Departmental Colloquium 2008-2009
Thursdays, 4:15 PM, JWB 335
October 2:
Speaker:
Jean-Francois Lafont,
The Ohio State University
Title:
Simplicial volume and locally symmetric spaces
Abstract:
The notion of simplicial volume was introduced by Gromov and
Thurston in the early 1980's, and measures how efficiently a manifold can
be "triangulated over the reals". I'll explain various results that follow
from positivity of this invariant. I will also outline the proof (joint
with B. Schmidt) of the following conjecture of Gromov: every closed
locally symmetric space of non-compact type has positive simplicial
volume.
October 9:
Speaker:
Jason Starr,
Stony Brook
Title:
The Weak Approximation Problem
Abstract:
Given a system of polynomial equations in several variables
and in 1 parameter, does there exist a "rational solution", i.e., a
family of solutions which is a rational function (fraction of
polynomials) in the parameter? Do there exist enough rational
solutions to approximate every power series solution in the parameter
to arbtirary order? The first problem, or rather the problem of
answering the first problem, is Hilbert's 10th problem for
$\mathbb{C}(t)$. It is expected there is no algorithm to answer the
first problem. The second problem, the "Weak Approximation Problem",
conjecturally has a very simple answer: there are enough rational
solutions precisely if after substituting a general value for the
parameter, the corresponding system is "rationally connected", i.e.,
every pair of solutions are common members of a family of solutions
which are the output of a rational function. I will discuss the
topological and number theoretic motivation of this conjecture, the
evidence for the conjecture due to Hassett -- Tschinkel, Hassett,
Knecht and Colliot-Th\'el\`ene -- Gille, and a new approach of Mike
Roth and myself putting this conjecture in the larger context of
"algebro-geometric analogues of topological obstruction theory".
November 13:
Speaker:
Christian Reidys,
The Center for Combinatorics at Nankai University
Title:
Tangled diagrams and beyond
Abstract:
In this talk we develop the notion of generalized vacillating tableaux from
molecular contact structures. We prove a bijection between tangled diagrams
(tangles) and vacillating tableaux, allowing for exact and asymptotic
enumeration. We show how tangles connect the concepts of partitions and
enhanced partitions, which are shown to be in bijection with a specific
subclass of tangles. We then show how tangles embed in a generalization of
the Brauer algebra and analyze the structure of the latter.
November 20:
Speaker:
J. Maurice Rojas, Texs A & M
Title:
ABCs of Real Algorithmic Geometry
Abstract:
We survey recent advances toward basic algorithmic questions
in real algebraic geometry. In particular, we consider the two questions
and their algorithmic complexity:
(1) Does a system of sparse polynomial equations have a real root?
(2) What is the topology of a real algebraic set defined by a set
of sparse polynomial equations?
A special case of Question (2) is counting real roots, and
even here optimal upper bounds are still an open question. So we
also review some concrete examples.
Along the way, we will see an unusual connection to the famous
Masser-Oesterle ABC-Conjecture, and an extension of Smale's 17th Problem
(on approximating complex roots of polynomial systems). We assume no
background in complexity theory or algebraic geometry.
December 4:
Speaker:
Christopher Hacon, Utah
Title:
Classification of Algebraic Varieties
Abstract:
A complex projective variety is a subset of complex projective
space
defined by a set of homogeneous polynomials. In this talk I will discuss
recent
results that describe the geometry of these varieties. The case of varieties
of
dimension 1, also known as Riemann surfaces, is classical. The geometry of
smooth varieties of dimension 2 was understood by the Italian school of
Algebraic Geometry at the beginning of the 20-th century. The Minimal Model
Program is an attempt to generalize these results to higher dimension. The 3
dimensional case was understood in the 1980's by celebrated work of Mori and
others. In this talk I will discuss joint work with Birkar, Cascini and
McKernan
towards extending the Minimal Model Program to arbitrary dimension. In
particular I will discuss the following: Theorem: The canonical ring of any
smooth projective algebraic variety is finitely generated. (Note that this
Theorem was independently proven by Y.-T. Siu.)
