I will present some results on discrete gravity and quantum gravity with a cosmological constant.

3d gravity is a topological theory which can be quantized in different equivalent formulations, such as Chern-Simons theory or Loop Quantum Gravity. I will give a quick overview of these different approaches and then focus on the Loop Quantum Gravity perspective. In this context, I will show how the basic building blocks of the theory are homogeneously curved polygons, which phase space is given in terms of Poisson Lie group structures. At the quantum level, I will discuss how the quantum states of these homogeneously curved polygons are labeled by quantum group representations.

I will finally discuss the generalization of this construction to the 4D gravity case with a cosmological constant. I will discuss how (Poisson) Lie cross module (aka Poisson Lie 2-group) structures become relevant.

Bio: Maïté Dupuis is the Director of Training, Educational Outreach and Scientific Programs at Perimeter Institute and an Adjunct Associate Professor in the Applied Mathematics and Physics and Astronomy departments at the University of Waterloo. She was an undergraduate and graduate student at the Ecole Normale Supérieure de Lyon (France) and was awarded a PhD in Physics in 2010. Maïté was a postdoctoral fellow at the Institute for Quantum Gravity of the Fredrich Alexander University Erlangen-Nüremberg in Germany. In 2013, she became a Banting postdoctoral fellow in the Applied Mathematics department of the University of Waterloo. In 2017, she joined the Perimeter Institute Academic Programs department as a lecturer, supervisor and mentor. She is working on Quantum Gravity, more particularly on non-perturbative approaches to Quantum Gravity.