

 Groupoidification 
 John Baez 
 University of California, Riverside, CA 
 Abstract
There is a systematic process that turns groupoids into vector spaces and
spans of groupoids into linear operators. "Groupoidification" is the
attempt to reverse this process, taking familiar structures from linear
algebra and enhancing them to obtain structures involving groupoids.
Like quantization, groupoidification is not entirely systematic. However,
examples show that it is a good thing to try! For example, groupoidifying
the quantum harmonic oscillator yields combinatorial structures associated
to the groupoid of finite sets. We can also groupoidify mathematics
related to quantum groups  for example, Hecke algebras and Hall algebras.
It turns out that we obtain structures related to algebraic groups defined
over finite fields. After reviewing the idea of groupoidification,
we shall describe as many examples as time permits.



 A Categorification of Hecke Algebras 
 Alex Hoffnung 
 University of California, Riverside, CA 
 Abstract
Given a Dynkin diagram and the finite field F_{q},
where q is a prime power, we get a finite algebraic group G_{q}.
We will show how to construct a categorification of the Hecke algebra
H(G_{q}) associated to this data. This is an example
of the Baez/Dolan program of "groupoidification", a method of
promoting vector spaces to groupoids and linear operators
to spans of groupoids. For example, given the A_{n} Dynkin
diagram, for which G_{q} = SL(n+1,q), the spans over the
G_{q}set of complete flags in F_{q}^{n+1}
encode the relations of the Hecke algebra associated to SL(n+1,q).
Further, we will see how categorified relations of this Hecke algebra
correspond to incidence relations in projective geometry.



 The EffrosHahn Conjecture for Groupoids 
 Marius Ionescu 
 University of Connecticut, StorrsMansfield, CT 
 Abstract
A dynamical system (A,G,\alpha), where A is a C*algebra, G is a locally compact group and \alpha is a strongly continuous
homomorphism of G into Aut(A), is called EHregular if every primitive ideal of the crossed product A \rtimes_\alpha G
is induced from a stability group. In their 1967 Memoir, Effros and Hahn conjectured that if (G,X) was a second countable
locally compact transformation group with G amenable, then (C_0(X),G,lt) should be EHregular. This conjecture, and its
generalization to dynamical systems, was proved by Gootman and Rosenberg building on results due to Sauvageot. In this talk,
which is based on joint work with Dana Williams, we present a proof of the generalization of the EffrosHahn
conjecture to groupoid C*algebras. Our work builds on groundbreaking results due to Jean Renault.



 Discrete Dirac Structures and Variational Discrete Dirac Mechanics 
 Melvin Leok 
 Purdue University, West Lafayette, IN 
 Abstract
Discrete Lagrangian and Hamiltonian mechanics can be expressed in
terms of Lie groupoids, in a manner that encompasses both the MoserVeslov
formulation of discrete mechanics, and the discrete analogue
of EulerPoincare reduction. As such, the correspondence between
discrete and continuous time variational mechanics is naturally
studied in the context of Lie groupoids and Lie algebroids.
Dirac structures generalize symplectic and Poisson structures, and a
unified treatment of Lagrangian and Hamiltonian mechanics can be
expressed either in terms of Dirac structures, or the HamiltonPontryagin
principle on the Pontryagin bundle $TQ \oplus T^*Q$.
Continuous Dirac structures are related to the geometry of
infinitesimally symplectic vector fields, and we introduce discrete
Dirac structures by considering the geometry of symplectic maps.
We provide a characterization of Dirac integrators in terms of
discrete Dirac structures, and a discrete HamiltonPontryagin
variational principle. Dirac integrators generalize both discrete
Lagrangian and Hamiltonian variational integrators, as well as
nonholonomic integrators.



 Groupoids and Structural Equivalence in Computational Calculi 
 Lucius Gregory Meredith 
 Biosimilarity LLC, Seattle, WA 
 Abstract
Since Milner's groundbreaking paper, Functions as processes, it is standard to present computational calculi
like the various lambda and picalculi in terms of a term grammar, structural equivalence and reduction relation.
This presentation allows computational calculi to make contact with more standard generatorsandrelations
presentations of algebras, and to extend them. In particular, standard algebras such as vector spaces, used to
model dynamical systems, encode dynamics in morphisms between such algebras; the computational calculi, on the
other hand, internalize symbolic dynamics as reduction relations on a given structure. In this talk we use groupoids
as a tool to investigate different aspects of this partitioning of dynamics, focusing on two important features:
repartitioning information from structural equivalence to behavioral equivalences and lifting structural and
behavioral information to "notions" of collections via a distribution law over monads.



 2Vector Spaces and Groupoids 
 Jeffrey Morton 
 University of Western Ontario, London, Ontario, Canada 
 Abstract
In this talk, I will describe a process which assigns a KapranovVoevodsky 2vector space to each (finite) groupoid,
and a "2linear map" to each span of groupoids. This process takes a 2category whose objects are groupoids and whose
arrows are spans of groupoids, into a 2category of 2vector spaces, in a way that crucially involves the representations
of the isotropy groups, and is functorial. I will describe this, and suggest how to extend this result to smooth groupoids.