December 11:
Speaker:
Roy Baty, Los Alomos
Title:
Nonstandard analysis and jump conditions for shock waves
Abstract:
Nonstandard analysis is a relatively new area of mathematics in which
infinitesimal numbers can be defined and manipulated rigorously like
real numbers. To demonstrate the power of the subject, the problem of
shock wave jump conditions is studied for a one-dimensional
compressible gas. It is assumed that the shock thickness occurs on an
infinitesimal interval and the jump functions in the thermodynamic
and fluid dynamic parameters occur smoothly across this interval. To
use conservation laws, pre-distributions of the Dirac delta measure
are applied whose supports are contained within the shock thickness.
Furthermore, piecewise differentiable pre-distributions of the
Heaviside function are applied which vary from zero to one across the
shock wave. It is shown that if the equations of motion are expressed
in non-conservative form then the relationships between the jump
functions for the flow parameters may be found unambiguously. The
analysis yields the classical Rankine-Hugoniot jump conditions for an
inviscid shock wave. Examples developed include a normal shock wave
and a radially symmetric shock wave.
January 15: (Special Colloquium)
Speaker:
Coralia Cartis,
University of Edinburgh
Title:
Adaptive regularization methods for nonlinear optimization
Abstract:
Nonlinear optimization problems represent the bed-rock of numerous real-life
applications, such as data assimilation for weather prediction, radiation
therapy treatment planning, optimal design of energy transmission networks,
and many more. The solution of these problems usually involves iteratively
constructing easier-to-solve local models of the function to be optimized,
with the optimizer of the model taken as an estimate of the sought-after
solution. Linear or quadratic models are usually employed locally in this
context; however, these approximations are often unsatisfactory either
because they are unbounded in the presence of nonconvexity and hence cannot
be meaningfully optimized, or they are accurate representations of the
function only in a small neighbourhood, yielding only small or no iterative
improvements. Hence such models require some form of regularization to
improve algorithm performance and avoid failure; traditionally, linesearchand trust-region
techniques have been employed for this purpose and represent the
state-of-the-art. Here, a new class of methods for nonlinear nonconvex
unconstrained problems will be presented that approximately globally minimize
a quadratic model of the
objective regularized by a cubic term, inspired by earlier regularization
approaches
of Nesterov (2007) and Griewank (1982). An overestimation property of
functions with Lipschitz-continuous Hessians underlies and justifies the
model construction
in the work to be presented. Preliminary numerical experiments show our
methods
to perform better than a trust-region implementation, while our convergence
and complexity results show it to be at least as reliable as the latter
approach.
Extensions to problems with simple constraints and a simplified application
to the
subclass of nonlinear least-squares problems will also be presented. This is
joint work with Nick Gould (Rutherford Appleton Laboratory, UK), Philippe
Toint (University of Namur, Belgium), and partly, also with Stefania Bellavia
and Benedetta Morini (University of Florence, Italy).
February 5: (Special Colloquium)
Speaker:
Mark Huber,
Duke University
Title:
Exact uniform generation of linear extensions of a poset
Abstract:
Suppose a user is interested in learning about the ranking of n items, where
partial information is known. For instance, perhaps it is known that 2
outranks 5, 5 outranks 7, and 5 outranks 8. As long as the conditions are
consistent (so no cycles like 2 outranks 3 outranks 4 outranks 2), such
conditions make the items a partially ordered set, or poset. With an
initial prior belief that all n! rankings are equally likely, the goal is to
answer questions such as: What is the chance that 2 is ranked first
conditioned on the partial information? The set of allowable rankings
subject to these conditions is formally known as the linear extensions of
the poset. In this talk, I will explain how to simulate random draws
exactly uniformly from this set of linear extensions in O(n^3 ln n) time,
which can then be used to approximate various probabilities in an unbiased
fashion. Linear extensions also arise in what are called convex rank tests
in nonparametric statistics, and I will discuss these applications as well.
