# Florent Schaffhauser

## Biography

### Department of Mathematics, University of Los Andes, Bogota, Colombia **& USIAS Fellow at the Institute for Advanced Mathematical Research (IRMA), University of Strasbourg**

Born in 1979, **Professor Florent Schaffhauser** obtained his PhD from the Parisian University Pierre and Marie Curie (UPMC) in 2005, and held a post-doctoral position at Keio University (Yokohama, Japan) over the next four years. After research stays at the French Institute of Advanced Scientific Studies (IHES) and the Max Planck Institute for Mathematics (MPIM) in Bonn in 2009-2010, he moved to the University of Los Andes (Bogota, Colombia) where he held the position of Assistant Professor and, since 2012, Associate Professor.

His research interests lie in the area between differential and algebraic geometry, specifically in studying algebro-geometric objects called moduli spaces using tools of differential geometry, in particular symplectic geometry and gauge theory - two domains that originate from theoretical physics (classical mechanics and field theory). A transverse theme in his research has been equivariant geometry, both in the differential and algebraic setting.

Since 2010, Florent Schaffhauser has been actively involved in transmitting mathematical knowledge to undergraduate and graduate students in Colombia and Latin America, through the organisation of scientific meetings and research schools. He has been invited to give research talks and lectures on his research at academic institutions in Canada, Colombia, France, India, Mexico, Spain and the United States, and to participate in thematic research programmes at the Newton Institute and the Henri Poincaré Institute (IHP).

## Project - The non-Abelian Hodge theory of an orbifold Klein surface

### September 2018 - December 2021

The starting point of this project is the 1965 Narasimhan-Seshadri theorem which, under certain specific assumptions, relates stable vector bundles over a compact Riemann surface to irreducible unitary representations of its topological fundamental group. Indeed, our project can be seen as part of a vast programme to generalise their results that was already initiated in the 1970s and 1980s by Ramanathan, Mehta and Seshadri himself. In the 1980s, the work of Kobayashi, Donaldson, Uhlenbeck, Yau, Hitchin, Corlette and Simpson culminated in what is perhaps the single most important generalisation of the Narasimhan-Seshadri *correspondence* briefly described above. It is a correspondence of the same type, known as *non-Abelian Hodge theory*, this time between stable *Higgs bundles* over a compact Riemann surface and irreducible *linear* representations of its topological fundamental group. Starting in the 1990s, the study of equivariant bundles over a curve endowed with the action of a finite group led to the replacement of the topological fundamental group with an orbifold one, which has notably allowed for applications to real algebraic geometry. Thus, the generalisation of the Narasimhan-Seshadri theorem to the real case was obtained by Biswas, Huisman and Hurtubise for bundles of degree 0 in 2010 and, in 2017, by Florent Schaffhauser to the general case (without forgetting Shuguang Wang’s prescient article from 1991). We now need a theory that would be valid for bundles over any compact *orbifold Klein surface* (a natural generalisation of the notion of a Riemann surface), if possible one that does not use equivariant methods, as those are non-intrinsic in general. This is the main goal of the present project.

More precisely, our project is part of a research field called *Higher Teichmüller Theory* that is represented by Professor Olivier Guichard at the Institute for Advanced Mathematical Research (IRMA) at the University of Strasbourg. Its main goals are the following:

**To introduce higher Teichmüller spaces for fundamental groups of orbi-surfaces and to study their geometry**; a first example of such a space was obtained by joint work with Daniele Alessandrini and Gye-Seon Lee, from Heidelberg University, Germany.**To extend the notion of positivity**, introduced by Olivier Guichard and Anna Wienhard (Heidelberg) in the case of usual higher Teichmüller spaces.-
**To explore the notion of higher Teichmüller space for (orbi-)surfaces of infinite type**, taking advantage of the proximity of experts on the subject such as Athanase Papadopoulos (Strasbourg). There too, new ideas will be needed to deal with analytical difficulties inherent to surfaces of infinite type.

The main collaborators on this project will be Dr. Daniele Alessandrini, Heidelberg University, Professor Olivier Guichard, University of Strasbourg, and Dr. Gye-Seon Lee, Heidelberg University.

## Links

Main publications:

**Decomposable representations and Lagrangian submanifolds of moduli spaces associated to surface groups**. Math. Annalen 342 (2008), no. 2, 405-447. doi:10.1007/s00208-008-0241-4.**Moduli spaces of vector bundles over a Klein surface**. Geom. Dedicata 151 (2011), no. 1, 187-206. doi:10.1007/s10711-010-9526-3.**Real points of coarse moduli schemes of vector bundles on a real algebraic curve**. J. Symplectic Geom. 10 (2012), no.4, 503-534. doi:10.4310/JSG.2012.v10.n4.a2.**With Chiu-Chu Melissa Liu. The Yang-Mills equations over Klein surfaces**. J. Topol. 6 (2013), no. 3, 569-643. doi:10.1112/jtopol/jtt001.**With Indranil Biswas. Vector bundles over a real elliptic curve**. Pacific J. Math. 283 (2016), no. 1, 43–62. doi:10.2140/pjm.2016.283.43.**On the Narasimhan-Seshadri correspondence for Real and Quaternionic vector bundles**. J. Differential Geom. (2017) 105 (1), 119-162. doi:10.4310/jdg/1483655861