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 Events
Mon 8 Jul 2013, 8:00am SPECIAL
Diff. Geom, Math. Phys., PDE Seminar
Earth Sciences Building Rm 2012- 2207 Main Mall
Analysis and Partial Differential Equations
Earth Sciences Building Rm 2012- 2207 Main Mall
Mon 8 Jul 2013, 8:00am-6:00pm

Abstract

Schedule to be posted in the conference webpage:

http://www.pims.math.ca/scientific-event/ghoussoub


This conference brings together world-renowned researchers in areas of mathematical analysis and PDE such as optimal transportation, the calculus of variations, convex analysis, elliptic systems, and geometric analysis, which are grounded in applications to the natural and social sciences while generating exciting new directions for mathematical research. Its primary aims are to survey the state-of-the-art in these interrelated fields, expand the connections between them, identify key future directions, and encourage a new generation of scientists to advance this fundamental area of mathematics.

 

The conference begins Monday 8th morning and ends Friday 12th evening.

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UBC
Wed 10 Jul 2013, 2:00pm
Probability Seminar
MATH 126
Pathwise non-uniqueness for the SPDE’s of some super-Brownian motions with immigration
MATH 126
Wed 10 Jul 2013, 2:00pm-3:00pm

Abstract

Uniqueness theory in stochastic partial differential equations (SPDE’s) concerns their completeness and can induce fundamental properties of solutions such as Markov property. Nonetheless, there remain no robust methods to determine uniqueness in general SPDE’s with non-Lipschitz diffusion coefficients. The most important problem, open for more than two decades, is whether pathwise uniqueness in the SPDE of one-dimensional super-Brownian motion holds. A recent work by Mueller, Mytnik, and Perkins sheds light on this difficult problem, proving, however, that pathwise uniqueness for some closely related SPDE’s fails. In contrast to these particular SPDE’s, the SPDE’s of one-dimensional super-Brownian motions with immigration share more properties with the SPDE of super-Brownian motion but, at the same time, raise additional difficulties in settling the question of pathwise uniqueness.
 
I will first review the SPDE of super-Brownian motion and some notions of uniqueness. I will then introduce the class of super-Brownian motions with immigration considered in our work and discuss our pathwise non-uniqueness result for their SPDE’s. In the rest of this talk, I will explain certain key arguments of our proof.
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Universität Heidelberg
Mon 15 Jul 2013, 12:00pm
Probability Seminar
MATH 126
BCS Models and Functional Integrals
MATH 126
Mon 15 Jul 2013, 12:00pm-1:00pm

Abstract

The theory of superconductivity of Bardeen, Cooper, and Schrie er (BCS) plays an important role in condensed-matter physics in understanding superconductors from fi rst principles. It is based on the idea that electrons in metals can form Cooper pairs, which then behave like bosons and become super fluid at very low temperatures. Mathematically, a proof of existence of a superconducting state as a charge-symmetry breaking state (in the quantum statistical sense of a positive linear functional on the observable algebra), remains a widely open problem.
In this talk I will briefly review the context of BCS theory, introduce BCS models as quantum many-fermion systems with an attractive interaction between Cooper pairs, and then focus on the so-called reduced BCS model, in which the interaction among the Cooper pairs does not decay with their distance. This is a vast simpli cation compared to the general situation, but the analysis of this model remains nontrivial because the underlying algebra is noncommutative. I will sketch a proof by functional integral methods that this model has a superconducting state at low enough temperatures in the thermodynamic limit
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David Kohler
Mon 15 Jul 2013, 12:30pm SPECIAL
One Time Event
Leon's Lounge, Graduate Student Center
Doctoral Exam
Leon's Lounge, Graduate Student Center
Mon 15 Jul 2013, 12:30pm-3:00pm

Details


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State University of New York at Buffalo
Mon 15 Jul 2013, 2:00pm
Probability Seminar
Math 126
The renormalization group according to Balaban
Math 126
Mon 15 Jul 2013, 2:00pm-3:00pm

