# Geometric Structures on Lie Groupoids (17w5023)

Arriving in Banff, Alberta Sunday, April 16 and departing Friday April 21, 2017

## Organizers

Eckhard Meinrenken (University of Toronto)

Henrique Bursztyn (Instituto Nacional de Matemática Pura e Aplicada)

(University of Illinois at Urbana-Champaign)

## Objectives

Over the last two decades, the theory of Lie groupoids and Lie algebroids has undergone several exciting developments. On the foundational side, there had been a complete solution of the problem of integrability ("Lie's 3rd theorem") for Lie groupoids, an improved understanding of the relationship between Lie algebroid and Lie groupoid cohomology groups ("Van Est theorems"), and deep results relating multiplicative differential forms on groupoids with their infinitesimal counterparts. On the other hand, Lie groupoids were found to be a natural framework for many applications, such as index theory of elliptic operators, Stokes phenomena in complex analysis, generalized complex geometry, geometric flows, and exterior differential systems. Typically, those applications involve additional geometric structures on groupoids:
• Poisson geometry: a symplectic form on the space G of arrows of a groupoid is said to be compatible with the groupoid structure if the graph of the groupoid multiplication is Lagrangian. In such a case, the manifold M of objects of the groupoid inherits a unique Poisson structure for which the source map is a Poisson map [36]. In this context, G is called the symplectic groupoid "integrating" the Poisson manifold M. It can be used to obtain information about the underlying Poisson manifold itself.
• Examples of non-trivial applications include the linearization of Poisson structures around leaves [16,19] and the linearization of Poisson Lie groups [1].

• Dirac geometry: Originally conceived as a framework for Dirac's theory of second class constraints in geometric mechanics,
• Dirac geometry has emerged as a flexible generalization of Poisson geometry. The global objects integrating Dirac manifolds are the presymplectic groupoids [11]. Lie algebroids and Lie groupoids enter many aspects of the theory, such as the construction of Dirac Lie groups [27].

• Generalized complex geometry: Mirror symmetry suggests interesting relations between complex manifolds with
• their symplectic `mirror' manifolds. Generalized complex structures treat symplectic and complex structures on the same footing [25,23]. The corresponding global objects are Lie groupoids with a multiplicative structure consisting of a symplectic form and a complex structure, satisfying certain compatibility relations [14].

• Multiplicative structures on Lie groupoids: generalizing the previous cases, the study of multiplicative structures (differential forms, multivector fields, connections, and so on) and their infinitesimal versions [8, 11, 31] has provided new insights into classical problems of differential geometry. For example, one can recast Cartan's work on Lie pseudogroups in the language of
• multiplicative forms on Lie groupoids, showing that the classical Spencer operator appears as the linearization data of the Cartan Pfaffian system [CrSaSt].

• Riemannian metrics on Lie groupoids: the notion of a Riemannian metric compatible with a groupoid structure is quite subtle and only recently was fully understood [21]. However, special cases of Riemannian groupoids have been studied before and applied with success, for example, in the study of the long time behavior of the Ricci flow [29]. More generally, one can use Riemannian structures on Lie groupoids to obtain deep linearization (or canonical forms) results [21].
• \item Index theory: Connes' approach to the Atiyah-Singer index theorem [12, 22], using the tangent groupoid of a manifold, has greatly influenced the subject and led to a number of advances, such as the Connes-Skandalis longitudinal index theorem for foliations [35]. Over the last few years, there has been a lot of activity towards extending these results to singular foliations [2], as well as developing a theory of (pseudo)-differential operators on groupoids [26,34]

The aim of this workshop is to bring together mathematicians working on the foundational aspects of Lie groupoids with researchers working on applications. We believe that there is a vast potential of this theory, much of which remains to be discovered. Hence, given the current interest by different groups of people and different viewpoints, there is a need for a focused workshop which can bring together the top researchers, promising young researchers, and students in these very active fields.

We plan to organize the workshop around five different topics, each day devoted to a main theme. Each day will start with a keynote lecture by a top researcher in the field. Possible topics for each day are:
1. Symplectic and Poisson structures on Lie groupoids;
2. Multiplicative structures on Lie groupoids;
3. Lie groupoids and differentiable stacks;
4. Riemannian structures on Lie groupoids;
5. Index theory and Lie groupoids.
% Special emphasis will be given to the most promising young researchers in the field. At the end of each day, a special session will be held to discuss open problems and major research directions in each topic.

Researchers in these areas come from many different parts of the world, as can be seen from the list of tentative participants. It is our conviction that BIRS offers an excellent environment and a unique location for gathering the experts and young researchers in the field.

#### Bibliography

1. A. Alekseev and E. Meinrenken, Linearization of Poisson Lie group structures, preprint arXiv:1312.1223, to appear in Journal of Symplectic Geometry.