 Equivariant Ktheory for Proper Groupoids 
 Alan Paterson 
 Oxford, MS 
 Abstract
A locally compact groupoid G is called proper if the map that sends g to (r(g),s(g)) is a proper map. Let G be
proper and with unit space Y. The talk discusses a topological equivariant Ktheory K_{G}(Y) and its identity with
analytic equivariant Ktheory for groupoid actions. The cocycles for the topological Ktheory are triples (E,F,t)
where E,F are GHilbert bundles over Y and t is GFredholm. In index theory, every equivariant family of elliptic
pseudodifferential operators determines such a cocycle. This topological Ktheory, in the case where G is a
transformation group HxY, was studied with H, Y compact by G. Segal and, more generally, when H acts properly on Y,
in the book of C. Phillips. Analytic equivariant Ktheory is just K(C*(G)).
The talk discusses the conjecture that the two Ktheories are identical: K_{G}(Y)=K(C*(G)).
In the case G=HxY with H, Y compact, this is a result of AtiyahGreenRosenbergJulg; Phillips in his book shows
more generally that it also holds for the case in which the locally compact group H acts properly on Y. It seems
very likely that the Phillips approach can be adapted to the general groupoid case. A key role in his approach is
played by a stabilization theorem for HxY, and, as a main step to proving the conjecture, we will present a general
groupoid form of this stabilization theorem.



 The C*Envelope of a Semicrossed Product 
 Justin R. Peters 
 Iowa State University, Ames, IA 
 Abstract
Given a dynamical system (X,h) with X compact Hausdorff and h:X>X continuous and surjective, we discuss the two distinct
relations that one could impose on the symbol U which generates the dynamics: either
(1) fU = Uf o h or (2) Uf = f o hU
for all f in C(X). The semicrossed product defined by relation (1) is
generally not isomorphic to that defined by (2), even in the case where
the map h is a homeomorphism.
However, both semicrossed products give rise to the same C* envelope.
The C* envelope is in fact a crossed product with respect to the
minimal lifting homeomorphism.
Two groupoid models for the C* envelope are considered; we indicate
why one model is the correct model.



 On the Fundamental Group of II_1 Factors and Equivalence Relations 
 Sorin Popa 
 University of California, Los Angeles, CA 
 Abstract
I will review some recent progress in calculating fundamental groups of II_1 factors
and equivaelnce relations arising from free ergodic measure preserving actions of countable groups
on probability spaces. One such result, obtained in joint work with Stefaan vaes, shows that
given any subgroup F in R_+ which is either countable or belongs to a certain class
of uncountable groups with arbitrary Hausdorff dimension (after taking the logarithm), there exist "many"
free ergodic probability measure preserving actions of the free group with infinitely many generators
F_\infty such that both the associated II_1 factors and equivalence relations
have fundamental group equal to F.



 Application of Coactions to Direct Integrals 
 John Quigg 
 Arizona state University, Tempe, AZ 
 Abstract
Let A>G be a Fell bundle over a group G. Then there are an associated transformation Fell bundle AxG>GxG and a coaction
of G on the crosssectional algebra C*(A). We (think we) have been able to prove that the crossed product C*(A)xG is
isomorphic to the crosssectional algebra C*(AxG). This has an application to representation theory: a representation of
C*(A)xG comes from a covariant pair (pi,mu) comprising a representation pi of A and a representation mu of C_0(G).
On the other hand, a representation of AxG involves a Borel Hilbert bundle H>G (because G is the unit space of the
transformation groupoid GxG), and linear maps sigma_(s,t):H_t>H_st among the fibers. Thanks to the C*isomorphism, the
representations correspond. Consequently, the representation pi of A decomposes over the Hilbert bundle. This does not
appear to follow directly (pun!) from standard direct integral theory.
This is work in progress, and is joint with Steve Kaliszewski (maybe others later). 


 Group Extensions and Duality of Gerbes 
 Xiang Tang 
 Washington University, St. Louis, MO 
 Abstract
In this talk, we will discuss a groupoid approach to
study a physics conjecture on duality of gerbes.



 Lie Groupoid Field Theories 
 Joris Vankerschaver 
 California Institute of Technnology, Pasadena, CA 
 Abstract
This talk is meant to give an overview of the use of Lie groupoids in
discrete classical field theories. I will elucidate what it means for a
discrete field theory to take values in a Lie groupoid, and why we would
want to study such a generalization in the first place. Secondly, I will
highlight some applications of these ideas, most notably to the nonlinear
sigma model, and I will show how the groupoid concept allows us to unify
and extend many previously known results in the theory of discrete fields.



 Groupoidified Linear Algebra 
 Christopher Walker 
 University of California, Riverside, CA 
 Abstract
Linear algebra is a core idea in almost all areas of mathematics. In this talk we will look at a process called
“groupoidification” which describes the basic structures of linear algebra in the language of groupoids.
Groupoidification is the reverse of a systematic process called “degroupoidification”, which turns groupoids into
vector spaces, and spans of groupoids into linear operators. Even though groupoidification itself is not a systematic
process, we will still be able to find analogs of the main operations in Hilbert spaces including addition, scalar
multiplication, and the inner product.



 The Volume of a Differentiable Stack 
 Alan Weinstein 
 University of California, Berkeley, CA 
 Abstract
We extend to the setting of Lie groupoids the notion of the cardinality of a finite groupoid
(a rational number, equal to the Euler characteristic of the corresponding discrete
orbifold). Since this quantity is an invariant under equivalence of groupoids, we call it the volume
of the associated stack rather than of the groupoid itself. Since there is no natural measure in the smooth case
like the counting measure in the discrete case, we need extra data to define the volume.
This data has the form of a section of a natural line bundle over the stack.
In the case of a group acting on itself by conjugation, or on its Lie algbera by the adjoint representation,
there is a canonical section of this line bundle and hence a canonical measure on the quotient stack.
The talk will not require prior knowledge of stacks.


