September 20:
Speaker:
Davesh Maulik,
Clay Mathematics Institute and Columbia University
Title:
Counting curves on threefolds
Abstract:
A classical question in algebraic geometry concerns counting
curves in some ambient space. That is, if we fix some discrete parameters
and geometric conditions, how many holomorphic curves are there on an
algebraic variety which satisfy these conditions? Motivated by
constructions in symplectic geometry and mathematical physics,
Gromov-Witten theory provides one technique for attacking these questions.
In the special case when the ambient space is three-dimensional, however,
there is an alternative approach to these questions known as
Donaldson-Thomas theory. In this talk, we will introduce these two
theories and explain recent progress in understanding their relationship
to each other.
October 4:
Speaker:
Graeme Walter Milton,
University of Utah
Title:
Cloaking: a New Phenomena in Electomagnetism and Elasticity
Abstract:
The making of an object invisible through some cloaking device
until recently was commonly regarded as science fiction. Two
quite different types of electromagnetic cloaking were proposed in
early 2006. In our cloaking scenario a collection of finitely many
polarizable dipoles becomes essentially invisible when
they are within a certain critical distance of a superlens.
Superlenses have attracted attention because they promise resolution on a
length scale finer than can be achieved using conventional lenses, i.e.,
finer than the wavelength. The radiation scattered by the polarizable
dipoles resonates with the superlens and acts back on the dipoles to
essentially cancel the field incident on them, which is why they become
invisible. Dipolar energy sources supplying constant power also become
invisible.
A second type of cloaking was proposed by Pendry, Schurig and Smith and
Leonhardt. In this scenario a shield cloaks objects to incident
electromagnetic waves by guiding the waves
around the object. This work is related
to the earlier work of Greenleaf, Lassas and Uhlmann, on cloaking
for conductivity. Here we will review these developments and also
discuss how cloaking might be extended to elasticity using these
ideas. This requires new materials, in particular materials with
anisotropic mass density and a constitutive law in which the
stress depends on the velocity and the momentum depends on the
displacement gradient. We sketch how such materials, with behavior
outside that of continuum elastodynamics, might be made.
This is joint work with Lindsay Botten, Mark Briane, Ross McPhedran,
Nicolae Nicorovici, and John Willis.
October 18:
Speaker:
Stephen DeBacker,
University of Michigan
Title:
Harmonic analysis on reductive $p$-adic groups
Abstract:
In recent years many old and cantankerous problems in $p$-adic
representation theory have fallen. In this talk, I will (eventually)
discuss some problems in the area which have been solved via
Bruhat-Tits theory. This beautiful theory was (re)introduced to the
subject through the fundamental work of Allen Moy and Gopal Prasad.
We shall attempt to dispel the notion that this is not good colloquium
material by using Michigan's ``time-tested algorithm'' for giving a
colloquium talk (see
http://www.math.lsa.umich.edu/seminars/colloq/visitor.html).
November 15:
Speaker:
Bumsig Kim,
KIAS
Title:
A generalization of Fulton-MacPherson Configuration Spaces
Abstract:
Let D be a nonsingular subvariety of a nonsingular variety X.
We present a wonderful compactification of n distinct labeled points in X away
from D. When D is empty, it is the Fulton-MacPherson configuration space.
This is a joint work with Fumitoshi Sato
January 10: SPECIAL COLLOQUIUM
Speaker:
German Enciso,
Department of Systems Biology,
Harvard University
Title:
Switches, oscillations, and the dynamics of monotone dynamical systems
Abstract:
Determining the long-term behavior of large biochemical models has
proved to be a remarkably difficult problem. Yet these models exhibit
several characteristics that might make them amenable to study under the
right perspective. One possible approach (first suggested by Sontag and
Angeli) is their decomposition in terms of so-called monotone systems,
which can be thought of as systems with exclusively positive feedback.
