Colloquium, Academic Year 2013-2014
Tea at 3:30 pm in 1425 Stevenson Center
|August 23, 2012|
|August 29, 2012||
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||
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||
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||
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|
|November 15, 2012||
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||
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||
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||
|January 10, 2013||
|January 15, 2013||
|January 18, 2013||
|February 7, 2013||
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||
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||
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||
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||
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||
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||
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||
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|
|April 11, 2013||
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|
Colloquium Chair (2013-2014): Denis Osin