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)

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

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

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

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)

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

Contact person: Jesse Peterson
November 20, 2014  

Faculty meeting, no colloquium

November 27, 2014  

Thanksgiving break

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

Contact person: Vaughan Jones
Colloquium Chair (2014-2015): Jesse Peterson

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