Colloquium, Academic Year 2014-2015
Tea at 3:30 pm in 1425 Stevenson Center
August 21, 2014 | Fall faculty assembly, no colloquium |
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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 | Contact person: Mark Sapir |
October 16, 2014 | Fall break |
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 | Contact person: Mark Sapir |
November 13, 2014 | Contact person: Jesse Peterson |
November 20, 2014 | Faculty meeting, no colloquium |
November 27, 2014 | Thanksgiving break |
December 4, 2014 | Contact person: Vaughan Jones |
Colloquium Chair (2014-2015): Jesse Peterson