We discuss Markov chain sampling problems arising in computational chemistry. In particular, we show how to sample functions of paths, especially hitting times, in the presence of practical difficulties like metastability and slow convergence. We show how a mathematical framework for these techniques allows for quantification of convergence, opening the way for improvement in efficiency and minimization of error.
Sampling distributions of Markov chains
Apr. 26, 2016 1pm (MATH 220)
Grad Algebra/Logic
Jakub Bulin (CU Boulder) Finitely related algebras, Part 2
In recent work with Laca, Neshveyev, Sims and Webster we investigate the factor types of the extremal KMS states for the preferred dynamics on the -algebra of a strongly connected finite k-graph. At inverse temperature 1, the prevalent outcome is a type III factor. We compute its Connes invariant in terms of the spectral radii of the coordinate matrices and the degree of cycles in the graph. Our analysis has two main steps: one is to analyze the von Neumann algebra of the equivalence relation of the -graph groupoid, the other is to formulate and prove a Frobenius-type result for the family of matrices associated to the -graph.
Von Neumann algebras of strongly connected higher-rank graphs Sponsored by the Meyer Fund
For this talk we will consider orbifolds as Lie groupoids with certain properties with generalized maps between them. The generalized maps are obtained by taking the bicategory of fractions of the 2-category of orbifold groupoids with groupoid homomorphisms. The fractions are needed because we want to consider Morita equivalences as invertible We will first consider what generalized maps between orbifolds are and how we can think of them in terms of a generalization of an atlas refinement together with a map on the refinement. We will then discuss the structure of the 2-cells between them (the arrows of the mapping groupoid) and find that this is simpler than what one would expect for an arbitrary bicategory of fractions, and we will give a nice geometric interpretation. We will end by calculating various examples, and in particular showing that the two ineffective groupoids that Laura Scull presented last week are indeed not Morita equivalent.
Mapping Spaces for Orbifolds Sponsored by the Meyer Fund
Sampling distributions of Markov chains