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  • Combinatorial evaluation of Hecke algebra traces. - Lie Groups and Representation Theory

    Speaker: Mark Skandera (Lehigh University) -

    When: Mon, May 1, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: The (type A) Hecke algebra $H_n(q)$ is a certain module over $\mathbb Z[q^{1/2},q^{-1/2}]$ which is a deformation
    of the group algebra of the symmetric group. The $\mathbb Z[q^{1/2},q^{-1/2}]$-module of its trace functions
    has rank equal to the number of integer partitions of $n$, and has bases which are natural deformations of
    those of the symmetric group algebra trace module. While no known formulas give the evaluation of these traces at
    the natural basis elements of $H_n(q)$, there are some nice combinatorial formulas for the evaulation of certain traces at certain Kazhdan-Lusztig basis elements. We will also discuss the open problem of evaluating these traces at other basis elements.
  • Optimal Transportation and Partial Differential Equations - RIT on Applied PDE

    Speaker: Martin Molina (University of Maryland) -

    When: Mon, May 1, 2017 - 3:00pm
    Where: Kirwan Hall 1311
  • Transversals to horocycle flow on the moduli space of translation surfaces - Geometry-Topology

    Speaker: Grace Work (Vanderbilt University) -

    When: Mon, May 1, 2017 - 3:15pm
    Where: Kirwan Hall 1313

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    Abstract: Computing the distribution of the gaps between slopes of saddle connections is a question that was studied first by Athreya and Cheung in the case of the torus, motivated by the connection with Farey fractions, and then in the case of the golden L by Athreya, Chaika, and Lelievre. Their strategy involved translating the question of gaps between slopes of saddle connections into return times under horocycle flow on the space of translation surfaces to a specific transversal. We show how to use this strategy to explicitly compute the distribution in the case of the octagon, how to generalize the construction of the transversal to the general Veech case (both joint work with Caglar Uyanik), and how to parametrize the transversal in the case of a generic genus 2 translation surface.
  • Variance reduction techniques for computing the trace of the inverse of a matrix - Numerical Analysis

    Speaker: Andreas Stathopolous (College of William and Mary) -

    When: Tue, May 2, 2017 - 3:30pm
    Where: 3258 AV Williams

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    Abstract: The computation of the trace of the inverse of a large matrix, A, appears
    in many statistics, machine learning, and quantum mechanics applications.
    Our driving application comes from Lattice Quantum Chromodynamics (LQCD).
    When the size of A does not allow the use of direct methods, most
    applications rely on the Hutchinson method, a Monte Carlo variant that
    requires the solution of a linear system with A per step.
    The variance of the estimator determines the accuracy and therefore the
    computational cost of the method. For Hutchinson, the variance equals the
    squared Frobenious norm of the matrix inverse, excluding its diagonal.

    In this talk, we present some variance reduction techniques we developed
    over the last few years that speed up the Hutchinson method. The goal is
    to approximate the off-diagonal elements of the inverse of A based either
    on structural or on spectral information.

    For LQCD, where the discretization space is a 4D regular lattice torus,
    our Hierarchical Probing method produces a sequence of Hadamard vectors
    that, if used in the Hutchinson method, hierarchically annihilate elements
    of the inverse of A whose vertices in the lattice are increasingly farther
    from each other. Based on the decay of Green's function, this approach
    has yielded significant variance reduction.

    Our second approach is to deflate A from its smallest singular triplets.
    The hope is that the variance of the deflated A is (much) smaller. Contrary
    to low rank matrix approximations, the above deflation may actually
    increase variance. We provide an analysis based on standard random unitary
    matrices, and derive criteria on when to expect improvement.

    Finally, we touch upon the daunting computational task of computing
    the smallest singular triplets of A, and the recent progress our group
    has made in this direction which are included in the software PRIMME.

  • Chen (TBA) - Lie Groups and Representation Theory

    Speaker: Tsao-Hsien Chen (University of Chicago) -

    When: Wed, May 3, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: TBA
  • Nonlinear Dynamics, Chaos and Complex Systems: A historical perspective - Applied Dynamics

    Speaker: Miguel Sanjuán ( Physics, Universidad Rey Juan Carlos) -

    When: Thu, May 4, 2017 - 12:30pm
    Where: ERF 1207
  • TBA - Dynamics

    Speaker: Leonid Koralov (UMD) -

    When: Thu, May 4, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: TBA