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 Events
Leevan Ling
Department of Mathematics, Hong Kong Baptist Univesity
Tue 4 Jul 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
Kernel based methods and some adaptive algorithms
ESB 4133 (PIMS Lounge)
Tue 4 Jul 2017, 12:30pm-1:30pm

Abstract

A brief introduction of translation-invariant kernel based methods, aka radial basis function methods, for function approximation and PDEs will be given. These methods do not require meshes, but in return, we have to deal with highly ill-conditioned linear systems. In this talk, we will introduce an adaptive algorithm that selects appropriate column subspaces that ensure linear independency.
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Ph.D. Candidate, Shaya Shakerian
Mathematics
Mon 10 Jul 2017, 9:00am SPECIAL
Room 202, Anthropology and Sociology Building, UBC
Borderline Variational Problems for Fractional Hardy-Scorödinger Operators
Room 202, Anthropology and Sociology Building, UBC
Mon 10 Jul 2017, 9:00am-11:00am

Details

Oral Defense Abstract: In this thesis, we investigate the existence of ground state solutions associated to the fractional Hardy-Schrödinger operator on Euclidean space and its bounded domains. In the process, we extend several results known about the classical Laplacian to the non-local operators described by its fractional powers. Our analysis show that the most important parameter in the problems we consider is the intensity of the corresponding Hardy potential. The maximal threshold for such an intensity is the best constant in the fractional Hardy inequality, which is computable in terms of the dimension and the fractional exponent of the Laplacian. However, the analysis of corresponding non-linear equations in borderline Sobolev-critical regimes give rise to another threshold for the allowable intensity. Solutions exist for all positive linear perturbations of the equation, if the intensity is below this new threshold. However, once the intensity is beyond it, we had to introduce a notion of Hardy-Schrödinger Mass associated to the domain under study and the linear perturbation. We then show that ground state solutions exist when such a mass is positive. We then study the effect of non-linear perturbations, where we show that the existence of ground state solutions for large intensities, is determined by a subtle combination of the mass (i.e, the geometry of the domain) and the size of the nonlinearity of the perturbations.

Note for Attendees

Latecomers will not be admitted.
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Ph.D. Candidate: Brett T Kolesnik
Mathematics
Mon 10 Jul 2017, 1:00pm SPECIAL
Room 203, Mathematics Building, UBC
Oral Defense: Geometry of Random Spaces: Geodesics and Susceptibility
Room 203, Mathematics Building, UBC
Mon 10 Jul 2017, 1:00pm-3:00pm

Details

ABSTRACT: This thesis investigates the geometry of random spaces.

Geodesics in Random Surfaces
The Brownian map, developed by Le Gall and Miermont, is a random metric space arising as the scaling limit of random planar maps. Its construction involves Aldous’ continuum random tree, the canonical random real tree, and Brownian motion, an almost surely continuous but nowhere differentiable path. As a result, the Brownian map is a non-differentiable surface with a fractal geometry that is much closer to that of a real tree than a smooth surface.

A key feature, observed by Le Gall, is the confluence of geodesics phenomenon, which states that any two geodesics to a typical point coalesce before reaching the point. We show that, in fact, geodesics to anywhere near a typical point pass through a common confluence point. This leads to information about special points that had remained largely mysterious.

Our main result is the almost everywhere continuity and uniform stability of the cut locus of the Brownian map. We also classify geodesic networks that are dense and find the Hausdorff dimension of the set of pairs that are joined by each type of network.

Susceptibility of Random Graphs
Given a graph G=(V,E) and an initial set I of active vertices in V, the r-neighbour bootstrap percolation process, attributed to Chalupa, Leath and Reich, is a cellular automaton that evolves by activating vertices with at least r active neighbours. If all vertices in V are activated eventually, we say that I is contagious. A graph with a small contagious set is called susceptible.