2. I. Androulidakis and G. Skandalis, The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626 (2009), 1--37.

3. M. Artin, A. Grothendieck, and J.L. Verdier. Seminaire de Geometrie Algebrique du Bois-Marie 1963-1964 (SGA 4). Lecture Notes in Mathematics, Vol. 269.

4. K. Behrend and P. Xu, Differentiable stacks and gerbes, J. Symplectic Geom.,9 Number 3 (2011), 285--341.

5. P. Bouwknegt, A. Carey, V. Mathai, M. Murray and D. Stevenson, Twisted K-Theory and K-Theory of Bundle Gerbes, Communications in Mathematical Physics 228 (2002), 17--49.

6. R.L. Bryant, Bochner-Kaehler Metrics, J. of Amer. Math. Soc. 14 (2001), 623--175.

7. R.L. Bryant, Notes on Exterior Differential Systems, preprint arXiv:1405.3116.

8. H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level, Math. Annalen 353 (2012), 663--705.

9. H. Bursztyn and M. Crainic, Dirac geometry, quasi-Poisson actions and D/G-valued moment maps, J. Differential Geom. 82 (2009), 501--566.

10. H. Bursztyn, G. Cavalcanti and M~Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), 726--765.

11. H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets, Duke Math. J. 123 (2004), 549--607.

12. A. Connes, Noncommutative Geometry, Academic Press, San Diego, 1994.

13. A.S. Cattaneo and G. Felder,Poisson sigma models and symplectic groupoids Quantization of singular symplectic quotients, Progr. Math., vol. 198, Birkhaeuser, Basel, 2001, pp. 61--93.

14. M. Crainic, Generalized complex structures and Lie brackets, Bulletin of the Brazilian Mathematical Society, 42 (2011), 559--578.

15. M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. 78 (2003), no. 4, 681--721.

16. M. Crainic and R.L. Fernandes, A geometric approach to Conn's linearization theorem. Annals of Math 173 (2011), 1119--1137.

17. M. Crainic and R.L. Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), no. 1, 71--137.

18. M. Crainic and R.L. Fernandes, Integrability of Lie brackets, Ann. of Math. 157 (2003), 575--620.

19. M. Crainic and I. Marcut, A normal form theorem around symplectic leaves. Journal of Differential Geometry 92 (2012), 417--461.

20. M. Crainic, A. Salazar and I. Struchiner, Multiplicative forms and Spencer operators. Mathematische Zeitschrift 279 (2015), 939--979.

21. M. del Hoyo and R.L. Fernandes, Riemannian Metrics on Lie Groupoids, preprint arXiv:1404.5989, to appear in: Journal fuer die reine und angewandte Mathematik (Crelle).

22. C. Debord and J.-M. Lescure : Index theory and groupoids. In: Geometric and topological methods for quantum field theory, 86-158, Cambridge Univ. Press, Cambridge, 2010

23. M. Gualtieri, Generalized complex geometry, Annals of Math. 174 (2011), 75--123.

24. A. Haefliger, Homotopy and integrability, Lecture Notes in Mathematics 197 (1971), 133--163.

25. N. Hitchin, Generalized Calabi-Yau manifolds, Quart. J.Math. 54 (2003), 281-308.

26. J.-M. Lescure, D. Manchon, S. Vassout, About the convolution of distributions on groupoids, preprint arXiv:1502.02002.

27. D. Li-Bland, E. Meinrenken, Dirac Lie groups, Asian Journal of Mathematics 18 (2014), 779--816

28. D. Li-Bland, E. Meinrenken, On the Van Est homomorphism for Lie groupoids, preprint arXiv:1403.1191, to appear in L'Enseignement Mathematique.

29. J. Lott, Dimensional reduction and the long-time behavior of Ricci flow, Comment. Math. Helv. 85 (2010), 485--534.

30. K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry Cambridge Univ. Press, 2005.

31. K. Mackenzie and P. Xu, Classical lifting processes and multiplicative vector gields, Quart. J. Math. 49 (1998), 59--85.

32. E. Martinez, Lagrangian Mechanics on Lie Algebroids, Acta Appl. Math. 67 (2001), 295--320.

33. I. Moerdijk. Orbifolds as groupoids: an introduction. In: Orbifolds in mathematics and physics (Madison, WI, 2001), Contemp. Math. 310 (2002), 205-222.

34. V. Nistor, A. Weinstein, and P. Xu, Pseudodifferential operators on differential groupoids, Pacific Journal of Mathematics 189 (1999), 117--152.

35. M. Pflaum, H. Posthuma, X. Tang, The localized longitudinal index theorem for Lie groupoids and the van Est map, Adv. Math. 270 (2015), 223--262.

36. A. Weinstein, Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. 16 (1987), no. 1, 101--104.