Ph.D. Candidate: Bernardo VillarrealHerrera
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, 12:30pm2:30pm
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 Gequivariant 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 nonunital 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 noncommuting qnilpotent ntuples. We prove this by showing that all these ntuples are conjugated to ntuples 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 (q1)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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Faculty of Mathematics, TU Chemnitz, Germany

Tue 25 Jul 2017, 12:30pm
Scientific Computation and Applied & Industrial Mathematics
ESB 4133 (PIMS Lounge)

A semiLagrangian scheme for the solution of HamiltonJacobiBellman equations

ESB 4133 (PIMS Lounge)
Tue 25 Jul 2017, 12:30pm1:30pm
Abstract
HamiltonJacobiBellman (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 nonexisting secondorder derivatives. In this presentation we focus on the discretization of HJB equations with a fully implicit timestepping scheme based on a semiLagrangian approach. We discuss the algorithmic idea in the context of a finite difference approximation and present numerical examples.
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