Bootstrap percolation has been analyzed on deterministic graphs, such as grids, lattices and trees. More recent work studies the model on random graphs, such as the fundamental Erdős–Rényi graph G(n,p).

We study thresholds for the susceptibility of G(n,p), refining approximations by Feige, Krivelevich and Reichman. Along the way, we develop large deviation estimates, which complement central limit theorems of Janson, Łuczak, Turova and Vallier. We also study graph bootstrap percolation, a variation due to Bollobás. Our main result identifies the sharp threshold for K4-percolation, solving a problem of Balogh, Bollobás and Morris.

Note for Attendees

Latecomers will not be admitted.
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Alexander Goncharov
Bilkent University
Fri 14 Jul 2017, 11:00am SPECIAL
Math 126
Equilibrium Cantor Sets
Math 126
Fri 14 Jul 2017, 11:00am-12:00pm

Details

We discuss a family of Cantor sets on which the equilibrium measure coincides with the corresponding Hausdorff measure. Several applications are presented. Some asymptotics for orthogonal polynomials (Widom factors) are given for non Parreau-Widom sets. Our main concern is the concept of the Szeg˝o class for singular measures.
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Ph.D. Candidate: Bernardo Villarreal-Herrera
Mathematics
Thu 20 Jul 2017, 12:30pm SPECIAL
Room 202, Anthropology and Sociology Building, UBC
A Simplicial Approach to Spaces of Homomorphisms
Room 202, Anthropology and Sociology Building, UBC
Thu 20 Jul 2017, 12:30pm-2:30pm

Details

 Oral Defense Abstract: Let G be a real linear algebraic group and L a finitely generated cosimplicial group. We prove that the space of homomorphisms Hom(Ln,G) has a homotopy stable decomposition for each n ≥ 1. When G is a compact Lie group, we show that
the decomposition is G-equivariant with respect to the induced action of conjugation by elements of G. In particular, under these hypotheses on G, we obtain stable decompositions for Hom(Fn /Γ,G) and Rep(Fn /Γq,G) respectively, where Fn /Γq are the finitely generated free nilpotent groups of nilpotency class q − 1. The spaces Hom(Ln,G) assemble into a simplicial space Hom(L,G). When G = U we show that its geometric realization B(L,U) has a non-unital E∞ -ring space structure whenever Hom(L0,U(m)) is path connected for all m ≥ 1. We describe the connected components of Hom(Fn /Γq,SU(2)) arising from non-commuting q-nilpotent n-tuples. We prove this by showing that all these n-tuples are conjugated to n-tuples consisting of elements in the the generalized quaternion groups Q2^q ⊂ SU(2), of order 2^q . Using this result, we exhibit the homotopy type of ΣHom(Fn/Γq,SU(2)) and a homotopy description of the classifying spaces B(q,SU(2)) of transitionally (q-1)-nilpotent principal SU(2)-bundles. The above computations are also done for SO(3) and U(2). We also include cohomology calculations for the spaces B(r,Q2^q ) for low values of r. Finally we compute the integral cohomology ring of BcomG_1 for the Lie groups G=SO(3), SU(2) and U(2)..
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Jan Blechschmidt
Faculty of Mathematics, TU Chemnitz, Germany
Tue 25 Jul 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)
A semi-Lagrangian scheme for the solution of Hamilton-Jacobi-Bellman equations
ESB 4133 (PIMS Lounge)
Tue 25 Jul 2017, 12:30pm-1:30pm

Abstract

Hamilton-Jacobi-Bellman (HJB) equations are nonlinear partial differential equation that arise as optimality conditions of stochastic control problems. HJB equations often possess a variety of difficulties, e.g., discontinuous coefficients, vanishing viscosity, unknown boundary conditions, etc. One particular issue is the handling of non-existing second-order derivatives. In this presentation we focus on the discretization of HJB equations with a fully implicit timestepping scheme based on a semi-Lagrangian approach. We discuss the algorithmic idea in the context of a finite difference approximation and present numerical examples.
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