
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, 8:00am6:00pm
Abstract
Schedule to be posted in the conference webpage:
http://www.pims.math.ca/scientificevent/ghoussoub
This conference brings together worldrenowned 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 stateoftheart 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 nonuniqueness for the SPDE’s of some superBrownian motions with immigration

MATH 126
Wed 10 Jul 2013, 2:00pm3:00pm
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 nonLipschitz diﬀusion coeﬃcients. The most important problem, open for more than two decades, is whether pathwise uniqueness in the SPDE of onedimensional superBrownian 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 onedimensional superBrownian motions with immigration share more properties with the SPDE of superBrownian motion but, at the same time, raise additional diﬃculties in settling the question of pathwise uniqueness.
I will ﬁrst review the SPDE of superBrownian motion and some notions of uniqueness. I will then introduce the class of superBrownian motions with immigration considered in our work and discuss our pathwise nonuniqueness 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, 12:00pm1:00pm
Abstract
The theory of superconductivity of Bardeen, Cooper, and Schrieer (BCS) plays an important role in condensedmatter physics in understanding superconductors from first 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 chargesymmetry 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 manyfermion systems with an attractive interaction between Cooper pairs, and then focus on the socalled reduced BCS model, in which the interaction among the Cooper pairs does not decay with their distance. This is a vast simplication 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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Mon 15 Jul 2013, 12:30pm
SPECIAL
Leon's Lounge, Graduate Student Center

Doctoral Exam

Leon's Lounge, Graduate Student Center
Mon 15 Jul 2013, 12:30pm3:00pm
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, 2:00pm3:00pm
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 first 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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Mon 15 Jul 2013, 4:00pm
SPECIAL
Graduate Student Center, Room 200

Doctoral Exam

Graduate Student Center, Room 200
Mon 15 Jul 2013, 4:00pm6:30pm
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 nonhyperbolic fixed point. These are two problems that
arise in a renormalization group method for random fields and
selfavoiding 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 semidefinite 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 thirdorder perturbations, of a certain class of dynamical
systems near a nonhyperbolic fixed point. We reformulate the stability
problem in terms of the well posedness of an infinitedimensional
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 longdistance behavior of four dimensional weakly selfavoiding
walks using this approach.
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Fri 19 Jul 2013, 9:00am
SPECIAL
Graduate Student Center, Room 200

Doctoral Exam

Graduate Student Center, Room 200
Fri 19 Jul 2013, 9:00am11:30am
Details
This thesis discusses various aspects of continuoustime simple random walks on measure weighted graphs, with a focus on behaviors related to largescale 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 offdiagonal estimates for heat kernels of continuoustime simple random walks; a Poissontype long range estimate which is valid unconditionally, and a stronger, Gaussiantype estimate which is valid in a restricted spacetime 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, 3:00pm4:00pm
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 welldefined "component" structure in our random porous set.
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