College of Arts and Science Vanderbilt University
Vanderbilt Univeristy College of Arts and Science
 

Colloquium, Academic Year 2014-2015

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

 

August 21, 2014  

Fall faculty assembly, no colloquium


August 28, 2014  
Amit Singer (Princeton University)

The mathematics of three-dimensional structure determination of molecules by cryo-electron microscopy.

Cryo-electron microscopy (EM) is used to acquire noisy 2D projection images of thousands of individual, identical frozen-hydrated macromolecules at random unknown orientations and positions. The goal is to reconstruct the 3D structure of the macromolecule with sufficiently high resolution. We will discuss algorithms for solving the cryo-EM problem and their relation to other branches of mathematics such as tomography, random matrix theory, representation theory, spectral geometry, convex optimization and semidefinite programming.

Contact person: Alex Powell
September 4, 2014  

Welcome event, no colloquium


September 11, 2014  
Martin Kassabov (Cornell University)

Dimensions of character varieties for finitely generated groups.

The character variety of a group describes all representations of a fixed degree. These varieties provide a convenient way to study all finite dimensional representations at once. I will outline a construction of family of finitely generated groups whose character varieties have some prescribed properties.

Contact person: Mark Sapir
September 18, 2014  
Uffe Haagerup (University of Copenhagen)

Approximation properties for groups and C*-algebras.

It is classical result in Fourier analysis, that the Fourier series of a continuous function my fail to converge uniformly or even pointwise to the given function. However if one use a summation method as e.g. convergence in Cesaro mean, one actually gets uniform convergence of the Fourier series. This result can easily be generalized to amenable locally compact groups, where in the non-abelian case, the continuous functions on dual group G^ must be replaced by the reduced group C*-algebra of G.
1994 Jon Kraus and I introduced a new approximation property (AP) for locally compact groups. The groups having (AP) is the largest class of locally compact groups for which a generalized Cesaro mean convergence theorem can hold. Amenable groups as well as the group SL(2,R) has property (AP), but it was proved by Vincent Lafforgue and Mikael de la Salle in 2011, that SL(n,R) fails to have (AP) for n = 3,4,... In two recent joint works with Tim de Laat we have extend their result by proving that Sp(2,R) and more generally all simple connected Lie groups of real rank >=2 does not have the (AP).
In the talk I will give an introduction to amenability and to the property (AP) for locally compact groups, and their relation to other group properties (e. g. weak amenability and Property T). The corresponding properties for C*-algebras will also be discussed.

Contact person: Dietmar Bisch
October 2, 2014  
Jared Speck (Massachusetts Institute of Technology)

Shock Formation in Solutions to 3D Wave Equations

I will provide an overview of the formation of shock waves, developing from small, smooth initial conditions, in solutions to quasilinear wave equations in 3 spatial dimensions. I will first describe prior contributions from many researchers including F. John, S. Alinhac, and especially D. Christodoulou. I will then describe some results from my recent book, in which I show that for two important classes of wave equations, a necessary and sufficient, condition for the phenomenon of small-data shock-formation is the failure of S. Klainerman's classic null condition. I will highlight some of the main ideas behind the analysis including the critical role played by geometric decompositions based on true characteristic hypersurfaces. Some aspects of this work are joint with G. Holzegel, S. Klainerman, and W. Wong.

Contact person: Marcelo Disconzi
October 9, 2014  
Efim Zelmanov (University of California, San Diego)


Infinite Dimensional Superalgebras

We will discuss basic examples of (superconformal) Lie algebras and superalgebras and their representation theory.

Contact person: Mark Sapir
October 16, 2014  

Fall break


October 23, 2014  

Faculty meeting, no colloquium


October 30, 2014  
Gilbert Strang (Massachusetts Institute of Technology)

Banded matrices and fast inverses.

