basic
understanding of these possibilities had been
available for some time, it is the success of recent
applications in seismology that have revealed the
great potential of correlation methods, passive sensors
and the constructive use of ambient noise in imaging. I will
introduce these ideas in an interdisciplinary, mathematical
way and show that a great deal can be done with them.
Things become more complicated, and a mathematically
more interesting, when the ambient medium is also
strongly scattering. I will end with a review of what is
known so far in this case, and what might be expected.
December 3: SPECIAL COLLOQUIUM
Speaker:
Florian Herzig,
Northwestern University
Title:
Modular representations of p-adic groups
Abstract:
The Langlands program relates complex representations of GL_n(Q_p) to Galois representations.
For n = 1 this is explained by class field theory and for n = 2 this is closely related to the theory of modular forms.
For general n, this is now understood by the work of Harris-Taylor and Henniart. In the last decade, a mod-p
(as well as a p-adic) version of the Langlands program have been emerging, and they have already played an
important role in some recent progress in number theory. But so far understanding has been limited to n = 1 and 2.
We survey some of the known story in the classical and in the mod p case,
and then discuss some recent progress on the classification of mod p representations of GL_n(Q_p), as time permits.
January 14: SPECIAL COLLOQUIUM
Speaker:
Christel Hohenegger,
Courant Institute of Mathematical Science
Title: Understanding the Dynamics and Mechanics of Complex Fluids
Abstract:
One of the challenges in modeling the transport properties of complex
fluids (e.g. many biofluids, polymer solutions, particle suspensions)
is describing the interaction between the suspended micro-structure
with the fluid itself. Here I will focus on my work in understanding
the dynamics of active suspensions -- motile bacterial baths are an
important example -- and also overview my work on characterization
and modeling of complex materials such as lung mucus.
Suspensions of active particles, like swimming bacteria or artificial
micro-swimmers, have been studied intensely over the past few years.
Using a recently derived kinetic model, I have investigated the
linearized structure of such an active system near a state of
uniformity and isotropy. I show that system instability can arise only
from the dynamics of the first azimuthal mode in swimmer orientation,
that the growth of fluctuations for a suspension of anterior
actuated swimmers is associated with a proliferation of oscillations
in swimmer orientation, and that at small-scales the system is
controlled independently of the nature of the suspension. Finally a
prediction about the onset of the instability as a function of the
volume fraction of anterior actuated swimmers can be made.
Einstein showed that the thermal fluctuations of tracer particles in
a fluid can be related to its bulk viscosity. In recent times this
observation has been extended and forms the basis of the field of
microrheology, which seeks to use statistical quantities to estimate
the viscous and elastic properties of materials from very small volume
samples. Following the basic model of two-point microrheology, I have
developed a Langevin-based model of the coupled fluctuations of two
beads in a viscoelastic liquid and from this derive new relations
between measurable quantities and fluid response properties. This
approach provides a new interpretation to memory response functions,
which play a dominant role in numerical simulation of such systems.
TUESDAY January 26: SPECIAL COLLOQUIUM
NOTE: Room JWB 335. Time 4:15 PM
Speaker:
Juan Souto,
University of Michigan
Title:
Relations between geometry and topology of hyperbolic 3-manifold.
Abstract:
By Mostow's rigidity theorem, geometric invariants of
hyperbolic 3-manifolds are in fact topological invariants. On the
other hand, it follows from the work of Thurston and Perelman that a
3-manifold is hyperbolic if and only if it satisfies some rather mild
conditions. In light of these results, it is an interesting question
to try to understand how topological conditions on a 3-manifold M
which admits a hyperbolic metric affect the geometry of the hyperbolic
metric. This question is rather imprecise. In other words, it has many
different incarnations. In this talk I will describe a few results on
some concrete formulations of the question above.
January 28: SPECIAL COLLOQUIUM
Speaker:
Alexandra Pettet,
University of Michigan
Title:
Out(F) and its relatives
Abstract:
The outer automorphism group Out(F) of a free group of finite rank
shares many properties with linear groups and the mapping class group
Mod(S) of a surface, although the techniques for studying Out(F) are
often quite different from the latter two. Motivated by analogy, I will
present some results about Out(F), previously well-known for the mapping
class group, and highlight some of the features in the proofs which
distinguish it from Mod(S). This is joint work with Matt Clay.
TUESDAY February 9: SPECIAL COLLOQUIUM
NOTE: Room JWB 335. Time 4:15 PM
Speaker:
Sunder Sethuraman,
Iowa State University
Title:
Nonequilibrium limit for a tagged particle in a zero-range system
Abstract:
The 'zero-range' interacting particle system, introduced in the 70's,
is a formal model of certain queues, traffic, and other physical phenomena. The
system follows a collection of random walks on a lattice which interact in that
the rate at which a particle jumps depends only on the occupation number of its
location.
