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)
This year it will take place in Sunriver, Oregon. Each participant will give a 2h30 lecture on a previously assigned topic. An excursion to Crater Lake National Park is also scheduled.

Scientific program:
1 - Lie groups and symmetric spaces
Speaker: Michelle Chu (UCSB)
References:
  • Helgason, "Differential Geometry, Lie groups, and symmetric spaces". Graduate Studies in Mathematics, AMS.
  • Tengren's notes.
2 - Introduction to non-abelian Hodge theory
Speaker: Lorenzo Ruffoni (FSU)
References:
  • 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
Speaker: Jane Wang (MIT)
References:
  • 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
Speaker: Xuesen Na (University of Maryland)
References:
5 - The geometry of convex real projective structures
Speaker: Martin Bobb (UT Austin)
References:
  • 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
Speaker: Xian Dai (Rice)
References:
  • 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
Speaker: Max Riestenberg (UT Austin)
References:
  • 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
Speaker: Giuseppe Martone (University of Michigan)
References:
  • 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
Speaker: Feng Zhu (University of Michigan)
References:
  • 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.

Tentative schedule:

Saturday 11 Sunday 12 Monday 13 Tuesday 14 Wednesday 15 Thursday 16 Friday 17 Saturday 18
9:30-10:45
A
R
R
I
V
A
L
Michelle
Jane
Martin
E
X
C
U
R
S
I
O
N
Max
Feng
D
E
P
A
R
T
U
R
E
10:45-11:15 Coffee-break
Coffee-break
11:15-12:30 Michelle
Jane
Martin
Max
Feng
12:30-3:00 Lunch
Lunch
3:00-4:15 Lorenzo
Xuesen
Dai
Giuseppe
F
R
E
E
4:15-4:45 Coffee-break
Coffee-break
4:45-6:00 Lorenzo
Xuesen
Dai
Giuseppe