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Higgs BUNDLES & HARMONIC MAPS WORKSHOP

January 3-11, 2015
Asheville, NC

Group Photo

We will emphasize and study the role of equivariant harmonic maps as a crucial link between Higgs bundles, geometric structures and representation varieties. After preliminary talks on harmonic maps, Higgs bundles and the geometry of symmetric spaces, further topics will include: Corlette's theorem on the existence of equivariant harmonic maps, relationships between harmonic maps and integrable systems, and harmonic maps to singular spaces. Subsequent talks will be based on current developments in the theory of harmonic maps, Higgs bundles and geometric structures. In particular, applications to studying minimal surfaces in symmetric spaces, deformation spaces of OPERS, the geometry of Anosov representations with quasi-Fuchsian representations and Hitchin representations serving as illustrative examples, and asymptotic analysis of such deformation spaces. The overarching theme may be summarized as follows: how does one unpack the data of Higgs bundles, and via a careful study of harmonic maps, start to understand the geometry of surface group representations in a more explicit way.

Scientific Program

The workshop will consist of whiteboard talks by the participants on the following topics.

Rob Maschal: Higgs bundles background

An outline of the nonabelian Hodge correspondence with focus on Higgs bundles, especially the Hitchin component and $SL(2,\mathbb{C})$.

Suggested literature

Tengren Zhang: Geometry of symmetric and homogeneous spaces

Semisimple Lie groups: Cartan decompositions, Riemannian geometry, and boundaries of the associated symmetric spaces.

Suggested literature

Jérémy Toulisse: Existence theory for harmonic metrics

Corlette's teorem

Suggested literature

Semin Kim: Harmonic maps to $\mathbb{R}$-trees and Morgan Shalen compactification

Suggested literature

Andrew Sanders: Background on Labourie's conjecture on existence and uniqueness of minimal surfaces for Hitchin representations

Suggested literature

Marco Spinaci: Labourie's recent paper: Cyclic surfaces and Hitchin components in rank 2

Suggested literature

Qiongling Li: Background on harmonic maps to metric spaces and survey of Katzarkov, Noll, Pandit, and Simpson's paper: Harmonic maps to Buildings and Singular perturbation theory

Suggested literature

Jorge Acosta: OPERS and complex projective structures

Suggested literature

Brian Collier: The relationship between Integrable systems and Harmonic maps with emphasis on the Toda lattice

Suggested literature

Andy Huang: The relationship between integrable systems and harmonic maps with emphasis on the Toda lattice

Suggested literature

Brice Loustau: Minimal surfaces in $\mathbb{H}^3$ and quasi-Fuchsian representations

Minimal surfaces as equivariant harmonic maps, Taubes moduli space of minimal hyperbolic germs, explicit examples of corresponding $SL(2,\mathbb{C})$ Higgs bundles.

Suggested literature

Jakob Blaavand: Exposition of “Ends of the Moduli Space of Higgs Bundles” by Mazzeo, Swoboda, Weiss, Witt

Suggested literature

Laura Fredrickson: Survey of Taubes’s paper $PSL(2,\mathbb{C})$-connections with $L^2$ bounds on curvature

Suggested literature

Daniele Alessandrini: Branched hyperbolic surfaces and nonmaximal $SL(2,\mathbb{R})$ representations/Higgs bundles

Suggested literature

Participants

Organizers