The inverse of a banded matrix A has a special form which we can find directly from the "Nullity Theorem." Then the inverse of that matrix A^-1 is the original A -- which can be found by a remarkable "local" inverse formula. This formula uses only the banded part of A^-1 and it offers a very fast algorithm to produce A.
That fast algorithm has a potentially valuable application. Start now with a banded matrix B (possibly the positive definite beginning of a covariance matrix C -- but covariances outside the band are unknown or too expensive to compute). It is a poor idea to assume that those covariances are zero. Much better to complete B to C by a rule of maximum entropy which for Gaussians means maximum determinant.
As statisticians and also linear algebraists discovered, the optimally completed matrix C is the inverse of a banded matrix. Best of all, the matrix actually needed in computations is that banded C^-1 (which is not B !).And C^-1 comes quickly and efficiently from the local inverse formula.
A very special subset of banded matrices contains those whose inverses are also banded. These arise in studying orthogonal polynomials and also in wavelet theory -- the wavelet transform and its inverse are both banded ( = FIR filters). We describe a factorization for all banded matrices that have banded inverses.

Contact person: Akram Aldroubi
November 6, 2014  
Mikhail Ershov (University of Virginia)

Golod-Shafarevich groups.

Golod-Shafarevich groups can be informally described as groups having a presentation with a small set of relators. They have been introduced almost 50 years ago as a tool for solving two outstanding problems: the existence of infinite class field towers and the existence of infinite finitely generated periodic groups. Since then Golod-Shafarevich groups have been used to settle many other problems in combinatorial and geometric group theory as well as some questions in three-manifold topology. I will give a survey of the main results about Golod-Shafarevich groups and the techniques used in the proofs and discuss some open problems.

Contact person: Mark Sapir
November 13, 2014  
Russell Lyons (Indiana University)


Random walks on groups and the Kaimanovich-Vershik conjecture.

Let G be an infinite group with a finite symmetric generating set S. The corresponding Cayley graph on G has an edge between x,y in G if y is in xS. Kaimanovich-Vershik (1983), building on fundamental results of Furstenberg, Derriennic and Avez, showed that G admits non-constant bounded harmonic functions iff the entropy of simple random walk on G grows linearly in time; Varopoulos (1985) showed that this is equivalent to the random walk escaping with a positive asymptotic speed. Kaimanovich and Vershik also presented the lamplighter groups (groups of exponential growth consisting of finite lattice configurations) where (in dimension at least 3) the simple random walk has positive speed, yet the probability of returning to the starting point does not decay exponentially. They conjectured a complete description of the bounded harmonic functions on these groups; in dimensions 5 and above, their conjecture was proved by Erschler (2011). I will discuss the background and present a simple proof of the Kaimanovich-Vershik conjecture for all dimensions, obtained in joint work with Yuval Peres.

Contact person: Jesse Peterson
November 20, 2014  
Imre Leader (University of Cambridge)

Partition Regular Equations.

A finite or infinite matrix M is called 'partition regular' if whenever the natural numbers are finitely coloured there exists a vector x, with all of its entries the same colour, such that Mx=0. Many of the classical results of Ramsey theory, such as van der Waerden's theorem or Schur's theorem, may be naturally rephrased as assertions that certain matrices are partition regular. While the structure of finite partition regular matrices is well understood, little is known in the infinite case. In this talk we will review some known results and then proceed to some recent developments. No knowledge of the subject will be assumed.

Contact person: Victor Falgas-Ravry
November 27, 2014  

Thanksgiving break


December 4, 2014  
Morwen Thistlethwaite (University of Tennessee, Knoxville)


Computer-assisted mathematics

The advent of cheap and sophisticated computing has often affected the way in which we do pure mathematics. We give some examples of this phenomenon, including (i) the determination of symmetry groups of knots and links, (ii) an extension of Lenstra's approach to the class number problem in number theory, and (iii) the exact determination of representation varieties of 3-manifold fundamental groups.

Contact person: Vaughan Jones
February 12, 2015  
Michael Lacey (Georgia Tech University)


The two weight inequality for Hilbert and Cauchy operators on the disk

Put one weight on the unit circle, and take a function f, square integrable with respect to this weight. For which weights in the disk is the analytic extension of f also square integrable? If one considers the Poisson extension, the question goes back 50 years to Carleson. And in the setting of the Cauchy operator, the question has deep implications in function and operator theory. We will survey the history, and recent solution of this question, reporting on work of the speaker, and Sawyer, Uriate-Tuero, C-Y Shen, and Wick.

