The n-Category Café

The fun starts with this colloquium talk on Friday:

• Alan Weinstein, Groupoid symmetry for Einstein’s equations?
  
  The Einstein equations for a Lorentz metric on 4-dimensional spacetime may be cast in hamiltonian form, where the configuration space is the space of riemannian metrics on a 3-dimensional manifold. It has been known since work of Dirac 50 years ago that the initial conditions for solutions of these equations are subject to constraints, but our geometric understanding of these constraints is not complete. It is generally felt that the constraints are related to conservation laws associated with the action of the group of diffeomorphisms of spacetime, but this group does not act on the initial data.
  In this talk, I will explain ongoing work with Christian Blohmann (Regensburg) and Marco Cezar Fernandes (Brasilia). We are attempting to show that the initial value constraints for the Einstein equations are associated with the action of a groupOID related to the groupoid of diffeomorphisms between all pairs of hypersurfaces in space-time.

We’ll all go out to dinner after this. The conference proper is on Saturday and Sunday, and then he’ll give a talk about his new paper:

• Alan Weinstein, The volume of a differentiable stack.
  
  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.

I’m looking forward to these talks on groupoids in discretized classical mechanics! The first is by a postdoc with Jerry Marsden’s group at Caltech. He came and visited UCR a while back:

• Joris Vankerschaver, Lie groupoid field theories.
  
  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.
  
  
• Melvin Leok, Discrete Dirac structures and variational discrete Dirac mechanics
  
  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-Poincaré 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.

There will also be three talks on groupoidification:

• Christopher Walker, Groupoidified linear algebra.
  
  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.
  
  
• Alex Hoffnung, A categorification of Hecke algebras.
  
  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.
  
  
• John Baez, Groupoidification.
  
  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.

Jeffrey Morton will be giving a talk that takes groupoidification to a higher level: instead of just turning spans of groupoids into operators between vector spaces, he’s turning spans of spans of groupoids into operators between operators between 2-vector spaces!

• Jeffrey Morton, 2-Vector spaces and groupoids.
  
  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.

I’m also looking forward to some of the talks involving operator algebras, in part because some of them use ‘fields of Hilbert spaces’, which are dear to my heart these days, since they show up in the study of infinite-dimensional 2-Hilbert spaces.