Talks and Abstracts

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.

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_qⁿ⁺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.

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 EH-regular 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 EH-regular. 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 Effros-Hahn conjecture to groupoid C*-algebras. Our work builds on groundbreaking results due to Jean Renault.

Discrete Lagrangian and Hamiltonian mechanics can be expressed in terms of Lie groupoids, in a manner that encompasses both the Moser-Veslov formulation of discrete mechanics, and the discrete analogue of Euler-Poincare 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 Hamilton-Pontryagin 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 Hamilton-Pontryagin variational principle. Dirac integrators generalize both discrete Lagrangian and Hamiltonian variational integrators, as well as nonholonomic integrators.

Since Milner's ground-breaking paper, Functions as processes, it is standard to present computational calculi like the various lambda and pi-calculi in terms of a term grammar, structural equivalence and reduction relation. This presentation allows computational calculi to make contact with more standard generators-and-relations 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.

In this talk, I will describe a process which assigns a Kapranov-Voevodsky 2-vector space to each (finite) groupoid, and a "2-linear map" to each span of groupoids. This process takes a 2-category whose objects are groupoids and whose arrows are spans of groupoids, into a 2-category of 2-vector 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.

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 K-theory K_{G}(Y) and its identity with analytic equivariant K-theory for groupoid actions. The cocycles for the topological K-theory are triples (E,F,t) where E,F are G-Hilbert bundles over Y and t is G-Fredholm. In index theory, every equivariant family of elliptic pseudodifferential operators determines such a cocycle. This topological K-theory, 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 K-theory is just K(C*(G)).
The talk discusses the conjecture that the two K-theories are identical: K_{G}(Y)=K(C*(G)).
In the case G=HxY with H, Y compact, this is a result of Atiyah-Green-Rosenberg-Julg; 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.

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.

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.

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 cross-sectional algebra C*(A). We (think we) have been able to prove that the crossed product C*(A)xG is isomorphic to the cross-sectional 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).

In this talk, we will discuss a groupoid approach to study a physics conjecture on duality of gerbes.

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 non-linear 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.

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.

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.