# Colloquia, AY 2012-2013

Thursdays 4:10 pm in 5211 Stevenson Center, unless otherwise noted
Tea at 3:30 pm in 1425 Stevenson Center

August 23, 2012

The Fall 2012 Faculty Assembly, no colloquium

August 29, 2012

Martin, Gaven (New Zealand Institute for Advanced Study)

New approaches to modelling nonlinear phenomena

This is a very general talk which will not be at all technical and will be painted with a broad brush.

When deforming a material body (e.g. heating, bending, stretching or otherwise stressing), physics
(basically the principle of least action) informs us that the final deformation rearranges itself so as to
minimise some energy or action functional. Modelling is often about trying to find the correct functional from other physical first principles.
The theory of nonlinear elasticity has been developed over a century to try and say important and generic things about the structure and regularity of
minimisers or minimal energy configurations for various classes of functionals. Of course minimisers of functionals satisfy differential equations (Euler-Lagrange) e There have been significant advances in solving these differential equations over the
years, particularly when they are nice (the technical term is elliptic). But to model more interesting
phenomena, like critical phase and transitions, supersonically moving objects and so forth, the
equations develop singularities and nonlinear terms can’t be ignored. In dealing with these ugly
equations (the technical term is nonlinear degenerate elliptic) it’s sometimes easier to go back to
the functionals themselves.

In this talk I will discuss some recent work with others about a special interesting case modelling nonlinear phenomena in elastic media by
minimising a scale invariant measure of the anisotropic properties of the material in the simplest 2D
case (with 3D applications). Surprisingly this is connected with a conjecture from J.C.C. Nitsche in
1962 (solved this year) concerning harmonic mappings and minimal surfaces. There is a wonderful
dichotomy in the solutions to these equations as one passes through a critical phase when one can
identify conformal invariants of the material (= geometric quantities derived from infinitesimal
information). This dichotomy shows, for instance, that materials can only be stretched so far before
breaking or tearing. There appear to be other applications in modelling cellular structures, foam
physics and tissues as well.

Contact person:
Vaughan Jones

September 13, 2012

Nigel Higson (Penn State University)

Contractions of Lie Groups and Representation Theory

Let K be a closed subgroup of a Lie group G. The contraction of G to K
is a
Lie group, usually more elementary in structure than G itself, that
approximates G to first order near K. The terminology is due to the
mathematical physicists, who examined the group of Galilean
transformations
as a contraction of the group of Lorentz transformations. My focus will
be
on a related but different class of examples, the prototype of which is
the
group of isometric motions of Euclidean space, viewed as a contraction
of
the group of isometric motions of hyperbolic space. It is natural to
expect some sort of limiting relation between representations of the
contraction and representations of G. But in the 1970s George Mackey
carried out a few calculations pointing to an interesting rigidity
phenomenon: as the contraction group is deformed back to G, the
representation theory remains in some sense unchanged. In particular
the
irreducible representations of the contraction group parametrize the
irreducible representations of G. I shall formulate a reasonably
precise
conjecture that was inspired by subsequent developments in C*-algebra
theory
and noncommutative geometry, and describe the evidence in support of it,
which is by now substantial. However a conceptual explanation for
Mackey’s
rigidity phenomenon remains elusive.

Contact person:
Dietmar Bisch

October 11, 2012

Nicolas Monod (ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE)

Fixed points and derivations

This expository lecture will lead to a new fixed point
theorem and discuss the relation with derivations. As an application,
we present the optimal answer to the “derivation problem” for group
algebras which originated in the 1960s.

Contact person:
Denis Osin

October 25, 2012

Klaus Boehmer (Philipps-Universität Marburg)

On Finite Element Methods for Fully Nonlinear Second Order
Elliptic Equations in R^n

This was an open problem for more than 20 years in the communities of Numerical
Analysis and Scientific Computing. Only the most important Monge-
Amore equation of order 2 in R^2 had been discussed without proofs, e.g. by
Dean/Glowinski. My paper in SINUM 2008 proved for the first time for the
general case of fully nonlinear elliptic differential equations (and systems of order
2m, m \geq 1 on C^2m) domains in R^2, the necessary stability and convergence
results. This includes convergence for the numerical method for solving these
equations and quadrature approximations. In the mean time several papers
have appeared for the most important spcial case, the Monge Ampere equation.
Brenner, S.C. and her group Gudi, T. and Neilan, M. and Sung, L. Y.studied
C^0 penalty methods and proved convergence for Monge- Amore equations.
A simplified approach for second order equations on convex polygonal domains
in Rn is presented in this lecture. It allows indicating the essential ideas
for the general case. We suggest a nonstandard nonconforming C^1 finite element
method. The classical theory of discretization methods is applied to the
differential operator. The consistency error vanishes, but the stability has to be
proved in an unusual way. This is the hard core of the paper. Essential tools
are linearization, a compactness argument, the interplay between the weak and
strong form of the linearized operator and a new regularity result for solutions
of finite element equations. An essential basis for our proofs are Davydov’s C^1
finite elements on polygonal. The method applies to non divergent quasilinear
elliptic problems as well. Algorithms are formulated to calculate the nonlinear
system and to solve it by a combination of continuation and discrete Newton
methods. The latter converges locally quadratically, essentially independently
of the actual grid size by the mesh independence principle. EXplicit numerical
results are due to Davydov/Saeed.

