As part of the activities of the Research Group in Poisson Geometry and Lie groupoids of IME-USP, this semester we will have a working group, whose main goal consists in the applications of simplicial techniques to Symplectic and Poisson Geometry.

Overview: Simplicial objects have appeared in different contexts related to the so-called Generalized Geometry, including: the integration of representations up to homotopy (C. Arias Abad and F. Schätz), stacky groupoids and the integration of Lie algebroids (C. Zhu) and more recently, the notion of symplectic 2-groupoid as the global object of a Courant algebroid (R. Mehta and X. Tang).

We will be specially concerned with simplicial objects in the category of smooth manifolds, namely, simplicial manifolds. Simplicial manifolds have appeared in connection with foliation theory, characteristic classes and cohomology theories, but also include Lie groupoids as special instances. Higher simplicial versions of Lie groupoids are called Lie n-groupoids. These are certain type of simplicial manifolds determined up to simplices of degree n, e.g.  a Lie 1-groupoid is just the nerve of a Lie groupoid.

Geometric structures on Lie groupoids, such as multiplicative symplectic forms, multiplicative Poisson bivector, foliations, among others, have many applications in Generalized Geometry. For instance, symplectic groupoids integrate Lie algebroids arising from Poisson manifolds, or more generally, pre-symplectic groupoids integrate Dirac structures. The generalized complex version has been achieved recently by M. Gualtieri and M. Bailey.

During this semester, we will study geometric structures on higher groupoids with focus on symplectic structures on Lie 2-groupoids and their relation with Courant algebroids.

We will study the recent preprint “Constant symplectic 2-groupoids” by R. Mehta and X. Tang. Also, we plan to study part of the notes “Differential forms on stacks” by E. Ezra Getzler.

Meetings: Every Monday, 16:00-17:00

Where: Room 241, Building A of the Institute of Mathematics and Statistics

Contact:: If you have any suggestion or you would like to give a talk, please contact one of the organizers Cristian Ortiz (cortiz@ime.usp.br) or Fernando Studzinski (fernando.studz@gmail.com)

Next Talk: April 03, 2017 - 16:00

Title: “The tangent complex of a simplicial manifold”

Speaker: Fernando Studzinski (IME-USP)

Previous Talks   

March 27

Title: “Simplicial manifolds and Lie n-groupoids”

Speaker: Fernando Studzinski (IME-USP)