Speaker: Jonathan Brown
(University of Dayton)
Title: Diagonal-Preserving
Ring $ \ast $-Isomorphisms of Leavitt Path Algebras
Abstract: In this talk, we show that the path groupoids of
directed graphs $ E $ and $ F $ are topologically isomorphic if
and only if there is a diagonal-preserving $ \ast $-isomorphism
between the Leavitt path algebras of $ E $ and $ F $. This helps
make progress on a conjecture by Abrams and Tomforde, which states
that if the Leavitt path algebras of $ E $ and $ F $ are
isomorphic as rings, then the $ C^{\ast} $-algebras of $ E $ and $
F $ are $ \ast $-isomorphic. This is joint work with L. Clark and
A. an Huef.
Speaker: Danny Crytser (Kansas
State University)
Title: Traces
Arising from Regular Inclusions
Abstract: In this talk, I will explain how to define and
parametrize tracial states using conditional expectations. The
basic question is: If $ E: A \to B $ is a conditional expectation
onto an abelian $ C^{\ast} $-subalgebra and $ \phi $ is a state on
$ B $, then what conditions must be met by $ \phi $ in order for
the composition $ \phi \circ E $ to be a tracial state on $ A $?
The answer involves normalizers of the $ C^{\ast} $-subalgebra and
is analogous to the notion of an invariant measure for a group
action on a topological space. We give an application to graph $
C^{\ast} $-algebras, parametrizing all tracial states and giving a
necessary and sufficient condition for all tracial states on a
graph $ C^{\ast} $-algebra to be gauge-invariant. This is joint
work with Gabriel Nagy.
Speaker: Valentin Deaconu
(University of Nevada, Reno)
Title: Cuntz-Pimsner
Algebras from Groupoid Actions
Abstract: We recall the concept of a groupoid action on
spaces, on $ C^{\ast} $-algebras, on graphs, and on $ C^{\ast}
$-correspondences. Groups act on objects, but groupoids act on
fibered objects. We construct several $ C^{\ast} $-correspondences
arising from groupoid actions and study the associated
Cuntz-Pimsner algebras. We discuss connections with crossed
products, graph $ C^{\ast} $-algebras, Doplicher-Roberts algebras,
and algebras of self-similar actions. We illustrate with some
examples.
Speaker: Elizabeth Gillaspy
(University of Münster, Germany)
Title: Twists
over Étale Groupoids and Twisted Vector Bundles
Abstract: Given a twist over an étale groupoid, one can
construct an associated $ C^{\ast} $-algebra that carries a good
deal of geometric and physical meaning; for example, the $ K
$-theory group of this $ C^{\ast} $-algebra classifies D-brane
charges in string theory. Twisted vector bundles, when they exist,
give rise to particularly important elements of this $ K $-theory
group. In this talk, we will explain how to use the classifying
space of the étale groupoid to construct twisted vector bundles,
under some mild hypotheses on the twist and the classifying space.
This is joint work with Carla Farsi.
Speaker: Leonard Huang
(University of Colorado, Boulder)
Title: Hilbert $
C^{\ast} $-Modules over Groupoid Dynamical Systems and
Square-Integrable Representations
Abstract: In a 2000 paper, Ralf Meyer gave a reformulation
of Rieffel’s theory of proper group actions on $ C^{\ast}
$-algebras in terms of Hilbert $ (G,A,\alpha) $-modules, where $
(G,A,\alpha) $ is a $ C^{\ast} $-dynamical system. Meyer’s results
were later generalized by Alcides Buss to dynamical systems for a
co-action of a locally compact quantum group on a $ C^{\ast}
$-algebra, and to continuous twisted $ C^{\ast} $-dynamical
systems by myself. In this talk, we outline how one may also
generalize his results to groupoid dynamical systems. To do this,
one works with an upper-semicontinuous bundle of Hilbert $
C^{\ast} $-modules corresponding to an upper-semicontinuous bundle
of $ C^{\ast} $-algebras. It is hoped that such a generalization
can be applied to $ C^{\ast} $-correspondences of directed graphs
equipped with a groupoid action.
Speaker: Daniel Ingebretson
(University of Illinois at Chicago)
Title: Hausdorff
Dimensions of Kuperberg Minimal Sets
Abstract: The Seifert Conjecture was answered negatively in
1994 by Kuperberg who constructed a smooth aperiodic flow on a $ 3
$-manifold. This construction was later found to contain a minimal
set with a complicated topology. This minimal set is embedded as a
lamination by surfaces with a Cantor transversal of Lebesgue
measure $ 0 $. In this talk, we will discuss the pseudogroup
dynamics on the transversal, the induced symbolic dynamics, and
the Hausdorff dimension of the Cantor set.
