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  • Twisted K-homology of compact Lie groups - Geometry-Topology

    Speaker: Jonathan Rosenberg (UMD) -

    When: Mon, October 30, 2017 - 3:15pm
    Where: Kirwan Hall 3206

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    Abstract: Computing the twisted K-homology of compact Lie
    groups is both a good test case for methods of topological K-theory
    and a subject of interest in physics (because of its connection with the
    WZW model). This problem was previously attacked by Moore,
    Hopkins, Braun, C. Douglas, and several others. We outline a new
    approach using a theorem of Khorami and the Segal spectral
    sequence. This leads to problems of computing the Hurewicz
    homomorphism in topological K-homology, which can be solved
    by standard methods in homotopy theory.
  • Adaptive FEM for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver - Numerical Analysis

    Speaker: Christian Glusa (Sandia National Laboratories) -

    When: Tue, October 31, 2017 - 3:30pm
    Where: Kirwan Hall 3206

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    Abstract: We explore the connection between fractional order partial differential
    equations in two or more spatial dimensions with boundary integral
    operators to develop techniques that enable one to efficiently tackle
    the integral fractional Laplacian. We develop all of the components
    needed to construct an adaptive finite element code that can be used to
    approximate fractional partial differential equations, on non-trivial
    domains in \(d\geq 1\) dimensions. Our main approach consists of taking
    tools that have been shown to be effective for adaptive boundary element
    methods and, where necessary, modifying them so that they can be applied
    to the fractional PDE case. Improved a priori error estimates are
    derived for the case of quasi-uniform meshes which are seen to deliver
    sub-optimal rates of convergence owing to the presence of singularities.
    Attention is then turned to the development of an a posteriori error
    estimate and error indicators which are suitable for driving an adaptive
    refinement procedure. We assume that the resulting refined meshes are
    locally quasi-uniform and develop efficient methods for the assembly of
    the resulting linear algebraic systems and their solution using
    iterative methods, including the multigrid method. The storage of the
    dense matrices along with efficient techniques for computing the dense
    matrix vector products needed for the iterative solution is also
    considered. Importantly, the approximation does not make any strong
    assumptions on the shape of the underlying domain and does not rely on
    any special structure of the matrix that could be exploited by fast
    transforms. The performance and efficiency of the resulting algorithm is
    illustrated for a variety of examples.

    This is joint work with Mark Ainsworth, Brown University.
  • Algebraic Combinatorics in Geometric Complexity Theory - Algebra-Number Theory

    Speaker: Greta Panova (Institute for Advanced Study) -

    When: Wed, November 1, 2017 - 2:00pm
    Where: Kirwan Hall 1311

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    Abstract: Some of the oldest classical problems in Algebraic Combinatorics concern finding a "combinatorial interpretation", or, more formally, a #P formula, of structure constants and multiplicities arising naturally in Representation Theory and Algebraic Geometry. Among them is the 80-year old problem of Murnaghan to find a positive combinatorial formula for the Kronecker coefficients of the Symmetric Group. More recently this and other related problems emerged in the newer area of Geometric Complexity Theory -- a program of Mulmuley--Sohoni aimed at resolving computational complexity problems like the P vs NP problem (or more precisely its algebraic version VP vs VNP) via Representation Theory and Algebraic Geometry.

    We will describe what this is all about from Combinatorics to Complexity, and show how the little we know about Kronecker coefficients can still be used to show that the P vs NP problem is even harder to solve than originally expected.

    [This talk will feature results from joint works with Peter B\"urgisser, Fulvio Gesmundo, Christian Ikenmeyer, Igor Pak.]
  • Multiscale convergence properties for spectral approximations of a model kinetic equation - CSCAMM

    Speaker: Dr. Zheng Chen (Computer Science and Mathematics Division, Oak Ridge National Laboratory) -

    When: Wed, November 1, 2017 - 2:00pm
    Where: CSIC 4122

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    Abstract: We prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form N−q, where N is the number of modes and q depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter ε, which measures the ratio of the mean-freepath to the characteristic domain length. In particular, we show that the error in the spectral approximation is O(εN+1). More surprisingly, for isotropic initial conditions, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the lth coefficient of the expansion scales like O(ε2N ) when l = 0 and O(ε2N+2−l) for all 1 ≤ l ≤ N. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on N, the time t, and the initial condition. We investigate specifically the dependence on N, in order to assess whether increasing N actually yields an additional factor of ε in the error. Numerical tests will also be presented to support the theoretical results.
  • TBA - Colloquium

    Speaker: Xuhua He (UMD) -

    When: Wed, November 1, 2017 - 3:15pm
    Where: Kirwan Hall 3206
  • Modeling the Network Dynamics of Pulse-Coupled Neurons - Applied Dynamics

    Speaker: Sarthak Chandra (IREAP, UMD)

    When: Thu, November 2, 2017 - 12:30pm
    Where: ERF 1027

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    Abstract: Computer modeling of neural dynamics is an important component of the long-term goal of understanding the brain. A barrier to such modeling is the practical limit on computer resources given the enormous number of neurons in the human brain (about 10^11.) Our work addresses this problem by developing a method for obtaining low dimensional macroscopic descriptions for functional groups consisting of many neurons. Specifically, we formulate a mean-field approximation to investigate macroscopic network effects on the dynamics of large systems of pulse-coupled neurons and derive a reduced system of ordinary differential equations describing the dynamics. We find that solutions of the reduced system agree with those of the full network. This dimensional reduction allows for more efficient characterization of system phase transitions and attractors. Our results show the utility of these dimensional reduction techniques for analyzing the effects of network topology on macroscopic behavior in neuronal networks.

  • Statistical properties of the Standard map with increasing coefficient - Dynamics

    Speaker: Alex Blumenthal (UMD) -

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

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    Abstract: The Chirikov standard map $F_L$ is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Analysis in this direction is notoriously difficult, and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any value of L.

    I will discuss two related results on a more tractable version of this problem. The first is a kind of ‘finite-time mixing estimate, indicating that for large L and on a suitable timescale, the map $F_L$ is strongly mixing. The second pertains to statistical properties of compositions of standard maps with increasing parameter L: when the parameter L increases at a sufficiently fast polynomial rate, we obtain asymptotic decay of correlations estimates, a Strong Law, and a CLT, all for Holder observables.
  • The bulk-boundary correspondence - RIT on Geometry and Physics

    Speaker: Freddy Cisneros (UMD Physics) -

    When: Thu, November 2, 2017 - 3:30pm
    Where: Physics Bldg 1117