# Higgs Bundles and Harmonic Maps of Riemann Surfaces (18w5198)

Arriving in Oaxaca, Mexico Sunday, July 1 and departing Friday July 6, 2018

## Objectives

Higgs bundles, arising from the self-dual Yang-Mills equations on a Riemannian 4-manifold by dimensional reduction to a Riemann surface, play an increasingly important role in pure mathematics and mathematical physics. Their study relates the areas of and provides profound insights in algebraic geometry, differential geometry, surface group representations, non-abelian Hodge theory, integrable systems, non-linear PDE, dynamics and theoretical physics. Harmonic maps are intricately connected to Higgs bundles: their existence via heat flow into non-positively curved manifolds provide the equivalence between the holomorphic description of a Higgs bundle and its realization as self-duality equations for a unitary connection and a Higgs field. For special low-rank Lie groups, the self-duality equations give familiar equations from mathematical physics, for instance Toda fields. These in turn describe surfaces with very special differential geometric properties, for instance affine spheres for the real three-dimensional unimodular group. Deep geometric-analytical results about those surfaces have recently been used to give detailed descriptions of certain components of the Higgs moduli space and the corresponding representation varieties. In recent years, there has been an explosion of interest and activity in so-called Higher (rank) Teichmuller theory, which explores the moduli spaces of surface group representations into a Lie group. Dynamical and synthetic geometric perspectives have enriched the study of the deformation spaces defined by Higgs bundles: participation overflowed programs at Institut Henri Poincare in 2012 at MSRI in 2015, and in the GEAR junior and senior retreats of 2012 and 2014.

Harmonic maps also arise in classical differential geometry as minimal surfaces and as Gauss maps of various special surface classes (such as constant mean curvature and Willmore surfaces). Those harmonic maps typically map into positively curved target spaces like spheres or complex projective spaces and as such cannot be obtained by heat flow methods. Nevertheless, these harmonic maps naturally give rise to Higgs bundles in the holomorphic description and holomorphic families of surface group representations. Over the past 10 years this observation, together with insights from the study of the Higgs bundle moduli space, provided a path towards the understanding of higher genus compact minimal surfaces (and more generally harmonic maps) into the 3-sphere. As it turns out the integrable systems description, which lays at the heart of the construction of harmonic tori, has a corresponding construction in the case of higher genus, namely the Hitchin system.

Over the past few years researchers from the two camps, one using harmonic maps to study the Higgs moduli space, and the other using Higgs bundles to study harmonic maps, began to communicate on an individual basis. Key concepts, such as spectral curves, opers, harmonic sequences, completely integrable PDEs, surface group representations, moduli of connections, holomorphic differentials and gluing methods are common to both groups and thus natural candidates for shared explorations. A challenge is to find bridges between the harmonic maps studied by one group -- typically into symmetric spaces of non-compact type -- and those studied by the other group -- typically compact minimal surfaces of higher genus into the 3-sphere and more general compact symmetric spaces. Some research themes of substantial prominence in the Higher (rank) Teichmueller theory realm are unknown to the classical differential geometry community, for example the uniqueness of minimal surfaces in quotients of symmetric spaces by surface groups, the problem of understanding limits of Higgs bundles (generalizations of the Thurston compactification of Teichmueller space), and the geometry of the pressure metric on Hitchin components of surface group representations. Similarly, the description of the moduli of compact minimal surfaces in the 3-sphere and the characterization of the holomorphic family of monodromy representations arising from the associated family of a minimal surface in the 3-sphere are examples of well-known difficult problems in classical differential geometry of surfaces but are unfamiliar to the Higher (rank) Teichmueller theory community. Bringing together key researchers from these groups for the first time will provide an ideal environment for participants of those two camps to interact, to exchange techniques, concepts, ideas, to learn about their respective important questions and open problems, and to foster long-term collaboration thereby advancing the theory of Higgs bundles and harmonic maps significantly.

