Colloquium, Academic Year 2014-2015
Tea at 3:30 pm in 1425 Stevenson Center
|August 21, 2014||
Fall faculty assembly, no colloquium
|August 28, 2014||
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||
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||
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||
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||
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||
|October 23, 2014||
Faculty meeting, no colloquium
|October 30, 2014||
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||
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||
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||
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||
|December 4, 2014||
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
Colloquium Chair (2014-2015): Jesse Peterson