Programme
Schedule
|
Monday, 12 |
Tuesday, 13 |
Wednesday, 14 |
Thursday, 15 |
Friday, 16 |
09:00-10:00 |
Registration |
|
10:00-10:50 |
S. Bradlow |
O. Biquard |
R. Wentworth |
T. Pantev |
S. Ramanan |
10:50-11:20 |
Coffee |
Coffee |
Coffee |
Coffee |
Coffee |
11:20-12:10 |
J. Hurtubise |
A. Dancer |
M. Garcia-Fernandez |
J. Heinloth |
P. Gothen |
12:20-13:10 |
C. Simpson |
D. Salamon |
B. Collier |
I. Biswas |
N. Hitchin |
13:20-15:00 |
Lunch |
Lunch |
Lunch |
Lunch |
End of conference |
15:00-15:50 |
A. Peón-Nieto |
J. E. Andersen |
|
I. Mundet i Riera |
|
15:50-16:20 |
Coffee |
Coffee |
Coffee |
16:20-17:10 |
A. Oliveira |
T. Hausel |
M. Santander |
17:15-18:45 |
Wine reception |
Lectures
Jørgen Ellegaard Andersen (QM, Univ. Southern Denmark): The Automorphism Equivariant Hitchin Index
Expand Abstract
We consider the action of an automorphism f of a Riemann surface on the determinant bundle L of the moduli space M of rank two Higgs bundles with fixed determinant. This action commutes with the action of the one-dimensional complex torus T, and we define and study the automorphism equivariant index t(M;Lk;f). We prove this is a topological invariant of the mapping torus Mf of f by proving a formula for it in terms of cohomological pairings of universal classes on moduli spaces of parabolic Higgs bundles over the quotient Riemann surface X and Seifert invariants of Mf: This invariant Z(Mf;k;t) can be seen via non-abelian Hodge theory as a version of the SL(2;ℂ)k quantum invariant of Mf, and it is a refinement of the SU(2)k WRT-invariant Z(Mf; k) of the mapping torus. This work is joint with Tamas Hausel and William Mistegaard.
Olivier Biquard (Sorbonne & Univ. Paris): Milnor–Wood inequalities and Higgs bundles
Expand Abstract
The Milnor–Wood inequality for representations of a surface group into a group
of Hermitian type was established a long time ago by Burger–Iozzi–Wienhard. In
this talk I shall explain the ideas of the Higgs bundle approach and their recent applications.
Indranil Biswas (TIFR, Mumbai): Holomorphic connections on the trivial vector bundle
Expand Abstract
Some properties of the space of holomorphic connection on the trivial vector bundles will be described. Also, some individual connections of this type will be explained. (Based on works with Sorin Dumitrescu, Lynn Heller, Sebastian Heller and Joao Pedro P. dos Santos.)
Steven Bradlow (Univ. Illinois Urbana-Champaign): From vortices to magical components: a 30-year ramble
Expand Abstract
On the occasion of Oscar’s 60th birthday celebration I will survey (most of) our
joint work, stretching (so far) over a 30-year span. Unwittingly starting from the same point (vortices) we jointly embarked on an exploration of the surrounding mathematical territory. In the process, together with with marvelous fellow travelers that we met along the way, we uncovered some pleasing features and stumbled on some unexpected byways.
Brian Collier (Univ. California Riverside): Holomorphic curves and cyclic G2-Higgs bundle
Expand Abstract
The 6-pseudosphere is the space of norm −1 vectors in ℝ4,3, and is a pseudo-Riemann analogue of the 6-sphere. Like the 6-sphere, the 6-pseudosphere has a non-integrable almost complex structure which arises from the split octonions; the split real form of G2 is the automorphism group of this structure. In this talk, we will describe a class of J-holomorphic maps from the upper half plane to the 6-pseudosphere which are equivariant for representations of the fundamental group of a closed surface into the split real form of G2 . We construct a moduli space of such objects which fibers over the Teichmuller space of the surface with fibers given by a certain G2 Higgs bundles fixed by a ℤ/6 action. It turns out the components of the moduli space are labeled by an integer which lies in a fixed interval, and at one extreme is the space of G2-Hitchin representations. Other than this extremal case, the moduli spaces do not define connected components of the character variety for the split real form of G2. This is joint work with Jeremy Toulisse.