In this talk I discuss some general properties of monotone dynamical
systems, especially classical and recent results regarding their generic
convergence towards an equilibrium. Then I will discuss the use of
monotone systems to model biochemical behaviors such as global
attractivity to an equilibrium, switches and oscillations under time
delays.
TUESDAY January 15: SPECIAL COLLOQUIUM
Speaker:
Patrick Shipman, University of Maryland
Title: Growth and Symmetry: Pattern Formation on Plants
Abstract: Tiling planforms dominated by diamonds (such as the diamond-shaped
seeds on a sunflower head), hexagons, or ridges (such as those on
saguaro cacti) are observed on many plants. We analyze PDE models for
the formation of these patterns that incorporate the effects of
growth and biophysical and biochemical mechanisms. The aim is to
understand both the underlying symmetries and the information
specific to the mechanisms. The patterns are compared to Voronoi
tessellations, and we will start to draw a bigger picture of growth
and symmetry in biological systems.
January 24: SPECIAL COLLOQUIUM
Speaker:
Richard Nickl, University of Connecticut
Title: Adaptive Estimation of the Distribution Function and the Density of a Random Variable
Abstract:
If X_1,...,X_n are i.i.d. random variables with arbitrary distribution function F,
the natural and classical estimator for F is the empirical distribution function F_n.
Classical results (in and before the 50s, e.g., Glivenko, Cantelli, Doob, Kolmogorov,
Donsker, Dvoretzky, Kiefer, Wolfowitz) show that F_n is an optimal estimator for F
in sup-norm loss. Most prominently, the process $\sqrt n(F_n-F)$ converges in law
in the Banach space of bounded functions on R to the Brownian bridge process.
On the other hand, if the true (but unknown) distribution function F has a
density f, the estimator d/dx(F_n(x)) fails badly as an estimator for f,
and it is known that density estimators can do much better under certain
assumptions on f. The question arises as to whether one can find a purely-data
driven estimator that optimally estimates both F and f (the latter only if it exists).
We show that one can indeed construct such estimators, based on classical
kernel or on more recent wavelet methods. The proofs require sharp exponential
bounds for wavelet and kernel estimators that can be established by using
empirical process theory (among others, Talagrands (1996) inequality),
as well as by employing statistical methods from adaptive estimation,
such as 'Lepski's (1990) method' or 'wavelet thresholding',
(Donoho, Johnstone, Kerkyacharian and Picard (1996)).
[This is joint work with Evarist Gine.]
February 7:
Speaker:
Ken Golden,
University of Utah
Title: Mathematics of Ice to Help Predict Climate Change
Abstract: Polar sea ice is both an indicator and agent of climate change. It also serves as
a primary habitat for robust algal and bacterial communities, sustaining rich marine
food webs. A new understanding of how salt water flows through porous sea ice
promises to improve forecasts of how global warming will affect earth's ice packs,
and how polar ecosystems may respond. We find that sea ice displays universal fluid
transport properties remarkably similar to crustal rocks. Mathematical techniques
from the physics of semiconductors are used to obtain the results. Related advances
on electrical properties will help in monitoring sea ice thickness. Biomedical
applications to lungs and bones will also be mentioned. Video from a 2007 Antarctic
expedition where we measured fluid and electrical transport in sea ice will be shown.
February 14:
Speaker:
Mircea Mustata,
University of Michigan
Title: Invariants of singularities: characteristic zero versus positive
charactarietic
Abstract: In characteristic zero, one way of looking at the
singularities of algebraic varieties involves valuations. This is the
point of view that is used in the study of higher-dimensional algebraic
varieties. On the other hand, in positive characteristic one can define
invariants of singularities using the Frobenius morphism. I will
discuss analogies, results and conjectures relating the two points of
view.
TUESDAY February 19: SPECIAL COLLOQUIUM
Speaker:
Dong Li, Institute for Advanced Study
Title: Singularities in fluid dynamics equations
Abstract: I will present some recent results on the construction of blow
up solutions to several equations in fluid dynamics.
