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Ph.D. Candidate: Bernardo Villarreal-Herrera
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


 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)..
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


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.