We will study geometric structures arizing from the theory of Higgs bundles and opers, such as complex projective structures, convex real projective structures, and pseudo-Riemannian geometry. We will also study the connections with Anosov and positive representations.

This log cabin Workshop is inspired by a series of similar events organized thanks to the GEAR Network:

- Relative character varieties and parabolic Higgs bundles (2018)
- Workshop on Compactification of moduli spaces of representations (2017)
- Workshop on Sp(4,R)-Anosov representations (2016)
- Workshop on Higgs bundles and harmonic maps (2015)
- Workshop on higher Teichmüller theory (2013)

1 - Lie groups and symmetric spaces

- Helgason, "Differential Geometry, Lie groups, and symmetric spaces".
*Graduate Studies in Mathematics, AMS.* - Tengren's notes.

2 - Introduction to non-abelian Hodge theory

- Hitchin, "The self duality equations on a Riemann surface".
*Proc. London Math. Soc. (3), 55(1):59-126, 1987.* - Hitchin, "Lie groups and Teichmüller space".
*Topology, 31(3):449-473,1992.* - Bradlows's lectures.

3 - Complex projective structures

- Dumas, "Complex projective structures".
*Handbook of Teichmüller Theory, Volume II. EMS, 2009.* - Gallo, Kapovich, Marden, "The monodromy groups of Schwarzian equations on closed Riemann surfaces".
*Annals of Math. 151 (2000), 625-704.* - Wentworth, Wolf, "Surface group representations to SL(2,C) and Higgs bundles with smooth spectral data".
*Geom. Top. 20 (2016), 3019-3032.*

4 - Opers

- Jorge's thesis.
- Dalakov's thesis.
- Sanders, "The presymplectic geometry of opers and the holonomy maps".
*arXiv:1804.04716.*

5 - The geometry of convex real projective structures

- Goldman, "Convex real projective structures on compact surfaces".
*J. Diff. Geom. 31 (1990), 791-845.* - Choi,Goldman, "The classification of real projective structures on compact surfaces".
*Bull. Amer. Math. Soc. 34 (1997),161-171.*

6 - Analytic aspects of convex real projective structures

- Baraglia's thesis.
- Loftin's thesis.
- Dumas, Wolf, "Polynomial cubic differentials and convex polygons in the projective plane".
*Geom. Funct. Anal. 25 (2015),1734-1798.* - Nie, "Poles of cubic differentials and ends of convex real projective structures".
*arXiv:1806.06319.*

7 - Anosov representations

- Guéritaud, Guichard, Kassel, Wienhard, "Anosov representations and proper actions".
*Geom. Top. 21. (2017), 485-584.* - Danciger, Guéritaud, Kassel, "Convex cocompactness in pseudo-Riemannian hyperbolic spaces".
*Geom. Dedicata 192 (2018), 87-126.* - Danciger, Guéritaud, Kassel, "Convex cocompact actions in real projective geometry".
*arXiv:1704.08711.*

8 - Positivity and Hitchin representations

- Fock, Goncharov, "Moduli spaces of local systems and higher Teichmüller theory".
*Publ. Math. Inst. Hautes Études Sci. (2006), 1-211.* - Guichard, Wienhard, "Positivity and higher Teichmüller theory".
*Proceedings of the 7th European Congress of Mathematics, 2016.* - Aparicio-Arroyo, Bradlow, Collier, Garcia-Prada, Gothen, Oliveira, "SO(p,q)-Higgs bundles and higher Teichmüller theory".
*arXiv:1802.08093.* - Aparicio-Arroyo, Bradlow, Collier, Garcia-Prada, Gothen, Oliveira, "Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars".
*Comptes Rendus Mathematiques, Vol. 356 (2018), 666-673.*

9 - Anosov representations in affine geometry

- Danciger, Zhang, "Affine actions with Hitchin linear part".
*arXiv:1812.03930.* - Ghosh, Treib, "Affine Anosov representations and proper actions".
*arXiv:1711.09712.* - Ghosh, "Avatars of Margulis invariants and proper actions".
*arXiv:1812.03777.*