February 21: SPECIAL COLLOQUIUM
Speaker:
Samuel Isaacson, University of Utah
Title: Mathematical Problems From Molecular Cell Biology
Abstract: We will explain the need for stochastic reaction-diffusion models
appropriate for studying the dynamics of gene and signaling networks
within biological cells. In particular, we will describe our work
developing a stochastic reaction-diffusion method that can
incorporate the complex geometry of cellular architecture, and the
application of this method to a model for eukaryotic gene expression
and nuclear transport. This work raised the question of what the
reaction-diffusion master equation (RDME), a lattice based stochastic
reaction-diffusion model, approximates as the lattice spacing is
decreased. We will discuss our recent work proving that in the
continuum limit reaction effects are lost in the RDME model. While
this may seem a negative feature, we will also show how the RDME for
finite lattice spacings may be interpreted as an asymptotic
approximation to a spatially-continuous stochastic reaction-diffusion
model due to Smoluchowski. We will conclude with a brief introduction
to a new, long term, modeling project we have begun, developing
stochastic-reaction diffusion models of gene/signaling networks
involved in several cardiac conditions.
February 28: DISTINGUISHED LECTURE
Speaker:
Benson Farb, University of Chicago
Title: Hidden symmetry
Abstract: Which contractible Riemannian manifolds (e.g. R^n with any metric) cover both
compact and noncompact, complete finite volume manifolds? Which Riemannian
products $X\times Y$ cover compact nonproducts? When do complex manifolds $M$
with $c_1(M)<0$ holomorphically split as a product of a locally symmetric
manifold and a ``rigid'' manifold? What are the isometries of Teichmuller
space?
These seemingly unrelated questions turn out to be instances of a single
underlying phenomenon, allowing for complete answers to each question. The
goal of this talk will be to explain some of the ideas behind this, which is
joint work with Shmuel Weinberger.
March 6:
Speaker:
Arun Ram,
University of Wisconsin
Title: Tantalizer algebras
Abstract: Tantalizer is short for tensor power centralizer. These
algebras often come as algebras of diagrams or of tangles, and so working
with them requires drawing lots of pictures. Their structure and
representation theory contains and immense amount of information about the
representation theory of groups and quantum groups of types GL, SO, Sp,
and they can be used to construct corresponding link polynomials and
3-manifold invariants. This talk will be a survey of some recent
developments in tantalizer algebras.
March 13:
Speaker:
Constantin Teleman, University of California, Berkeley
Title: Semi-simple topological field theories in 2 dimensions
Abstract: Ruan and Tian defined a quantum multiplication
on the cohomology of a compact, symplectic manifold, which
deforms the cup-product by counting the rational curves.
This is part of a larger structure of a "Cohomological
Field Theory" in 2 dimensions. In important special
cases, the quantum cohomology ends up being a semi-simple
algebra (isomorphic to a direct sum of copies of C).
In the talk, I will outline how semi-simplicity interacts
with the field theory structure and the recently proved
Mumford conjecture to show that the full field theory
is determined by the quantum multiplication alone.
March 27:
Speaker:
Pavle Pandzic, University of Zagreb
Title: Dirac operators in representation theory
Abstract: We will first review the original Dirac's construction, as well as a few
basic facts and examples of representations of reductive Lie groups.
Then we will describe a version of the Dirac operator related to
representations and explain its properties and relevance. One of the
main results says that in certain cases the action of the Dirac operator
determines the infinitesimal character, the most basic invariant of a
representation. We will also comment on relationships with Lie algebra
cohomology and finish with mentioning some open problems.
