In fact, though groups are sufficient to characterize the symmetries of homogeneous structures, there are many objects (such as finite parts of a crystal) which exhibit what is clearly recognized as symmetry, but admit few automorphisms. The correct way to describe these symmetries is to use groupoids. Also, to classify certain mathematical objects (say, Riemann surfaces or Platonic solids) without losing track of their symmetries, groupoids must be used. In this context, the groupoid refines the concept of moduli space, parameterizing the objects to be classified while incorporating their symmetries.
The theory of Lie groupoids is one approach to the problem of endowing an abstract groupoid with geometric structure. The theory of stacks is another such approach.
Groupoids were originally introduced by Brandt in 1926 in a paper on the composition of quadratic forms in four variables. Ehresmann added further structures to groupoids and used them as a tool in differential topology and geometry. In analysis, Mackey used groupoids under the name of virtual groups to allow the treatment of ergodic actions of groups and observed that the convolution operation extends from groups to groupoids allowing one to construct many interesting noncommutative algebras. In this context, groupoid convolution algebras, substituting for algebras of functions (on badly behaved quotient spaces), plays a central role in the noncommutative geometry of Connes; this theory provides a unified study of operator algebras, foliations, and index theory. In algebraic topology, the fundamental groupoid of a topological space has been exploited by Higgins, Brown, and others.
Grothendiek introduced stacks initially to give geometric meaning to higher noncommutative cohomology classes. This is also the context in which gerbes first appeared. During the 1960s, Artin, Deligne, and Mumford worked out the basics of the theory of algebraic stacks, which has been very successful in the theory of moduli. One way to say what a stack is, is that it is a Morita equivalence class of groupoids. Groupoids relate to stacks like open covers relate to manifolds. Just like there are many ways to describe the same manifold by open covers and gluing data, there are many groupoids describing the same stack. Grothendieck had the insight that the correct way to make this notion precise is by using fibered categories.
Though the theory of stacks laid almost dormant for many years, the last decade has seen an explosion of activity. Intersection theory on stacks was pioneered by Gillet and Vistoli. Later, the work of Kontsevich and Manin on quantum cohomology required a highly sophisticated intersection theory machinery on algebraic stacks. The localization formula for virtual fundamental classes plays a critical role in Givental's proof of the mirror conjecture. Kontsevich's recent work on mirror symmetry has revealed deep, and as yet poorly understood, connections between quantization, deformation theory, and stacks.
The theory of Lie groupoids has also undergone rapid development in the last fifteen years. It played a fundamental role in the recent development of Poisson geometry. In a certain sense, Poisson manifolds can be considered to be nonlinear generalizations of Lie algebras. The search for group-like objects for Poisson manifolds lead to the discovery of symplectic groupoids by Karasev and Weinstein. Symplectic groupoids are expected to play an essential role in quantization of Poisson manifolds, just as Lie groups do for Lie algebras.
Field-dependent gauge symmetries appear in several field theories. Among them, considerable attention has recently been directed at the Poisson sigma model of Schaller, Strobl, and Ikeda, who studied nonlinear gauge theory. Roughly speaking, this is a generalization of usual gauge theory where the gauge group is replaced by a Lie groupoid (in this case, the symplectic groupoid of the Poisson manifold). The Poisson sigma model was employed by Cattaneo and Felder in depth, and they showed that the perturbative quantization of this model yields Kontsevich's celebrated star-products. A significant generalization occurs in the Berends-Burgers-van Dam approach to " particles of spin 2 ". They attacked the problem of particles of higher spin by letting the gauge parameters act in a field-dependent way. Fulp-Lada-Stasheff showed that their theory is related to L -algebras.
Groupoids have also been extensively studied by operator algebraists in the last twenty years, not only for their own sake, by extending known properties for groups (the C*-algebra of a groupoid, amenability, cohomology, etc.), but also in a variety of applications. In the domain of operator algebras, and in particular in the classification program of simple C*-algebras, groupoids are an important source of examples.
Locally compact groupoids also provide a unifying framework in which to understand the various forms of the Baum-Connes conjecture for groups, group actions, and foliations. Let us add that the K-theory of groupoids also turns out to be an important tool for studying the physics of aperiodic media, as shown in the work of Bellissard and Kellendonk on gap labeling of quasicrystals.
