Mathematical Physics
2016-2017

Regular time and location: Wednesdays 2:00-3:00pm, 326 Kerchof Hall

September 7 Organizational Meeting
September 14 Speaker: Sergio Yuhjtman, University of Buenos Aires
Title: Cluster expansion using Penrose tree-graph identity
Abstract: We consider the cluster expansion for polymer systems with stable two-body interactions. By applying the Penrose tree-graph identity (1967) in a novel way, we obtain improved conditions for the convergence of the expansion. For a classical gas of point particles in the continuum, we improved significantly the lower bound on the convergence radius. Furthermore, the method seems to be useful for a very general class of systems. This is a joint work with Aldo Procacci.
October 12 Speaker: Abdelmalek Abdesselam
Title: A second-quantized Kolmogorov-Chentsov theorem for random Schwartz distributions
Abstract: A long-standing issue in the study of nonlinear phenomena using Schwartz distributions is that pointwise products and powers in general do not make sense. In the random case, the situation is better and there are techniques for, almost-surely, constructing such products via a renormalization procedure. An example of such a procedure is the notion of Wick products for Gaussian random distributions. In this series of talks I will present a new result which is a vast generalization of this construction to the non-Gaussian case relevant, e.g., for scaling limits of statistical mechanics models. The hypothesis of this new theorem is a precise form of Wilson's operator product expansion which is an algebraic structure constraining the behavior of multilinear distributional kernels of correlations near the diagonal.
October 19 Speaker: Abdelmalek Abdesselam
Title: A second-quantized Kolmogorov-Chentsov theorem for random Schwartz distributions (cont.)
Abstract: A long-standing issue in the study of nonlinear phenomena using Schwartz distributions is that pointwise products and powers in general do not make sense. In the random case, the situation is better and there are techniques for, almost-surely, constructing such products via a renormalization procedure. An example of such a procedure is the notion of Wick products for Gaussian random distributions. In this series of talks I will present a new result which is a vast generalization of this construction to the non-Gaussian case relevant, e.g., for scaling limits of statistical mechanics models. The hypothesis of this new theorem is a precise form of Wilson's operator product expansion which is an algebraic structure constraining the behavior of multilinear distributional kernels of correlations near the diagonal.
October 26 Speaker: Abdelmalek Abdesselam
Title: A second-quantized Kolmogorov-Chentsov theorem for random Schwartz distributions (cont.)
Abstract: A long-standing issue in the study of nonlinear phenomena using Schwartz distributions is that pointwise products and powers in general do not make sense. In the random case, the situation is better and there are techniques for, almost-surely, constructing such products via a renormalization procedure. An example of such a procedure is the notion of Wick products for Gaussian random distributions. In this series of talks I will present a new result which is a vast generalization of this construction to the non-Gaussian case relevant, e.g., for scaling limits of statistical mechanics models. The hypothesis of this new theorem is a precise form of Wilson's operator product expansion which is an algebraic structure constraining the behavior of multilinear distributional kernels of correlations near the diagonal.
November 9 Speaker: Abdelmalek Abdesselam
Title: A second-quantized Kolmogorov-Chentsov theorem for random Schwartz distributions (cont.)
Abstract: A long-standing issue in the study of nonlinear phenomena using Schwartz distributions is that pointwise products and powers in general do not make sense. In the random case, the situation is better and there are techniques for, almost-surely, constructing such products via a renormalization procedure. An example of such a procedure is the notion of Wick products for Gaussian random distributions. In this series of talks I will present a new result which is a vast generalization of this construction to the non-Gaussian case relevant, e.g., for scaling limits of statistical mechanics models. The hypothesis of this new theorem is a precise form of Wilson's operator product expansion which is an algebraic structure constraining the behavior of multilinear distributional kernels of correlations near the diagonal.
February 22 Speaker: David Brydges, UBC
Title: The Lace expansion for the $|\varphi|^4$ model
Abstract: Akira Sakai has shown that a convergent lace expansion exists for the Ising and $\varphi^4$ models. He uses the current representation for the Ising model to convert the system to a percolation. In work with Tyler Helmuth and Mark Holmes we give a different expansion based on the Symanzik local time isomorphism. This expansion exists for $|\varphi|^4$, O(n) models and the continuous time lattice Edwards model (n=0), but we can only prove convergence for n=0,1,2 because the GHS inequalities are not known to hold for n>2. As in all other lace expansions, for convergence a small parameter is required. Thus the method gives information on critical exponents for the listed models in high dimensions, or for finite but sufficiently long range coupling.
April 12 Speaker: John Imbrie
Title: Localization and Eigenvalue Statistics for Lattice Schroedinger Operators with Discrete Disorder
Abstract: Convergent expansions for eigenvalues and eigenvectors lead to new insights into the way randomness localizes eigenfunctions, smooths out eigenvalue distributions, and produces eigenvalue separation.
May 11 Speaker: Almut Burchard, University of Toronto
Title: Eulerian calculations in the Wasserstein space
Abstract: The optimal transport problem defines a notion of distance in the space of probability measures over a manifold, the *Wasserstein space*. In his thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called *displacement interpolants*. In contrast with classical function spaces, the Wasserstein space is not a linear space, but rather an infinite-dimensional analogue of a Riemannian manifold. In this talk, I will describe how to differentiate functionals along displacement interpolants, using an Eulerian formulation for the underlying optimal transportation problem. Time permitting, I will sketch how to justify the calculations under minimal regularity assumptions, discuss geometric implications, and mention open questions. (Joint work with Benjamin Schachter)
*indicates different day, time, and/or location


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Contact: John Imbrie (imbrie @ virginia.edu)