Saturday December 2, 2017

10-10:30 Coffee & light breakfast

10:30-11:20 Markus Pflaum (University of Colorado at Boulder)

Title: The formal solution space and formal integrability of non-linear PDE's

Abstract: In the talk, we explain how to formulate PDE's within the framework of jet spaces. This allows the definition of the so-called formal solution space of a non-linear PDE. In case the PDE is formally integrable, the formal solution space carries in a natural way the structure of a profinite dimensional manifold. We also explain the fundamentals of this particular category of infinite dimensional manifolds, and show that in many ways the profinite dimensional manifolds appearing as formal solution spaces of formally integrable PDE's are easier to deal with than the real solution spaces of these PDE's. In addition, we prove a new criterion for formal integrability formally integrable PDE's and derive from it that the Euler-Lagrange Equation of a relativistic scalar field with a polynomial self-interaction is formally integrable. The talk is on joint work with Batu Gueneysu, Humboldt University, Berlin.

11:40 - 12:30 Ivan Contreras (UIUC)

Title: Integration of poly-Poisson structures.

Abstract: Poly-Poisson geometry can be traced back to de-Donder and Weyl in 1930's. This approach leads to a poly-symplectic formulation of Lagrangian field theories, with several applications to mechanics. In this talk we address the problem of integration of poly-Poisson manifolds via Lagrangian field theories with boundary, which is a natural extension of the path space construction for the Poisson sigma model. Joint work with N. Martinez Alba (arXiv: 1706.0614)

2:30-3:20 Junwu Tu (University of Missouri Columbia)

Title: Categorical primitive forms and Gromov-Witten invariants of matrix factorizations

Abstract. We generalize Saito’s definition of primitive forms in singularity theory to the categorical setup. We also extend Li-Li-Saito’s correspondence between the set of primitive forms and the set of certain splittings of the Hodge filtration to the categorical setup. This result, when applied to the category equivariant matrix factorizations, yields a definition of B-model genus zero Gromov-Witten type invariants of Landau-Ginzburg orbifolds, conjecturally mirror to the FJRW theory. This is a joint work with Andrei Caldararu.

3:30-4:20 Daan Michiels (UIUC)

Title: Associativity of local Lie groupoids

Abstract. A theorem by Mal'cev characterizes the local Lie groups that can be embedded into a global Lie groups. The only obstruction turns out to lie in the (higher) associativity of the local groups. In work by Olver, local Lie groups are classified in terms of Lie groups. We generalize these two results to the setting of local Lie groupoids.

Furthermore, the associativity of a local Lie groupoid turns out to be intimately related to the integrability of its Lie algebroid. We clarify this relationship, showing that, in some sense, the lack of higher associativity is a combinatorial counterpart to the monodromy groups of Crainic and Fernandes.

This is joint work with Rui Fernandes.

4:20-4:50 Coffee break

4:50-5:40 Hessel Posthuma (University of Amsterdam)

Title: Resolutions of proper Lie groupoids

Abstract: Proper Lie groupoids are generalizations of proper actions of Lie groups and many of the good properties of such actions generalize to the setting of proper Lie groupoids. In this talk I will discuss an example of such a property: the existence of resolutions. I will define the notion of a resolution of a proper Lie groupoid, and after this I will show that such a resolution always exists by a sequence of blow-ups. Further topics to be discussed are: Morita invariance of this blow-up, and the extension of this result to proper Riemannian Lie groupoids, i.e., equipped with a metric. This is joint work with X. Tang and K. Wang.

Host: Xiang Tang