Cantor sets and Cantor measures

Speaker: 

David Damanik

Institution: 

Rice University

Time: 

Thursday, April 3, 2014 - 4:00pm

Host: 

Location: 

RH306

A subset of the real line is called a Cantor set if it is compact,
perfect, and nowhere dense. Cantor sets arise in many areas; in this
talk we will discuss their relevance in the spectral theory of
Schr\"odinger operators. We discuss several results showing that the
spectrum of such an operator is a Cantor set, from the discovery of the
first example by Moser to a genericity result by Avila, Bochi, and
Damanik. A Cantor measure is a probability measure on the real line
whose topological support is a Cantor set. A primary example in the
spectral theory context is the density of states measure in situations
where the spectrum is a Cantor set. A conjecture of Simon claims a
strict inequality between the dimensions of the set and the measure for
the Fibonacci potential. If time permits, we will discuss a recent
result of Damanik, Gorodetski, and Yessen, which establishes this
conjecture in full generality.

Missing values of polynomial maps

Speaker: 

Hendrik Lenstra Jr.

Institution: 

Universiteit Leiden

Time: 

Tuesday, January 7, 2014 - 3:00pm to 4:00pm

Host: 

Location: 

NSII 1201

Given a polynomial map between two vector spaces over a field,
how many values can it miss? The lecture will present a number of
new results on this question. They were inspired by the work of
Daqing Wan, and obtained jointly with Michiel Kosters (Leiden).

Modern Geometry: from Local to Global, from Smooth to Rough, from Static to Dynamic

Speaker: 

Jean-Pierre Bourguignon

Institution: 

IHES and Stanford

Time: 

Thursday, November 7, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

NSII 1201

In the last century, Geometry underwent several
substantial extensions and revisions based on
the fundamental revolutions that it lived through
in the XIXth century.

The purpose of the lecture is to discuss several
aspects of these transformations: the new
concepts that emerged from these new points
of view, the new perspectives that could be
drawn from bringing together the continuous and
discrete viewpoints, some classical problems that could
be solved, and the new interactions with other
disciplines that went along.

It includes the presentation of the views of the late Professor
Chern Shiing Shen on some of these issues.

CONSENSUS and FLOCKING in SELF-ALIGNMENT DYNAMICS

Speaker: 

Eitan Tadmor

Institution: 

University of Maryland, Mathematics

Time: 

Wednesday, June 5, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

We discuss self-organized dynamics of agent-based models
with focus on a prototype model driven by non-symmetric self-alignment
introduced in [1].
Unconditional consensus and flocking emerge when the self-alignment is
driven by global interactions with a sufficiently slow decay rate.  In
more realistic models, however, the interaction of self-alignment is
compactly supported, and open questions arise regarding the emergence
of clusters/flocks/consensus, which are related to the propagation of
connectivity of the underlying graph.
In particular, we discuss heterophilious self-alignment: here, the
pairwise interaction between agents increases with the diversity of
their positions and we assert that this diversity enhances
flocking/consensus. The methodology carries over from agent-based to
kinetic and hydrodynamic descriptions.

[1] A new model for self-organized dynamics and its flocking behavior, J.
Stat. Physics 144(5) (2011) 923-947.
 

Spectral theory of multidimensional periodic and almost-periodic operators: Bethe-Sommerfeld conjecture and the integrated density of states.

Speaker: 

Leonid Parnovsky

Institution: 

UCL

Time: 

Thursday, April 4, 2013 - 4:00pm

Host: 

Location: 

RH 306

I will make a survey of recent results on the spectrum of periodic and, to a smaller extent, almost-periodic operators. I will consider two types of results:

1. Bethe-Sommerfeld Conjecture. For a large class of multidimensional periodic operators the numbers of spectral gaps is finite.

2. Asymptotic behaviour of the integrated density of states of periodic and almost-periodic operators for large energies.

Cutting and pasting in algebraic geometry

Speaker: 

Ravi Vakil

Institution: 

Stanford University

Time: 

Thursday, April 25, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Given some class of "geometric spaces", we can make a ring as follows.

  1. (additive structure)  When U is an open subset of such a space X, [X] = [U] + [(U)];
  2. (multiplicative structure)  [X x Y] = [X] [Y].

In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be the case of "points on a line" --- polynomials in one variable.  (This talk is intended for a broad audience.)  This is joint work with Melanie Matchett
Wood.

Atomistic-to-Continuum Coupling Methods

Speaker: 

Mitchell Luskin

Institution: 

University of Minnesota

Time: 

Thursday, November 15, 2012 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

Many materials problems require the accuracy of atomistic modeling in small regions, such as the neighborhood of a crack tip. However, these localized defects typically interact through long ranged elastic fields with a much larger region that cannot be computed atomistically. Many methods have recently been proposed to compute solutions to these multiscale problems by coupling atomistic models near a localized defect with continuum models where the deformation is nearly uniform. During the past several years, we have given a theoretical structure to the description and formulation of atomistic-to-continuum coupling that has clarified the relation between the various methods and the sources of error. Our theoretical analysis and benchmark simulations have guided the development of optimally accurate and efficient coupling methods.

Tits geometry and positive curvature

Speaker: 

Fuquan Fang

Institution: 

Capital Normal Univ. Beijing and Notre Dame

Time: 

Thursday, January 17, 2013 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but one that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a topological building in the sense of Burns and Spatzier. Using this structure we classify all polar actions on (simply connected) positively curved manifolds of cohomegeneity at least two.
(Joint work with K.Grove and G. Thorbergsson)

CR geometry in dimension three

Speaker: 

Paul Yang

Institution: 

Princeton University

Time: 

Thursday, December 6, 2012 - 4:00pm to 5:00pm

Host: 

Location: 

RH306

There is a good deal of resemblence of CR geometry in dimension three with conformal geometry
in dimension four. Exploiting this resemblence is quite fruitful. For instance, the presence of
several conformally covariant operators in both geometries allows us to formulate correct
conditions for the embedding problem as well as the CR Yamabe problem. There is also
large difference in the presence of pluriharmonic functions. I will also describe a new
operator which gives control of the pluriharmonics and allows a formulation of a
sphere theorem in this geometry.

The mathematics of Multiwave Imaging

Speaker: 

Professor Plamen Stefanov

Institution: 

Purdue University

Time: 

Thursday, March 1, 2012 - 4:00pm

Location: 

RH 306

 
Thermoacoustic (TAT) and Photoacoustic Tomography (PAT) are examples of multiwave imaging methods allowing to combine the high imaging contrast of one wave (an electromagnetic or a photoacoustic one) with the high resolution of ultrasound. We present recent results obtained in collaboration with Gunther Uhlmann, Jianliang Qian and Hongkai Zhao on the mathematical theory behind TAT, PAT and other multiwave methods. We allows the acoustic speed to be variable, and consider the partial data case as well. We will also discuss the case of a discontinuous speed modeling brain imaging.  Numerical reconstructions will be shown as well.
Most of the progress is due to the use of microlocal methods. One of the goals of the talk is to show the usefulness of microlocal methods to solving real life problems.

Pages

Subscribe to RSS - Colloquium