Of interest is the asymptotic behavior of a distinguished, or tagged particle in
the system. In this talk, we discuss a nonequilibrium limit for the tagged
particle in one dimension when the transition probabilities are mean-zero. The
limit process is a diffusion whose coefficients depend on the underlying
hydrodynamic evolution of the mass density.
February 11: SPECIAL COLLOQUIUM
Speaker:
Hai Long Dao,
University of Kansas
Title:
Playing with Ext and Tor: study singularities via homological algebra
Abstract:
To understand an algebraic object (groups, rings, etc.) it
often pays to understand some category such object acts on. For
example, when our object is a commutative ring R, we wish to explore
the category mod(R) of (finitely generated) modules over R. Two
fundamental operations in mod(R) are tensor product and Hom. Studying
these operations naturally leads to their respective derived functors,
namely Tor and Ext. Despite the relatively simple definitions
involving these functors, their actual behaviour remain quite
mysterious. In this talk I will describe how understanding simple
questions such as when the Ext and Tor modules vanish can lead to
concrete results on seemingly unrelated topics such as non-commutative
crepant resolutions or torsions in Picard groups of certain
singularities. Most of the talk will be accessible to non-experts.
TUESDAY February 16: SPECIAL COLLOQUIUM
NOTE: Room JWB 335. Time 4:15 PM
Speaker:
Cecilia Diniz Behn,
University of Michigan
Title:
Investigating the dynamics of REM sleep
Abstract:
Microdialysis and microinjection experiments suggest that
neurotransmitter dynamics play an important role in the initiation and
maintenance of sleep/wake states. However, the synaptic coupling in
traditional population firing rate models does not explicitly
incorporate the dynamics of neurotransmitter concentrations. We
introduce a novel network modeling framework that includes
neurotransmitter concentrations, and we use this framework to model
interactions among primary brainstem nuclei involved in rat sleep/wake
regulation. Analysis of the bifurcation structure underlying model
dynamics provides insights into the mechanisms governing REM sleep,
particularly those associated with circadian modulation.
February 18: SPECIAL COLLOQUIUM
Speaker:
Yekaterina Epshteyn,
Carnegie Mellon University
Title: Numerical Methods for Chemotaxis Models
Abstract:
In this talk, I will first discuss several chemotaxis models including
the classical Keller-Segel model.
Chemotaxis is the phenomenon in which cells, bacteria, and
other single-cell or multicellular organisms direct their movements
according to certain chemicals (chemoattractants) in their
environment. The mathematical models of chemotaxis are usually described
by highly nonlinear time dependent systems of PDEs. Therefore, accurate
and efficient numerical methods are very important for the validation and
analysis of these systems. Furthermore, a common property of all existing
chemotaxis systems is their ability to model a concentration phenomenon
that mathematically results in solutions rapidly growing in small
neighborhoods of concentration points/curves. The solutions may blow up or
may exhibit a very singular, spiky behavior. In either case, capturing
such solutions numerically is a challenging problem.
In our work we propose a family of stable (even at times near blow up)
and highly accurate numerical methods, based on interior penalty
discontinuous Galerkin schemes (IPDG) for the Keller-Segel chemotaxis
model with parabolic-parabolic coupling. This model is the basic step in
the modeling of many real biological processes and it is described by a
system of a convection-diffusion equation for the cell density, coupled
with a reaction-diffusion equation for the chemoattractant concentration.
We prove theoretical hp error estimates for the proposed discontinuous
Galerkin schemes. Our proof is valid for pre-blow-up times since we
assume some regularity of the exact solution.
Numerical experiments to demonstrate the stability and high accuracy
of the proposed methods for chemotaxis models and comparison with other
methods will be presented. Ongoing research projects will be discussed as
well.
February 25:
Speaker:
Charles Favre,
École Polytechnique
Title: TBA
Abstract: TBA
March 4:
Speaker:
John R. Parker,
University of Durham
Title:
Complex hyperbolic lattices
Abstract:
A complex hyperbolic lattice is a discrete group of isometries
of the unit complex ball whose quotient has finite volume (with respect to the Bergman metric). There are relatively few examples of
complex hyperbolic lattices known, but these examples may be
described from several different points of view. Namely, one may
use hyperbolic geometry and a fundamental polyhedron; one may
use methods of algebraic geometry, in particular line arrangements;
one may describe many of them as monodromy groups of hypergeometric
functions and one may use techniques from number theory to give
many of them as arithmetic groups. In this talk I will explain the
relation between these points of view for a family of lattices constructed
by Deligne and Mostow in the 1980s.
March 11:
Speaker:
Gunther Uhlmann,
University of Washington
Title: TBA
Abstract: TBA
March 18:
Speaker:
Anne Thomas,
University of Oxford
Title: TBA
Abstract: TBA