Geometric aspects of Higgs bundles

11-18 May 2019
Organizers: Xian Dai, Charles Ouyang, Andrea Tamburelli.

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.

Here you may find the notes from the talks.

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