February 10, Tuesday: (Special Colloquium)
Speaker:
Yuliya Gorb,
Texas A and M University
Title:
Discrete Network Approximation for Determining Asymptotics of
Effective Properties of High Contrast Concentrated
Composites
Abstract:
A blow up of effective properties of high contrast composites with
particles close to touching is of interest. The goal is to derive and
justify asymptotic formulas for effective properties of such composites as
a characteristic interparticle distance tends to zero. A derivation and
justification is done by constructing a so-called discrete network
approximation. The main idea of this approximation is based on a reduction
of the original continuum problem described by partial differential
equations with rough coefficients to a discrete problem on a graph, called
a discrete network. The approach is illustrated by considering a highly
packed suspension of rigid particles in a Newtonian fluid (vectorial
problem), and a medium of finite conductivity with perfectly conducting
particles (scalar problem).
February 12: (Special Colloquium)
Speaker:
Margaret Beck,
Brown University
Title:
Nonlinear stability of time-periodic viscous shocks
Abstract:
If a given solution of a PDE is stable, then, roughly speaking, any
other solution that
starts near it, stays near it for all time. This is an important concept in
applications, because it
is typically only the stable solutions that are observed in practice. I will
outline two key
mathematical difficulties that one can encounter when analyzing the stability of
time-periodic
solutions of dissipative PDEs on unbounded domains. Briefly, they are the
presence of zero
eigenvalues that are embedded in the continuous spectrum and the
time-periodicity of the associated
linear operator. In the context of viscous shocks in systems of conservation
laws, I will show how
these difficulties can be overcome. The method involves the development of a
contour integral
representation of the linear evolution, similar to that of a strongly continuous
semigroup, and
detailed pointwise estimates on the resultant Greens function, which are
sufficient for proving
nonlinear stability under the necessary assumption of spectral stability.
February 19: (Special Colloquium)
Speaker:
Pak-Wing Fok,
California Institute of Technology and UCLA
Title:
Mathematical aspects of epitaxial growth: asymptotics, conservation laws
and multiscale modeling
Abstract:
A crystal lattice with a slight miscut results in a surface covered with
atomic height steps. The study of epitaxial growth commonly involves
analyzing the motion of these steps. In this talk, three systems will be
discussed: a single straight step, a system of parallel straight steps and
a system of concentric circular steps.
The single straight step is studied as a Stefan problem driven by
convection. In this system, the effect of kinetics at the interface
regularizes an otherwise divergent step velocity at short times while a
finite layer thickness regularizes an otherwise divergent acceleration.
The nature of long time solutions is determined by the convective forcing.
For a system of straight steps interacting through elastic dipoles, the
problem is formulated as a system of coupled ODEs. The relaxation of these
steps is studied via continuum coarse-graining that results in a
Boltzmann-type conservation law coupled to two moving boundaries.
Similarity solutions are found that match the ODE data.
A system of concentric circular steps serves as a model for a simple
axisymmetric nanostructure. The evolution of circular steps is complicated
by the effect of step line tension that gives rise to step bunching
instabilities and expanding circular facets. The ODEs for the step radii
are solved numerically using a novel fourth order adaptive-in-time
multirate scheme. The step bunches are analyzed using a nonlinear PDE. In
a Lagrangian setting, the facet is modeled as a shockwave and studied
using the method of characteristics. In Eulerian coordinates, a
fourth-order PDE is used to study the slope profile of the nanostructure.
The solution of the PDE requires boundary conditions at the facet edge
that account for the fast time scales of the inner most step.
March 5:
Speaker:
Thomas Haines,
University of Maryland
Title:
Survey of affine Deligne-Lusztig varieties
Abstract:
The classical Deligne-Lusztig varieties are used to study
representations of a finite group of Lie type. They are subvarieties of
the flag variety of the group in question. Affine Deligne-Lusztig
varieties (ADLVs) are "loop group" counterparts, and can live in the
affine Grassmannian or the affine flag variety associated to the group.
They arise naturally in the study of certain Shimura varieties over finite
fields. They have their origin in the theory of isocrystals which was
initiated by Dieudonne, Manin, Katz, and Mazur, and which was later
generalized by Kottwitz to the context of general reductive groups.
In this talk I will explain the background and basic questions about
ADLVs from an elementary point of view. I will then review the current
state of knowledge about their dimensions and structure. Open questions
will also be discussed. I will mention joint work with Goertz,
Kottwitz, and Reuman.
