Geometry Conference: 50 years of Nigel Hitchin's Mathematics

Jorgen Andersen, Danish Institute for Advanced Study: Geometric Quantization of general Kähler manifolds

We will consider Geometric Quantization on general Kähler manifold and propose a program for compatible constructions of the quantization of functions, the Hilbert space structure, and the dependence on the choice of the Kähler structure, fixing only the underlying symplectic manifold and a prequantum line bundle.  Video

Gil Cavalcanti, Universiteit Utrecht: T-duality with fixed points

I will recap some of the history of T-duality from the very biased point of view of a mathematician, including its successes and a few still standing problems. I will then explain how a new differential geometric approach to log-geometry can be used to shed new light into how to do T-duality in the presence of fixed points for the torus action without the need to look for non-compact spaces or introduce ad-hoc potentials. The material of this talk is joint work with Aldo Witte.  Video

Jose Seade, UNAM: Discrete group actions on the complex projective plane CP^2

Classical Kleinian groups are discrete groups of automorphisms of the projective line CP^1. The study of their geometry and dynamics has been for decades the paradigm of complex geometry and holomorphic dynamics. In this talk I will speak about discrete group actions on the projective plane CP^2. This is joint work with several colleagues in Mexico.  Video

Richard Wentworth, University of Maryland: Asymptotics in Hitchin's moduli space

The moduli space of rank 2 Higgs bundles has a much studied very rich structure related to integrable systems, hyperkaehler reduction, mirror symmetry, and supersymmetric gauge theory. The space admits different notions of ideal points at infinity, arising from its various incarnations via the nonabelian Hodge theorem. In this talk, I will present results comparing this asymptotic behavior. One is a refinement of the Morgan-Shalen compactification of the Betti moduli space.

A second comes from the algebraic geometry of the C-star action on the moduli space. And third arises from analytic "limiting configurations" of solutions to the Hitchin equations. I will discuss how the nonabelian Hodge correspondence extends as a map between the latter two (partial) compactifications. Somewhat surprisingly, the extension is not continuous.  Video

Marcos Jardim, UNICAMP: Instanton sheaves, the next frontier

I will outline the history of instanton sheaves on 3 dimensional varieties, from the Atiyah--Ward correspondence and the ADHM construction in the 1970s until the most recent advances involving Bridgeland stability conditions. This is based on joint work with G. Comaschi, C. Martinez, and D. Mu.  Video

Nigel Hitchin, Oxford University: Click here for more information  Video

Carlos Simpson, University of Nice: Parabolic Higgs bundles corresponding to the Hecke eigensheaves appearing in the geometric Langlands program

In this talk we'll describe current work with Ron Donagi and Tony Pantev on their general conjectural program for the construction of Hecke eigensheaves using nonabelian Hodge theory. Their previous work treated the case of the projective line minus 5 points. Here, we consider a curve X of genus 2. Starting from a rank 2 local system on X, the geometric Langlands program associates a local system of rank 8 on a Zariski open subset of the moduli stack Bun of rank 2 bundles. We show how to describe the associated parabolic logarithmic Higgs bundle. In particular, the spectral variety is a very natural one: it is the Hitchin fiber corresponding to the input local system, viewed as a subvariety of T^*(Bun).  Video

Bảo Châu Ngô, University of Chicago: Generalized Hitchin systems  Video

Tamas Hausel, Institute of Science and Technology Austria: Mirror symmetry and big algebras

I will explain how to model the Hitchin system on certain very stable upward flows by the spectrum of equivariant cohomology of a Grassmannian and its mirror by the spectrum of the Kirillov algebra of a minuscule representation of the Langlands dual group. Then we generalise this to non-minuscule representations using a big commutative subalgebra of the Kirillov algebra, ringifying the equivariant intersection cohomology of affine Schubert varieties. I will finish by showing some neat pictures of the skeletons of the big and medium algebras of the octet and decuplet of SL(3).  Video

Karen Uhlenbeck, University of Texas at Austin: Click here for more information  Video

Anna Wienhard, Max Planck Institute for Mathematics in the Sciences: Positivity and higher Teichmüller spaces

About thirty years ago, Hitchin introduced the Teichmüller components in varieties of representations of surface groups into split real Lie groups of higher rank. Over the past twenty years we learned that these are in fact one family in a larger class of examples, which are realted to positivity in Lie groups.