Contact person: Alexander Reznikov
February 19, 2015  
Jacques Verstraëte (UC San Diego)


Semirandom methods in combinatorics

The development of the probabilistic method in combinatorics since its inception by papers of P. Erdős has led to groundbreaking results across a broad mathematical landscape. In this talk, I will survey a technique which has come to be known as the semirandom method, starting with the ideas of V. Rödl. Some of the highlights include applications to combinatorial and projective geometry, and most notably the recent proof of the existence of combinatorial designs. The main ideas will be discussed, without delving too far into the technical details, and a number of open problems will be presented.

Contact person: Victor Falgas-Ravry
February 26, 2015  

Faculty meeting, no colloquium


March 5, 2015  

Spring break


March 19, 2015  
David Kerr (Texas A&M University)


Entropy inside out.

In the late 1950s Kolmogorov introduced the concept of entropy into ergodic theory, and since that time entropy has become a pervasive presence in the theory of dynamical systems with applications to various areas including Riemannian geometry, analytic number theory, and Diophantine approximation. Kolmogorov's approach is based on Shannon's theory of information from the 1940s and is most generally applicable to actions of groups satisfying a kind of internal finite approximation property called amenability.
In the last few years a new approach to entropy in dynamics was pioneered by Lewis Bowen and further developed by Hanfeng Li and myself. Here one externalizes the finite approximation of the dynamics so that it occurs outside the acting group, and then counts these models in the spirit of Boltzmann's work in statistical mechanics. This notion of entropy applies to the much larger class of acting groups satisfying the property of soficity, which includes free groups. In fact it is not known whether non-sofic groups exist.
I will discuss all of these developments, and describe how the passage from single transformations to actions of general amenable and sofic groups marks a shift in applications away from geometry and smooth dynamics and more towards noncommutative harmonic analysis and operator algebras.

Contact person: Jesse Peterson
March 26, 2015  
Alexander Volberg (Michigan State University)


Why the oracle may not exist: ergodic families of Jacobi matrices, absolute continuity without almost periodicity.

We will explain the recent solution of Kotani's problem pertinent to the existence/non-existence of ``oracle" (almost periodicity) for the ergodic families of Jacobi matrices (discrete Schroedinger operators). Kotani suggested that such families are subject to the following implication: if family has a non-trivial absolutely continuous spectrum (this happens almost surely) then almost surely it consists of almost periodic matrices (hence the possibility to predict the future by the past). Kotani proved an important positive result of this sort. Recently independently Artur Avila and Peter Yuditskii--myself disproved this conjecture of Kotani (by two different approaches). We will show the hidden singularity that defines when such Kotani's oracle exists or not.

Contact person: Ed Saff
April 2, 2015  
Ross Willard (University of Waterloo)


The finite basis problem for equational theories

Given an algebraic structure (such as a group, ring, semigroup etc.), one can ask which facts in the form of equational laws (such as the associative law, or a distributive law) hold true in the structure. A second-order question one can ask is whether all the true equational laws of the structure follow logically from some finite number of them. This is the finite basis problem of universal algebra. In my talk I will illustrate the problem with examples, and describe some recent progress on an old problem of Bjarni Jonsson.

Contact person: Ralph McKenzie
April 9, 2015  
Slava Kharlamov (Strasbourg University)


Abundance Phenomena in Real Enumerative Geometry.

Surprisingly, in quite a few real enumerative problems the number of real solutions happens to satisfy high lower bounds. For the moment, such a phenomenon is rather deeply studied in the following cases: in counting real lines and higher dimensional linear subspaces on projective hypersurfaces and complete intersections (a typical problem from a kind of Real Schubert Calculus); and in counting real rational curves interpolating real points on rational and K3 surfaces (a typical problem from a kind of Real Gromov-Witten Calculus). In this talk we will explain how in this context one can create a sign count that makes the algebraic number of real solutions independent on input data (Welschinger invariant is one of the most prominent examples) and how one can deduce from such an invariant count an abundance of real solutions. Namely, we see that in all these cases, independently on input data, the number of real solutions taken in the logarithmic scale happens to be approximately equal to the number of complex ones.

Contact person: Rares Rasdeaconu
 
Colloquium Chair (2014-2015): Jesse Peterson

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