March 12: (Distinguished Colloquium)
Speaker:
Allen Moy
University of Science and Technology, Hong Kong
Title:
Buildings - What they are and some of their uses
Abstract:
The notion of a building is a simplicial complex
introduced by Jacques Tits to study simple algebraic groups over
arbitrary fields. We give a brief introduction to buildings
and then describe some of their uses.
March 26:
Speaker:
Michael Vogelius,
Rutgers
Title:
A survey of results concerning existence and blow up for some nonlinear
elliptic and parabolic problems related to corrosion modelling
Abstract:
TBA
April 9:
Speaker:
Gang Bao,
Michigan State
Title:
Distinguishability via Uncertainty Principle for inverse Scattering
Abstract:
The inverse scattering problem arises in diverse
areas of industrial and military applications, such as
nondestructive testing, seismic imaging, submarine detections,
near-field or subsurface imaging, and medical imaging.
A general model is concerned with a time-harmonic electromagnetic
plane wave incident on a medium enclosed by a bounded domain.
Given the incident field, the direct problem is to determine the scattered
field for the known scatterer. The inverse medium scattering problem
is to determine the scatterer from the boundary measurements of near
field currents densities. Although this is a classical problem in
mathematical physics, numerical solution of the inverse problem
remains to be challenging since the problem is nonlinear, large-scale, and
most of all ill-posed! The severe ill-posedness has thus far limited in
many ways
the scope of inverse problem methods in practical applications.
In this talk, our recent results in mathematical analysis and
computational studies of the inverse boundary value problems for the
Maxwell equations will be reported. A novel continuation approach based on
the uncertainty principle will be presented. By using multi-frequency
or multi-spatial frequency boundary data, our approach is shown to
overcome
the ill-posedness for the inverse medium scattering problems.
Convergence issues for the continuation algorithm will be examined. Our
most recent progress on inverse source problems will also be discussed.
April 16:
Speaker:
Robert V. Kohn,
Courant Institute, NYU
Title:
Price Bubbles from Heterogeneous Beliefs
Abstract:
Harrison and Kreps showed in 1978 how the heterogeneity of
investor beliefs can drive speculation, leading the price of an asset to
exceed its intrinsic value. By focusing on an extremely simple market
model -- a finite-state Markov chain -- the analysis of Harrison and Kreps
achieved great clarity but limited realism. My talk discusses joint work
with Xi Chen, which achieves similar clarity with greater realism by
considering an asset whose dividend rate is a mean-reverting stochastic
process. Our investors agree on the volatility, but have different beliefs
about the mean reversion rate. We determine the minimum equilibrium price
explicitly; in addition, we characterize it as the unique classical
solution of a certain linear differential equation. Our example shows, in
a simple and transparent manner, how heterogeneous beliefs about the mean
reversion rate can lead to everlasting speculation and a permanent
"price bubble".
April 23: In LCB 215 with refreshments in the LCB loft at 3:45
Speaker:
Dan Barbasch,
Cornell
Title:
Theunitarydualoftherationalpointsofarealorp-adiclinearreductivegroup
Abstract:
While maybe as hard to parse, sadly it is not as catchy
as Mark Twain's
"Constantinopolitanischerdudelsackspfeifenmachersgesellschafft".
In the 1930's I.M. Gelfand outlined a program of abstract harmonic analysis, which offered a paradigm for the use of symmetry to study a very wide class of mathematical problems. A key technical step is the following:
Problem: For every locally compact group G, determine the set G^_u of irreducible unitary representations G.
The group G is usually the symmetry group of a problem. In mathematical physics, differential geometry, or differential equations it is a real Lie group. In number theory, the group may be an algebraic group over a local fields. In combinatorics, it is often a finite group.
In Gelfand's program, G is acting (as a symmetry group) on a measure space X, preserving the measure. In this setting there is a Hilbert space H = L^2(X) of (complex-valued) square-integrable functions on X. Then G acts linearly on H by [p(g)f](v):=f(g^{-1}v), and the fact that the action preserves the measure amounts to the fact that p(g) is unitary. The first step in Gelfand's program is to express questions about X (related to the symmetry group G) as questions about L^2(X) (and the linear operators p(g)). Knowledge of the unitary dual G^_u is a crucial ingredient.
In this talk I will explain the nature of the answer of the unitary dual of a reductive real or p-adic group, and give some examples of its uses.