I will introduce the relevant notion of positivity, discuss positive representations and why they form connected components, and in the end will discuss a few open questions and conjectures.  Video

Dan Freed, Harvard University: Complex Chern-Simons invariants of 3-manifolds via abelianization

A hyperbolic 3-manifold M carries a flat PSL(2;C)-connection whose Chern-Simons invariant has been much studied since the early 1980's.  For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm.  In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the Chern-Simons invariant, expressed in the language of topological field theory, plays an important role. The dilogarithm arises from a novel construction involving Chern-Simons invariants of flat C*-connections over a 2-torus.

Ernesto Lupercio, CINVESTAV: Hodge Numbers in Quantum Toric Geometry

In this talk, I will begin by introducing Quantum Toric Geometry (QTG), a natural non-commutative analog of classical toric geometry. Subsequently, I will explore how QTG enriches the study of Hodge numbers in the context of non-algebraic complex manifolds. This collaborative research is a joint effort with Katzarkov, Lee, Meersseman, and Verjovsky.  Video

Ron Donagi, UPenn: Variations on a theme of Hitchin’s

We explore several recent developments involving Hitchin’s system. These include connections to SCFTs of Class S, extensions to families of integrable systems over the Deligne-Mumford moduli of stable curves, geometry of the base, and connections to the Deligne-Simpson problem.  Video

Brian Collier, University of California: Higgs bundles, convex cocompact subgroups of SU(1,n) and Slodowy slices

Given a stable Higgs bundle, it is often very difficult to say something concrete about the holonomy group of the associated local system. In this talk I will explain how certain SU(1,n) Higgs bundles on a compact Riemann surface define convex cocompact subgroups of SU(1,n) which are holonomies of complex variations of Hodge structure. Our method produces such representations in every component of the SU(1,n) character variety. From the structure of the Higgs bundles the associated complex hyperbolic manifolds are described as certain fibrations over the surface.   A key input to our approach is Filip's recent work proving certain SO(2,3)-Higgs bundles have holonomy representations which are Anosov.  This is joint work with Zach Virgilio.  Video

Steven Bradlow, University of Illinois: Components of Higgs bundle moduli spaces as a legacy of Hitchin’s ideas

In three groundbreaking papers in 1987 and 1992 Hitchin defined the objects we now know as Higgs bundles on a Riemann surface and explored a remarkable range of features of their moduli spaces.

We will survey what has been learned since that time about the components of these moduli spaces, highlighting  - in the spirit of this meeting - how the seeds of ideas and techniques introduced over thirty years ago continue to bear fruit.  Video

Round Table: Future of Hitchin Mathematics - Ron Donagi, Jorgen Andersen, Oscar Garcia-Prada, Nigel Hitchin, Carlos Simpson, Dan Freed & Marco Gualteri: Future of Hitchin Mathematics

Moderators: Oscar Garcia-Prada & Carlos Simpson  Video

Marco Gualteri, University of Toronto: Double groupoids and Generalized Kahler structures

The underlying holomorphic structure of a generalized Kahler structure has been recently understood to be a square in the double category of holomorphic symplectic groupoids (or (1,1)-shifted symplectic stacks). I will explain what this means and how it allows us to describe the generalized Kahler metric in terms of a single real scalar function. This is joint work with Yucong Jiang and Daniel Alvarez.  Video

Ljudmila Kamenova, Stony Brook: Twistor spaces and compact manifolds admitting both Kaehler and non-Kaehler structures

Let's recall the classical problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were first constructed independently by M. Atiyah, A. Blanchard and E. Calabi in the 1950's. In the 1980's Tsanov gave the first example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a K3 surface. We show that the quaternion twistor space of a hyperkähler manifold has the same property - it is a simply connected manifold of higher dimension that admits both Kähler and non-Kähler complex structures.  Video

Ana Peón-Nieto, University of Santiago de Compostela: Wobbly fixed points, equivariant multiplicities and U(p,q)-Higgs bundles

The geometry of the moduli space of Higgs bundles is governed by the so-called global nilpotent cone, the fiber over 0 of the Hitchin map. A particularly meaningful subset of the nilpotent cone is given by fixed points of a C*action, which determine important geometric information. The C* dynamics within the nilpotent cone yield a natural classification of fixed points into wobbly and very stable.
When very stable fixed points exist, Hausel and Hitchin proved that the multiplicities of the corresponding irreducible component of the nilpotent cone matches the degree of a suitable restriction of the Hitchin map. This is moreover captured by the virtual equivariant multiplicities defined by these two authors. In this talk, I will analyse the case of nilpotent order two fixed points, a particular case of U(p,q)-Higgs bundles, showing that, in most cases, they do not contain very stable points.  I will contrast these results with the computation of the aforementioned invariants. Finally, I will show that general points are wobbly if and only if they are U(p,q)-wobbly, an à priori stronger notion.  Video