April 3: DISTINGUISHED LECTURE
Speaker:
Daniel Gillespie, Gillespie Consultants
Title: Stochastic Chemical Kinetics
Abstract: The time evolution of a well-stirred chemically reacting system
is traditionally modeled by a set of coupled ordinary differential equations
called the reaction rate equation (RRE). The resulting picture of
continuous deterministic evolution is, however, valid only for infinitely
large systems. That condition is usually well approximated in laboratory
test tube systems. But in biological systems formed by single living cells,
the small population numbers of some reactant species can result in
dynamical behavior that is noticeably discrete rather than continuous, and
stochastic rather than deterministic. In that case, a more physically
accurate mathematical modeling is obtained by using the machinery of Markov
process theory, specifically, the chemical master equation (CME) and the
stochastic simulation algorithm (SSA). After reviewing the theoretical
foundations of stochastic chemical kinetics, we will describe a way to
approximate the SSA by a faster simulation procedure, and then show how this
way also provides a logical bridge between the CME/SSA description and the
RRE description.
April 10:
Speaker:
Frank Wattenberg, United States Military Academy
Title: Modeling and Simulation in the Space Program and Undergraduate Mathematics and Science Education
Abstract: This talk is really an invitation to join what we hope will
become a massive Open Source/Open Content project in the spirit of Linux
and other Open Source software initiatives. This report will show some of
the preliminary work being done by folks at the United States Military
Academy and Tietronix Software working with folks at NASA. We are
developing software that enables Mathematics, Science, and Engineering
faculty and students to develop models using the ordinary language of
mathematics (for example, differential equations) and then at the click
of a mouse button see an immersive, three-dimensional, almost gamelike,
simulation. Students and faculty can focus on the science and
mathematics without being distracted by the details of programming.
The project is also building a library of components that can be used by
students and faculty. Each component contains two kinds of information.
Most importantly, it contains technical information (for example, the
thrust and burn rate of a thruster) so that models are scientifically
accurate. In addition, it contains the three-dimensional modeling
information required for realistic three-dimensional simulations.
This talk will focus on our two initial domains -- orbital mechanics and
NASA's Autonomous Lunar Outpost Project.
April 17, at 2PM (NOTE UNUSUAL TIME!): DISTINGUISHED LECTURE
Speaker:
Daniel Allcock, University of Texas at Austin
Title: Recent Progress in Hyperbolic Reflection Groups
Abstract: A lot of progress has been made recently concerning discrete groups
generated by reflections, acting on hyperbolic space. A sort of
classification may be taking shape. We will survey some of this
progress, including work by Agol, Nikulin, the speaker, and others.
MONDAY April 21, at 3PM (NOTE UNUSUAL TIME!): SPECIAL COLLOQUIUM
Speaker:
Daniela Ferrero, Texas State University
Title: Connectivity of Path Graphs
Abstract:
A major problem in the design of interconnection networks consists of
linking an arbitrarily large number of nodes, each of them subject to different
physical constraints, in such a way that the communication delay between any two
nodes is minimized. In order to solve this problem, as well as a broad range of
other problems in different disciplines, the line graph has been proven to be a
useful graph-valued function. The path graph generalized the line graph while
enhancing some of its good properties with regard to the design of interconnection networks.
It is also desirable that a network still communicates, and with reasonable
efficiency, under the presence of failures in some of its components. The
study of the fault-tolerance of interconnection networks modeled by graphs
is essentially based on the concept of connectivity. From counting alternatives
routes between nodes, this notion has evolved into more sophisticated measures that
provide a much fuller characterization of the network.
In this talk we will explore the developments on the notion of connectivity in the
framework of interconnection networks obtained by the iteration of the path graph.
All necessary background on interconnection networks and graph theory will be presented.
April 24:
Speaker:
Thang T. Q. Le, Georgia Institute of Technology
Title: On the quantum MacMahon Master Theorem.
Abstract: The celebrated MacMahon Master theorem is a matrix generalization of
the identity
1 + x + x^2 + x^3+ .... = 1/(1-x).
The MacMahon Master Theorem helped to solve many conjectures in combinatorics.
In 1975 G. Andrews asked for a natural q-analog of
the MacMahon Master Theorem. Here we answered Andrews' question. Our quantum
MacMahon Master Theorem has origin in knot theory.
This is joint work with S. Garoufalidis and D. Zeilberger.
May 22:
Speaker:
Jared Tanner, University of Utah and University of Edinburgh