Contact person:
Larry Schumaker

November 1, 2012

Department Meeting, no colloquium

November 15, 2012

Scott Morrison (the Australian National University )

Small index subfactors

Over the last two decades our understanding of small index subfactors
has improved substantially. We have discovered a slew of examples,
some related to finite groups or quantum groups, and other sporadic’
examples. At present we have a complete classification of
(hyperfinite) subfactors with index at most 5, and a few results that
push past 5. I’ll explain the main techniques behind these
classification results, and also spend a little time describing how we
construct the sporadic examples. (Joint work with many people!)

Contact person:
Vaughan Jones

November 29, 2012

Justin Tatch Moore (Cornell University)

Nonassociative Ramsey Theory and the amenability problem for Thompson’s
group

In 1973, Richard Thompson considered the question of whether his newly
defined group $F$ was amenable. The motivation for this problem stemed
from his observation — later rediscovered by Brin and Squire — that
$F$ did not contain a free group on two generators, thus making it a
candidate for a counterexample to the von Neumann-Day problem. While the
von Neumann-Day problem was solved by Ol’shanskii in the class of finitely
generated groups and Ol’shanskii and Sapir in the class of finitely
presented groups, the question of $F$’s amenability was sufficiently basic
so as to become of interest in its own right.

In this talk, I will analyze this problem from a Ramsey-theoretic
perspective. In particular, the problem is related to generalizations of
Ellis’s Lemma and Hindman’s Theorem to the setting of nonassociative
binary systems. The amenability of $F$ is itself equivalent to the
existence of certain finite Ramsey numbers. I will also discuss the
growth rate of the F\olner function for $F$ (if it exists).

Contact person:
Ralph N. McKenzie

December 6, 2012

Kasso A. Okoudjou (University of Maryland )

Scalable frames

Frames provide a mathematical framework for stably representing signals as linear combinations of basic building blocks that constitute an overcomplete
collection. Finite frames are frames for finite dimensional spaces, and are especially suited for many applications in signal processing. The inherent redundancy of frames can be exploited to build compression and transmission algorithms that are resilient not only to lost of information but also to noise. For instance, tight frames constitute a particular class of frames that play important roles in many applications.

After giving an overview of finite frame theory, I will consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors. A frame that positively answers this question will be called scalable. I will give various characterizations of the set of scalable frames, and present some topological descriptions of this set. (This talk is based on joint work with G. Kutyniok, F. Philipp and E. Tuley).

Contact person:
Akram Aldroubi

January 8, 2013

A Special Lecture

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January 10, 2013

A Special Lecture

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January 15, 2013

A Special Lecture

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January 18, 2013

A Special Lecture

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February 7, 2013

Barry Simon (Cal Tech)

Tales of our Forefathers

This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians
as only names assigned to theorems. With vignettes and anecdotes, I’ll convince you they were also human beings and that,
as the Chinese say, “May you live in interesting times” really is a curse.

Contact person:
Edward B. Saff

February 14, 2013

Chris Heil (Georgia Institute of Technology)

Music, Time-Frequency Shifts, and Linear Independence

Fourier series provide a way of writing almost any signal as a
superposition of pure tones, or musical notes. Unfortunately, this
representation is not local, and it does not reflect the way that music
is actually generated by instruments playing individual notes at different
times. We will discuss time-frequency representations, which are a type
of local Fourier representation of signals. While such representations
have limitations when it comes to music, they are powerful mathematical
tools that appear widely throughout mathematics (e.g., partial differential
equations and pseudodifferential operators), physics (e.g., quantum
mechanics), and engineering (e.g., time-varying filtering). We ask one
very basic question: are the notes in this representation linearly
independent? This seemingly trivial question leads to surprising
mathematical difficulties. This talk is intended to be introductory
and accessible to beginning graduate students.

Contact person:
Alex Powell

February 21, 2013

Loukas Grafakos (University of Missouri)

Gabor ridge functions: theory and applications

We discuss a directionally sensitive time-frequency decomposition and representation of functions.
The coefficients of this representation allow one to measure the “amount” of frequency the function
(signal, image) contains in a certain time interval, and also in a certain direction.
This has been previously achieved using a version of wavelets called ridge lets, but in this work we discuss
an approach based on time-frequency or Gabor elements.
Applications to image processing are discussed.