Andrew Dancer (Univ. Oxford): Implosion and contraction in hyperkahler geometry
Expand Abstract
We discuss implosion and contraction constructions in symplectic and hyperkahler
geometry, and how the Moore–Tachikawa category can provide a useful organising principle for such constructions. (Joint work with Frances Kirwan and Johan Martens.)
Mario Garcia-Fernandez (ICMAT–UAM Madrid):
Futaki invariants and harmonic metrics for the Hull–Strominger system - Slides
Expand Abstract
The Hull–Strominger system has been proposed as a geometrization tool for under-
standing the moduli space of Calabi–Yau threefolds with topology change (‘Reid’s
Fantasy’). In this talk we will introduce obstructions to the existence of solutions for these equations which combine infinite-dimensional moment maps, ‘harmonic metrics’, and a holomorphic version of generalized geometry. We will discuss the implications of these new obstructions in relation to Reid’s Fantasy and a conjecture by S.-T. Yau for the Hull-Strominger system. This is joint work with Raúl Gonzalez Molina.
Peter Gothen (Univ. Porto):
Components of G-Higgs bundle moduli and the Cayley correspondence - Slides
Expand Abstract
Let M be the moduli space of surface group representations in a real reductive Lie group G. The most basic topological question about M is the determination of its connected components. Much progress has been made on this problem over the last 30–40 years but it is, in general, still open. A conjectural answer to the question — based on Hitchin’s approach via non-abelian Hodge Theory — can be stated in terms of the generalised Cayley correspondence for moduli spaces of G-Higgs bundles, and is closely related to higher Teichmüller theory.
The talk based on joint work with Steve Bradlow, Brian Collier, Oscar García-Prada, and André Oliveira.
Tamás Hausel (IST Austria):
Hitchin map on very stable and even very stable upward flows - Slides
Expand Abstract
I will recall recent results with Hitchin on very stable upward flows and in some cases an explicit description of the Hitchin system on them as the spectrum of equivariant cohomology of Grassmannians and other cominuscule flag varieties. At the end I will mention very recent work with González González on even very stable bundles and how the spectrum of equivariant cohomology of quaternionic Grassmannians, spheres and the real Cayley plane appear to describe the Hitchin map on even cominuscule upward flows. The even upward flows in question are the same as upward flows in Higgs bundle moduli spaces for quasi-split inner real forms. The latter spaces have been pioneered by Oscar García-Prada and his collaborators.
Jochen Heinloth (Univ. Duisburg-Essen): Proper quotients for torus actions
Expand Abstract
For moduli problems it is often not so hard to show that they define algebraic stacks, but to obtain moduli spaces often requires the choice of a “semistable” part of the moduli problem admits proper coarse moduli spaces. While we have necessary and sufficient conditions that an open the “semistable” part needs to satisfy to have a proper quotient, much fewer techniques are known to find such open subsets. In the talk I will try to explain how looking at this problem for algebraic stacks, gives an approach to the problem to characterize all open subsets of smooth projective varieties with an action of a torus that admit a proper quotient space. Not all of these come from the choice of a line bundle, but a slightly different cohomological ingredient can be used here.
Nigel Hitchin (Univ. Oxford): Higgs fields in low genus - Slides
Expand Abstract
The talk will address the issue of describing more concretely the Higgs fields for stable bundles given by the classical constructions of moduli spaces in genus 2 and 3. This is partly an exercise to understand better the objects which have been studied for 35 years or more, and partly to answer questions arising from mirror symmetry.
