68 research outputs found
Asymptotic Hodge Theory of Vector Bundles
We introduce several families of filtrations on the space of vector bundles
over a smooth projective variety. These filtrations are defined using the large
k asymptotics of the kernel of the Dolbeault Dirac operator on a bundle twisted
by the kth power of an ample line bundle. The filtrations measure the failure
of the bundle to admit a holomorphic structure. We study compatibility under
the Chern isomorphism of these filtrations with the Hodge filtration on
cohomology.Comment: 23 page
Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations
We consider the weak convergence of numerical methods for stochastic
differential equations (SDEs). Weak convergence is usually expressed in terms
of the convergence of expected values of test functions of the trajectories.
Here we present an alternative formulation of weak convergence in terms of the
well-known Prokhorov metric on spaces of random variables. For a general class
of methods, we establish bounds on the rates of convergence in terms of the
Prokhorov metric. In doing so, we revisit the original proofs of weak
convergence and show explicitly how the bounds on the error depend on the
smoothness of the test functions. As an application of our result, we use the
Strassen - Dudley theorem to show that the numerical approximation and the true
solution to the system of SDEs can be re-embedded in a probability space in
such a way that the method converges there in a strong sense. One corollary of
this last result is that the method converges in the Wasserstein distance,
another metric on spaces of random variables. Another corollary establishes
rates of convergence for expected values of test functions assuming only local
Lipschitz continuity. We conclude with a review of the existing results for
pathwise convergence of weakly converging methods and the corresponding strong
results available under re-embedding.Comment: 12 pages, 2nd revision for IMA J Numerical Analysis. Further minor
errors correcte
Geometrical Frustration and Static Correlations in a Simple Glass Former
© American Physical Society, Charbonneau, B., Charbonneau, P., & Tarjus, G. (2012). Geometrical Frustration and Static Correlations in a Simple Glass Former. Physical Review Letters, 108(3). Published by APS and available at: https://doi.org/10.1103/PhysRevLett.108.035701We study the geometrical frustration scenario of glass formation for simple hard-sphere models. We find that the dual picture in terms of defects brings little insight and no theoretical simplification for the understanding of the slowing down of relaxation, because of the strong frustration characterizing these systems. The possibility of a growing static length is furthermore found to be physically irrelevant in the regime that is accessible to computer simulations.National Science Foundation [NSF PHY05-51164, NSF DMR-1055586]; NSER
Analytic aspects of periodic instantons
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (p. 131-134) and index.The main result is a computation of the Nahm transform of a SU(2)-instanton over R x T³, called spatially-periodic instanton. It is a singular monopole over T³, a solution to the Bogomolny equation, whose rank is computed and behavior at the singular points is understood under certain conditions. A full description of the Riemannian ADHMN construction of instantons on R⁴ is given, preceding a description of the heuristic behind the theory of instantons on quotients of R⁴. The Fredholm theory of twisted Dirac operators on cylindrical manifolds is derived, the spectra of spin Dirac operators on spheres and on product manifolds are computed. A brief discussion on the decay of spatially-periodic and doubly-periodic instantons is included.by Benoit Charbonneau.Ph.D
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