Contact person:
Akram Aldroubi

February 28, 2013

Yves de Cornulier (Universite Paris-Sud 11)

Actions on trees and ends of groups

A metric space is multiended if it admits a bounded subset whose
complement has at least two unbounded connected components. For
instance, the line is multiended but higher-dimensional
Euclidean spaces are not. In the late sixties, Stallings has
given a remarkable characterization of those finitely generated
groups whose Cayley graph is multiended; the only such
torsion-free groups are free products of two nontrivial groups!
A key part of the proof is the construction of a action on a
tree; in the seventies, the study of general group actions on
trees was achieved by Bass and Serre. In the meantime, the study
of multiended Schreier graphs was started by Abels and Houghton,
and a remarkable connection with nonpositively curved cube
complexes was discovered by Sageev twenty years later. While
outstanding applications of cube complexes have been made since
then, I will try to focus on the question of understanding which
finitely generated groups admit a multiended Schreier graph.

Contact person:
Vaughan Jones

March 14, 2013

Marcus Khuri (SUNY at Stony Brook)

Geometric Inequalities in General Relativity

Perhaps the most outstanding open problem in mathematical relativity is the so called Cosmic Censorship conjecture,
which roughly asserts that singularities in the evolution of spacetime must always be hidden inside black holes,
and moreover that spacetime must eventually settle down to a stationary final state. Based on heurisitc physical arguments,
R. Penrose derived a series of geometric inequalities relating total mass, area of the event horizon, electricromagnetic charge,
and angular momentum, all of which serve as necessary conditions for the validity of Cosmic Censorship. In this talk,
we will detail recent advances in the rigorous mathematical formulation and proofs of some of these inequalities.

Contact person:
Gieri Simonett

March 18, 2013,
Special Colloquium
SC 1308

Xiu-Xiong Chen (SUNY at Stony Brook)

Kaehler Einstein metrics on Fano manifolds

In 1980s, Yau conjectured that the existence of Kaehler Einstein metric on Fano manifold is related to an algebraic geometric condition of “stability”.
The recent work with Donaldson, Sun Song confirmed this conjecture. In the talk, we will review history of this problems as well as this subject,
and we also will review earlier work of G. Tian and others on this problems. We will outline the strategy of proof,
which involves deforming through metrics with cone singularities. If time permits, we will give more details about various aspects of the proof.

Contact person:
Ioana Suvaina

March 21, 2013

Paul Schupp (UIUC)

Asymptotic density and the theory of computability

Eighty years after the beginning of the general theory of computability, ideas from the “asymptotic
point of view” prevalent in several areas of mathematics have begun to interact with computability theory.
This will be a very general talk, developing the necessary ideas from scratch. I will try to give an
idea of how this point of view leads to new questions and new answers.

Contact person:
Denis Osin

March 28, 2013

Marston Conder (University of Auckland, New Zealand),
AMS-NZMS Maclaurin Lecturer 2013

Discrete objects with maximum possible symmetry

Symmetry is pervasive in both nature and human culture. The notion
of chirality (or handedness’) is similarly pervasive, but less well understood.
In this lecture, I will talk about a number of situations involving discrete
objects that have maximum possible symmetry in their class, or maximum
possible rotational symmetry while being chiral. Examples include
geometric solids, combinatorial graphs (networks), maps on surfaces,
dessins d’enfants, abstract polytopes, and even compact Riemann surfaces
(from a certain perspective). I will describe some recent discoveries about
such objects with maximum symmetry, illustrated by pictures as much
as possible.

Contact person:
Vaughan Jones

April 4, 2013

The Spring 2013 Faculty Assembly, no colloquium

April 11, 2013

Oleg R. Musin
(University of Texas at Brownsville)

The kissing problem in three and four dimensions

The kissing number k(n) is the maximal number of equal nonoverlapping spheres in n-dimensional space that can touch another sphere of the same size. This problem in dimension three was the subject of a famous discussion between Isaac Newton and David Gregory in 1694. In three dimensions the problem was finally solved only in 1953 by Sch\”utte and van der Waerden. It was proved that the bounds given by Delsarte’s method are not good enough to solve the problem in 4-dimensional space.
Delsarte’s linear programming method is widely used in coding theory. In this talk we will discuss a solution of the kissing problem in four dimensions which is based on an extension of the Delsarte method. This extension also yields a new proof of k(3)<13.
We also going to discuss our recent solution of the strong thirteen spheres problem. It is a joint work with Alexey Tarasov.

Contact person:
Doug Hardin

April 18, 2013

Math Awards ceremony, no colloquium

Colloquium Chair (2012-2013): Dechao Zheng

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