Abstract

Over the years Balaban has developed a powerful renormalization group technique that is applicable to many problems in quantum field theory and statistical mechanics. In this expository talk we fi rst review his results for various gauge theories and the linear sigma model. Then we describe in some detail how the method can be applied to a simpler case, namely to the ultraviolet problem for the \phi^4_3 model. The treatment deals with the basic renormalization problem in a somewhat novel way which fits naturally with Balaban's scheme. This is a discrete dynamical systems method that makes no reference to perturbation theory.
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Roland Bauerschmidt
Mon 15 Jul 2013, 4:00pm SPECIAL
One Time Event
Graduate Student Center, Room 200
Doctoral Exam
Graduate Student Center, Room 200
Mon 15 Jul 2013, 4:00pm-6:30pm

Details

The main results of this thesis concern the spatial decomposition of
Gaussian fields and the structural stability of a class of dynamical
systems near a non-hyperbolic fixed point. These are two problems that
arise in a renormalization group method for random fields and
self-avoiding walks developed by Brydges and Slade. This renormalization
group program is outlined in the introduction of this thesis with
emphasis on the relevance of the problems studied subsequently. 

The first original result is a new and simple method to decompose the
Green functions corresponding to a large class of interesting symmetric
Dirichlet forms into integrals over symmetric positive semi-definite and
finite range (properly supported) forms that are smoother than the
original Green function.  This result gives rise to multiscale
decompositions of the associated free fields into sums of independent
smoother Gaussian fields with spatially localized correlations. Such
decompositions are the point of departure for renormalization group
analysis. The novelty of the result is the use of the finite propagation
speed of the wave equation and a related property of Chebyshev
polynomials. The result improves several existing results and also gives
simpler proofs. 

The second result concerns structural stability, with respect to
contractive third-order perturbations, of a certain class of dynamical
systems near a non-hyperbolic fixed point. We reformulate the stability
problem in terms of the well- posedness of an infinite-dimensional
nonlinear ordinary differential equation in a Banach space of carefully
weighted sequences. Using this, we prove the existence and regularity of
flows of the dynamical system which obey mixed initial and final
boundary conditions. This result can be applied to the renormalization
group map of Brydges and Slade, and is an ingredient in the analysis of
the long-distance behavior of four dimensional weakly self-avoiding
walks using this approach. 
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Matthew Folz
Fri 19 Jul 2013, 9:00am SPECIAL
One Time Event
Graduate Student Center, Room 200
Doctoral Exam
Graduate Student Center, Room 200
Fri 19 Jul 2013, 9:00am-11:30am

Details

This thesis discusses various aspects of continuous-time simple random walks on measure weighted graphs, with a focus on behaviors related to large-scale geometric properties of the underlying graph.  In contrast to previous work in this area, the majority of the results presented here are applicable to random walks with unbounded generators.  A recurring theme in this research is the use of novel distance functions for graphs known as adapted metrics, which are demonstrated to be a powerful tool for studying random walks on graphs.

Using adapted metrics, we prove two off-diagonal estimates for heat kernels of continuous-time simple random walks; a Poisson-type long range estimate which is valid unconditionally, and a stronger, Gaussian-type estimate which is valid in a restricted space-time region.These results hold under mild geometric hypotheses and are applicable to many models of random walks in random environments.

Subsequently, we prove sharp upper bounds for the bottom of the essential spectrum of graph Laplacians in terms of the adapted volume growth.  We prove two bounds, depending on whether the generator is bounded or not.  Our estimate for the bounded case generalizes prior results of Fujiwara for graphs, and our estimate for the unbounded case is analogous to a result of Brooks for Riemannian manifolds.

Finally, we prove sharp criteria relating adapted volume growth to stochastic completeness of graphs.  To do this, we construct a diffusion on a metric graph which behaves very similarly to the random walk under consideration, which makes it possible to use techniques from the theory of strongly local Dirichlet forms.  This result is a significant improvement over the best previous results on this problem, and is analogous to the sharp result of Grigor'yan for stochastic completeness of Riemannian manifolds.
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Universiteit Leiden
Wed 31 Jul 2013, 3:00pm
Probability Seminar
Math 126
The gaps left by a Brownian motion
Math 126
Wed 31 Jul 2013, 3:00pm-4:00pm

Abstract

Run a Brownian motion on a torus for a long time. How large are the random gaps left behind when the path is removed? In three (or more) dimensions, we find that there is a deterministic spatial scale common to all the large gaps anywhere in the torus. Moreover, we can identify whether a gap of a given shape is likely to exist on this scale, in terms of a single parameter, the classical (Newtonian) capacity. I will describe why this allows us to identify a well-de fined "component" structure in our random porous set.
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