Speaker: Marius Ionescu (United
States Naval Academy)
Title: Obstructions
to Lifting Cocycles on Groupoids and the Associated $ C^{\ast}
$-Algebras
Abstract: Let $ G $ be an amenable locally compact
groupoid, and let $ A $ be a closed subgroup of a locally compact
abelian group $ B $. Given a $ B / A $-valued $ 1 $-cocycle $ \phi
$ on $ G $, there is a central extension $ \Sigma_{\phi} $ of $ G
$ by $ A $ that is trivial if and only if $ \phi $ lifts to a $ B
$-valued cocycle. We prove that $ {C^{\ast}}(\Sigma_{\phi}) $ is
isomorphic to the induced algebra of the natural action $
\widehat{B / A} $ on $ {C^{\ast}}(G) $. We also consider a simple
class of examples arising from Čech $ 1 $-cocycles. This is joint
work with Alex Kumjian.
Speaker: Olga Lukina (University
of Illinois at Chicago)
Title: Wild
Solenoids
Abstract: The dynamics of weak solenoids is described in
terms of a pseudogroup action on a transversal to the foliation on
the solenoid. A localization of the pseudogroup is the étale
groupoid, associated to the transversal of the foliation. In weak
solenoids, the transversal is totally disconnected, and the action
of the holonomy pseudogroup is equicontinuous. Our goal is to find
invariants that classify solenoids up to a homeomorphism type. The
choice of a transversal in a weak solenoid is not well-defined,
and so it is natural to ask for characteristics of the action that
are invariant under restriction to a clopen set. We show that
there exist actions that never ‘stabilize’ under such
restrictions. To classify such actions, we introduce an invariant,
called the ‘asymptotic discriminant’. We also construct an
uncountable collection of wild solenoids with pairwise distinct
asymptotic discriminant.
Speaker: Alan Paterson
(University of Mississippi (Professor Emeritus))
Title: The Dual of $ {C^{\ast}}(G) $
Abstract: The talk describes the dual of the $ C^{\ast}
$-algebra of a locally compact groupoid in terms of
positive-definite measures.
Speaker: Laura Scull (Fort Lewis
College)
Title: Mapping
Spaces for Orbispaces
Abstract: Orbifolds, and more generally orbispaces, are a
class of spaces that have well-behaved singularities. These are
often modeled using topological groupoids. The category of
orbispaces can be described as a bicategory of fractions of
groupoids, where a certain class of maps — the Morita equivalences
— have been inverted. Using this approach, we can define a mapping
object that is another groupoid. Our goal is to give this groupoid
a topology so that it becomes another orbispace and (with certain
compactness conditions) becomes an exponential object in the
category of orbispaces. I will discuss the main difficulties in
doing this and the results that allow us to overcome them. This is
joint work with D. Pronk at Dalhousie University.
Speaker: Jack Spielberg (Arizona
State University)
Title: Groupoids from Permutations, Continued
Abstract: Given a permutation $ f $ of a set $ S $, one can
define a semigroup (with identity) by the presentation: $ \langle
S \mid \forall a,b \in S: ~ a f(a) = b f(b) \rangle $. It turns
out that this semigroup is left-cancellative, and hence there is
well-oiled machinery for producing $ C^{\ast} $-algebras from it.
However, it seems that the groupoid approach is particularly
well-suited for sorting out the details. This is (still) work in
progress, joint with Tron Omland, David Pask and Adam Sørensen.
Speaker: Thomas Timmermann
(University of Münster, Germany)
Title: Quantum
Groupoids — Why, What and How
Abstract: We indicate how the notion of a quantum groupoid
appears in various situations and how this notion can be made
precise, and discuss some examples including quantum
transformation groupoids. Along the way, we try to assume the
perspective of a groupoids person (rather than that of a
quantum-groups person).
Speaker: Jordan Watts (University
of Colorado, Boulder)
Title: Symplectic
Quotients by Circle Actions Are Not Representable
Abstract: Let the circle act on a symplectic manifold in a
Hamiltonian fashion, and fix a value of the momentum map. There is
a question of whether the symplectic quotient at this value is
diffeomorphic to the orbit space of some proper Lie-group action.
Besides defining what exactly we mean by ‘diffeomorphism’, we will
prove that this only occurs if the symplectic quotient is
diffeomorphic to an effective orbifold. In this case, the
corresponding level set of the momentum map has at most one
positive weight or at most one negative weight about each fixed
point.
Last updated: May 15,
2017.