In noncommutative geometry, pseudodifferential calculus is intimately related to Lie groupoids. For instance, Alain Connes showed that the classical pseudodifferential calculus on a smooth manifold is given by the groupoid MxM and that the analytic index is related to the tangent groupoid of M. In fact, a number of index theorems concern G-invariant elliptic operators on a proper G-manifold; in these cases, the analytic index is an element of the K-theory group of C*(G).
Recently, motivated by physics, there has been an increased interest in gerbes. For instance, in string theory, space-time comes equipped with an additional geometric structure called a B-field, which is indeed a gerbe. In this case, the field strength is represented by a closed 3-form on the space-time, and is the curvature of the gerbe. A B-field naturally defines a twisted version of the K-theory of space-time. According to Witten, the D-brane charges define classes in the twisted K-theory. The relation between gerbes and string theory has been explored by many physicists and mathematicians including Atiyah-Segal-Moore, Freed, and Bouwknegt-Mathai, to name a few. Applications of gerbes to the Wess-Zumino-Witten model have been studied, for instance, by Gawedzki.
In quantum field theory, gerbes arise when one asks whether a given bundle of quantum mechanical projective spaces is a projectivization of a Hilbert space bundle. Nontrivial obstructions to the existence of Hilbert bundles are generated by quantum field theory anomalies. The study of a quantum field theory problem involving massless fermions leads naturally to gerbes, as shown by Carey-Mickelsson and many others in the Adelaide school.
One objective of the proposed program is to bring together the groupoid community (both differential geometers and operator algebraists) and the stack community to encourage more interaction and cross-fertilization between these fields. This is becoming more and more urgent as both fields progress deeper. To this purpose we plan to include a series of introductory graduate courses and lectures on both theories (groupoids and stacks), to facilitate communication between the fields, and write-ups of these lectures might be of interest to a more general audience.
On the other hand, we plan to have the main emphasis of a series of workshops on expositions of recent research. We also wish to bring to the program physicists working on quantum field theory and string theory, since physics has been a steady source of ideas in these subjects.
For instance, in the study of Poisson groupoids, the notion of Courant algebroid has arisen, and is in some sense a homotopy Lie algebra (according to Roytenberg-Weinstein). There is much evidence that there is a close connection between gerbes and Courant algebroids. Courant algebroids are also likely to be connected with chiral algebras and vertex algebras, which have been studied mainly by algebraic geometers, for example, Beilinson and Drinfeld. It is very important to bring together the two groups of mathematicians to clarify these connections.
L^infinity -algebras occur in physics in the framework of the Batalin-Vilkovisky procedure for quantizing gauge theories. On the other hand, the Courant algebroids seem to provide a geometric framework for constrained Hamiltonian systems. It is known that gauge Lagrangians lead to constrained theories in the Hamiltonian formalism. This suggests that homotopy Lie algebras arising in the Batalin-Vilkovisky formalism and those in the Courant formalism might be related. It is thus important to clarify their precise connection. Another example where Courant algebroids come into picture is the generalized complex structure introduced by Hitchin-Gualtieri, which is a Dirac structure of a Courant algebroid. The application of generalized complex structures to questions in geometry and physics has yet to be explored. We hope that these questions will be addressed in the program.
For another example, consider mirror symmetry, which is currently arguably the most exciting web of conjectures in geometry. Stacks are the fundamental tool on the algebra-geometric side of the theory, but deformation quantization is also a central concept. We expect a much better understanding of the intricacies of mirror symmetry to emerge by bringing together experts in the two fields.
Recently, motivated by D-branes and string theory, there has been a great deal of interest in the study of twisted K-theory. The twisted K-theory of a topological space by a torsion class was first studied by Donovan-Karoubi in the early 1970s. Twisted K-theory by a general integer third cohomology class was introduced by Rosenberg using operator algebras. Freed-Hopkins-Teleman recently announced that the twisted equivariant K-theory of a compact Lie group is a Verlinde algebra. Operator algebra K-theory provides a basic tool to understand the twisted K-theory of stacks. We expect a better understanding of the subject by bringing together experts in different fields such as mathematical physics and operator algebras.