The format of the workshop is designed to enable and enhance this dialogue: The conference will feature early overview lectures by prominent researchers with a reputation for accessible talks by each group; we have included in the list of invitees some mathematicians who are particularly good at building communities of researchers. These initial lectures and discussions will provide the common backdrop for the later more research-oriented talks and also provide junior researchers with a bird's eye view of the subject areas. These will be followed by more traditional and narrowly focused talks on contemporary research topics. Now, researchers like to talk about problems they have solved, but we will nevertheless ask those people presenting research talks to end their presentations with a discussion of important open problems. Due to the efforts of a few of the researchers in the subject (notably Wienhard, Iozzi), this area is already quite welcoming to women mathematicians: we aim to make a special effort to promote this conference among women mathematicians and under-represented minorities. (Among the list of invitees, we list fifteen women and one Mexican-American.) We also expect to have a number of Ph. D. students attend for whom there will be supplementary support by grants of senior participants.

Harmonic maps also arise in classical differential geometry as minimal surfaces and as Gauss maps of various special surface classes (such as constant mean curvature and Willmore surfaces). Those harmonic maps typically map into positively curved target spaces like spheres or complex projective spaces and as such cannot be obtained by heat flow methods. Nevertheless, these harmonic maps naturally give rise to Higgs bundles in the holomorphic description and holomorphic families of surface group representations. Over the past 10 years this observation, together with insights from the study of the Higgs bundle moduli space, provided a path towards the understanding of higher genus compact minimal surfaces (and more generally harmonic maps) into the 3-sphere. As it turns out the integrable systems description, which lays at the heart of the construction of harmonic tori, has a corresponding construction in the case of higher genus, namely the Hitchin system.

Over the past few years researchers from the two camps, one using harmonic maps to study the Higgs moduli space, and the other using Higgs bundles to study harmonic maps, began to communicate on an individual basis. Key concepts, such as spectral curves, opers, harmonic sequences, completely integrable PDEs, surface group representations, moduli of connections, holomorphic differentials and gluing methods are common to both groups and thus natural candidates for shared explorations. A challenge is to find bridges between the harmonic maps studied by one group -- typically into symmetric spaces of non-compact type -- and those studied by the other group -- typically compact minimal surfaces of higher genus into the 3-sphere and more general compact symmetric spaces. Some research themes of substantial prominence in the Higher (rank) Teichmueller theory realm are unknown to the classical differential geometry community, for example the uniqueness of minimal surfaces in quotients of symmetric spaces by surface groups, the problem of understanding limits of Higgs bundles (generalizations of the Thurston compactification of Teichmueller space), and the geometry of the pressure metric on Hitchin components of surface group representations. Similarly, the description of the moduli of compact minimal surfaces in the 3-sphere and the characterization of the holomorphic family of monodromy representations arising from the associated family of a minimal surface in the 3-sphere are examples of well-known difficult problems in classical differential geometry of surfaces but are unfamiliar to the Higher (rank) Teichmueller theory community. Bringing together key researchers from these groups for the first time will provide an ideal environment for participants of those two camps to interact, to exchange techniques, concepts, ideas, to learn about their respective important questions and open problems, and to foster long-term collaboration thereby advancing the theory of Higgs bundles and harmonic maps significantly.

The format of the workshop is designed to enable and enhance this dialogue: The conference will feature early overview lectures by prominent researchers with a reputation for accessible talks by each group; we have included in the list of invitees some mathematicians who are particularly good at building communities of researchers. These initial lectures and discussions will provide the common backdrop for the later more research-oriented talks and also provide junior researchers with a bird's eye view of the subject areas. These will be followed by more traditional and narrowly focused talks on contemporary research topics. Now, researchers like to talk about problems they have solved, but we will nevertheless ask those people presenting research talks to end their presentations with a discussion of important open problems. Due to the efforts of a few of the researchers in the subject (notably Wienhard, Iozzi), this area is already quite welcoming to women mathematicians: we aim to make a special effort to promote this conference among women mathematicians and under-represented minorities. (Among the list of invitees, we list fifteen women and one Mexican-American.) We also expect to have a number of Ph. D. students attend for whom there will be supplementary support by grants of senior participants.