Karen Uhlenbeck, University of Texas at Austin: Click here for more information  Video

Vestislav Apostolov, Université du Québec à Montréal: The Calabi problem in generalized Kahler geometry

The notion of a generalized Kahler (GK) structure was introduced in the early 2000’s by Hitchin and Gualtieri in order to provide a geometric framework of certain nonlinear sigma model theories that has been studied in physics. Since then, the subject developed rapidly. It is now realized, thanks to Hitchin,  that GK structures are naturally attached to Kahler manifolds endowed with a holomorphic Poisson structure. Inspired by Calabi’s program in Kahler geometry,  which aims at finding a ''canonical” Kahler metric in a fixed deRham class, and recent works by Goto and Gualtieri,  I will present in this talk an approach towards a “generalized Kahler” version of Calabi’s problem motivated by an infinite dimensional moment map formalism. As an application, I will  give an essentially complete resolution of this problem in the case of a toric complex Poisson variety. Based on a joint work with J. Streets and Y. Ustinovskiy.  Video

Yuri Tschinkel, New York University: Equivariant geometry of cubics (joint with Boehning--von Bothmer and Cheltsov--Zhang)

I will discuss applications of new invariants in equivariant birational geometry to cubic threefolds and fourfolds.  Video

Anna Fino, Florida International University: An overview on closed G_2-structures

Closed G_2-structures on 7-manifolds are defined by  closed positive 3-forms and constitute the starting point in various known and potential methods to obtain holonomy G_2-metrics. The fact that G_2 might  be the holonomy group of certain Riemannian 7-manifolds was first suggested by the 1955 classification theorem of Marcel Berger, but the construction of holonomy G_2-metrics turned out to be  a very difficult task.  One method to obtain  local metrics with holonomy  in G_2  starting from a 6-manifold  is by  solving the so-called Hitchin flow equations. Although linear, the closed condition for a G_2-structure is very restrictive, and no general results on the existence of closed G_2-structures on compact 7-manifolds are known. In the seminar I will review known examples of compact 7-manifolds admitting a closed G_2-structure. Moreover, I will discuss some results on exact G_2-structures and the behaviour of the Laplacian G_2-flow starting from a closed G_2-structure whose induced metric satisfies suitable extra conditions. This flow can be interpreted in the  compact setting  as the gradient flow of the Hitchin volume functional.  Video

Brent Pym, McGill University: Hodge theory for Poisson varieties and nonperturbative quantization

Kontsevich's deformation quantization formula associates to any Poisson manifold an algebra of "quantum observables", defined as a noncommutative deformation of the product of functions. The formula is Feynman-style series expansion, whose coefficients are multiple zeta values, making it intractable for direct calculation.  Following a suggestion of Kontsevich, I will explain how K-theory and mixed Hodge structures can be used to construct natural "period coordinates" on the moduli space of smooth Poisson varieties, in which the quantization can often be computed simply, explicitly and nonperturbatively as the exponential map for a complex torus. This gives a conceptual explanation for the appearance of various classical transcendental functions in the relations defining well-known noncommutative algebras. This talk is based on forthcoming joint work with A. Lindberg.  Video

Ruxandra Moraru, University of Waterloo: Commuting pairs of generalized structures, para-hyper-Hermitian geometry and Born geometry

Let $M$ be a smooth manifold with tangent bundle $T$ and cotangent bundle $T^*$. By a generalized structure on $M$, we mean an endomorphism of $T \oplus T^*$ that squares to $\pm Id_{T \oplus T^*}$. In this talk, we consider pairs of generalized structures on $M$ whose product is a generalized metric. An example of such commuting pairs is given by generalized K\"ahler structures. There are three other types of such commuting pairs: generalized para-K\"ahler, generalized chiral and generalized anti-K\"ahler structures. We discuss the integrability of these structures and explain how para-hyper-Hermitian and Born geometry fit into this generalized context.  Video

Jeffrey Streets, University of California-Irvine: Calabi-Yau theory in generalized Kahler geometry

The Calabi-Yau theorem plays a central role in Kahler geometry.  In this talk I will survey various recent results extending aspects of this theory to the setting of generalized Kahler geometry.

We will exhibit a relationship between Gualtieri’s Calabi-Yau equation and certain supergravity equations of motion.  I will then discuss a geometric flow approach to constructing solutions using the generalized Kahler-Ricci flow, and describe various global existence and convergence results for this flow.  Video