Elizabeth Gillaspy (University of Münster, Germany)
X
Between 2012 and 2015, Kumjian, Pask and Sims introduced cohomology theory for higher-rank graphs (-graphs). Higher-rank graphs provide concrete examples of -algebras that are relatively easy to compute and to study, in particular because one can also view these -algebras as arising from groupoids. Group cohomology theories often generalize to groupoids, leading one to ask how the cohomology of a -graph relates to the cohomology of its associated groupoid. This natural question was only answered in a few special cases by Kumjian, Pask and Sims, using ad hoc arguments that seemed unlikely to generalize.
In joint work with Alex Kumjian, we have found a new way to describe the cohomology of a -graph, which is equivalent to the original definition of Kumjian, Pask and Sims, but which enables us to describe a homomorphism from -graph cohomology to groupoid cohomology in full generality. Along the way, we discovered that for -graph groupoids (and in fact, for a much more general class of groupoids, namely those associated to actions of directed semigroups), the groupoid cohomology can be computed from a relatively small, friendly subset of : the Renault-Deaconu semigroupoid of . These insights all rely on recasting the cohomology of a -graph, and of its associated groupoid, in category-theoretic terms.
In this talk, we will begin by presenting the category-theoretic definition of groupoid cohomology, which we relied on for results mentioned above. After showing that this perspective is equivalent to the usual definition of the continuous cocycle cohomology, we observe that the new definition generalizes easily to the setting of -graphs and of Renault-Deaconu semigroupoids, and again matches with the more familiar definitions of cohomology in the literature. These preliminary results, together with an inductive limit functor connecting -graph modules and groupoid sheaves, combine to prove our main results: the homomorphism from -graph cohomology to groupoid cohomology, and the isomorphism between groupoid cohomology and Renault-Deaconu-semigroupoid cohomology.
Cohomology for -Graphs, Groupoids, and Renault-Deaconu Semigroupoids Sponsored by the Meyer Fund
Oct. 06, 2016 3pm (MATH 350)
Probability
Vladimir Rotar (San Diego State University)
X
The talk presents a survey and some remarks regarding the notion of pathwise asymptotic optimality in infinite horizon stochastic optimization problems: setups, approaches, results.
Retrospective Remarks on Pathwise Asymptotic Optimality with an infinite horizon
Oct. 06, 2016 11pm (Math 350)
Noncomm Geometry
John Lott (University of California, Berkeley)
X
It is an invariant of smooth manifolds that combines differential forms with K-theory. I'll give background material and describe some models for differential K-theory. One of them, based on Hilbert bundles, is joint work with Alexander Gorokhovsky.
What is differential K-theory?" Sponsored by the Meyer Fund
Cohomology for -Graphs, Groupoids, and Renault-Deaconu Semigroupoids