**Friday,
January 3, 2014**
09:00 - 10:00 **Paul F. Baum** (Penn State)

The Peter-Weyl-Galois theorem for compact principal bundles

Let G be a compact Hausdorff topological group, and let X be
a compact Hausdorff topological space with a given continuous
action of G. The talk will prove that the action of G on X is
free if and only if the canonical map resulting from viewing an
appropriate algebra of functions on X as a comodule algebra over
the Hopf algebra of polynomial functions on G is an isomorphism.

This is joint work with Piotr M. Hajac and Kenny De Commer.

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10:30 - 11:30** Piotr M. Hajac** (Warsaw, Poland)

*Free actions of compact quantum group on unital C*-algebras*

Let F be a field, G a finite group, and Map(G,F) the Hopf algebra
of all set-theoretic maps G - F. If E is a finite field extension
of F and G is its Galois group, the extension is Galois if and
only if the canonical map resulting from viewing E as a Map(G,F)-comodule
is an isomorphism. Similarly, a finite covering space is regular
if and only if the analogous canonical map is an isomorphism.
The main result to be presented in this talk is an extension of
this point of view to arbitrary actions of compact quantum groups
on unital C*-algebras. I will explain that such an action is free
(in the sense of Ellwood) if and only if the canonical map (obtained
using the underlying Hopf algebra of the compact quantum group)
is an isomorphism. In particular, we are able to express the freeness
of a compact Hausdorff topological group action on a compact Hausdorff
topological space in algebraic terms. Also, we can apply the main
result to noncommutative join constructions and coactions of discrete
groups on unital C*-algebras. (Joint work with Paul F. Baum and
Kenny De Commer.)

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13:30 - 14:30 **Jonathan Rosenberg **(Maryland)

*Levi-Civita connections for noncommutative tori *

We show how to define Riemannian metrics and connections on
a noncommutative torus in such a way that an analogue of Levi-Civita's
theorem on the existence and uniqueness of a Riemannian connection
holds. The major novelty is that we need to use two different
notions of noncommutative vector field. Levi-Civita's theorem
makes it possible to define Riemannian curvature using the usual
formulas.

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15:00 - 16:00 **Mira Peterka** (Kansas)

*Stable Rank of the Theta-Deformed Spheres.*

We show that any theta-deformed sphere (of arbitrary dimension)
has topological stable rank equal to 2 in the case that all deformation
parameters are irrational. We also show that the stable rank can
exceed 2 if some of the deformation parameters are irrational
and others are rational. We compare these results to some related
results of T. Sudo. Time permitting, we will go in a different
direction and discuss some very preliminary results concerning
modules over theta-deformed complex projective spaces.

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### Saturday January 4, 2014

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09:00 - 10:00 **Tomasz Brzezinski **(Swansea, Wales)

* Noncommutativity and resolution: Quantum teardrops and the noncommutative
pillow.*

We discuss recently studied examples of quantum or noncommutative
orbifolds and argue that they provide one with an explicit illustration
of noncommutative resolution of singularities. This resolution
can be understood on several levels: as separation of roots in
polynomial equations defining algebraic varieties; finiteness

of the projective dimension in the category of finitely generated
bimodules; isomorphism of integral and differential forms; change
of the C*-description. On the algebraic level, the key observation
is that, when the noncommutativity is introduced, the actions
of groups on manifolds by which these spaces are defined become
free. In these way deformed orbifolds become bases of quantum
(compact) principal bundles. Examples include quantum teardrops,
weighted real projective planes and the noncommutative pillow
manifold.

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11:00 - 12:00 **Albert Sheu** (Kansas)

*Quantum lens space as a groupoid C*-algebra*

Using the classification of all irreducible *-representations
of the quantum lens space obtained by Brzezinski and Fairfax in
their study of quantum line bundles over a quantum teardrop, we
construct directly a concrete groupoid whose groupoid C*-algebra
is the C*-algebraic quantum lens space. This facilitates a way
to explicitly identify those quantum line bundles, found by Brzezinski
and Fairfax inside the quantum lens space, among the well-classified
finitely generated projective modules over the quantum teardrop.

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14:00 - 15:00 **Olivier Gabriel** (Goettingen, Germany)

*A case study of noncommutative U(1)-principal bundles*

We discuss noncommutative U(1)-principal bundles from the topological
and cohomological points of view. After reviewing the C*-algebraic
setting, we shift attention to Fréchet algebras. Under
certain assumptions, we prove a version of the Pimsner-Voiculescu
exact sequence for periodic cyclic cohomology. We discuss applications
and establish that in the commutative case, our assumptions are
satisfied by smooth U(1)-principal bundles.

**Participant List:**

Paul F. Baum (Penn State University, USA)
Tomasz Brzezinski (University of Swansea, Wales)
George Elliott (University of Toronto, Canada)
Olivier Gabriel (University of Gottingen, Germany)
Piotr M. Hajac (University of Warsaw and IMPAN, Poland)
Byung-Jay Kahng (Canisius College, USA)
Mira Peterka (University of Kansas, USA)
Jonathan Rosenberg (University of Maryland, USA)
Albert Sheu (University of Kansas, USA)
Karen Strung (University of Munster, Germany) |

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