Jacques Hurtubise (McGill Univ.): Degenerations of bundle moduli
Expand Abstract
We build an explicit degeneration of the moduli of SL(n) bundles on a Riemann surface associated to a degeneration to a nodal curve. The degeneration works well in both the holomorphic and the symplectic context. This explains a result of Jeffrey and Weitsman on the ‘not quite toric’ structure of the moduli of SU(2) representations of the fundamental group of a Riemann surface, by giving the actual toric variety that appears in the limit. (joint with I. Biswas)
Ignasi Mundet i Riera (Univ. Barcelona):
Actions on topological manifolds: finite abelian groups vs tori - Slides
Expand Abstract
Let X be a closed and connected topological manifold. Let D(X)≥0 be the greatest integer m such that X supports effective actions of (ℤ/r)m for arbitrarily large integers r. Let A(X)≥0 be the greatest integer m such that X supports an effective action of (S1)m. It is well known that A(X)≤dim X, with equality only if X is homeomorphic to a torus. In view of this, one may ask whether D(X)≤dim X is true for every X, and whether D(X)=dim X implies that X is homeomorphic to a torus. We will present some evidence towards a positive answer.
André Oliveira (Univ. Porto & UTAD): Lagrangians of Hecke cycles
Expand Abstract
I will describe a class of complex Lagrangians in the moduli space M of Higgs bundles, mapping to the discriminant locus of the Hitchin base, which are being considered in ongoing work with E. Franco, R. Hanson and J. Horn. These are generalizations of the Lagrangians which appear as the support of the dual brane of a hyperholomorphic bundle over the locus in M of fixed points under tensorization by a finite order line bundle, and which are also under study together with E. Franco, P. Gothen and A. Peón-Nieto.
Tony Pantev (Univ. Pennsylvania): Geometry and Topology of Wild Character Varieties - Slides
Expand Abstract
Wild character varieties parametrize monodromy representations of flat meromorphic connections on compact Riemann surfaces. They are classical objects with remarkable geometric and topological properties. In the past twenty years new insights from algebraic geometry lead to precise conjectures on the topological structure and complexity of character varieties. I will recall some of these conjectures and will sketch a strategy for approaching them. In particular I will describe recent joint works with Chuang, Diaconescu, Donagi, and Nawata in which we use dualities in geometry and physics to extract cohomological invariants of wild character varieties from enumerative Calabi–Yau geometry and refined Chern–Simons invariants of torus knots.
Ana Peón-Nieto (Univ. Birmingham & Côte d'Azur): Geometry of the nilpotent cone
Expand Abstract
In this talk I will explain joint work with C. Pauly on the geometry of the nilpotent cone in rank three. The nilpotent cone, and more specifically, moduli spaces of chains, are amongst Oscar García-Prada’s many contributions. In this work, we take a different perspective which recovers some of Oscar and collaborator’s past results, as well as new ones. I will explain some key characters like (higher) wobbly divisors, and potential extensions of our techniques to higher rank.
S. Ramanan (CMI Chennai): Some rationality results on Hyper-elliptic curves - Slides
Expand Abstract
Classically, one associates to a number field K, a group called the Brauer group. This can be generalised to the notion over a scheme, in particular over a hyper-elliptic curve C over K. To each element of the Brauer group one may associate two numbers, called period and index. It is known that the period divides the index and has the same prime divisors. But the exact relation between them in each case, is a delicate question. When the genus of C is 2, and the period is 2, one shows the index is also 2 when K is a totally imaginary field, using the theory of vector bundles over C.
Dietmar Salamon (ETZ Zürich):
Almost complex structures and Hamiltonian symplectomorphisms - Slides
Expand Abstract
The space of compatible almost complex structures on a closed symplectic manifold can be viewed as an infinite-dimensional Kaehler manifold, equipped with a Hamiltonian group action by the group of Hamiltonian symplectomorphisms, where the
moment map is the scalar curvature (Quillen–Fujiki–Donaldson). This talk will explore to what extent the familiar parallel between finite-dimensional GIT and this infinite-dimensional setting carries over from integrable to non-integrable almost complex structures, and pose some open questions.
Mariano Santander (Univ. Valladolid):
Newtonian gravitation as curvature of a space-time: Newton-Cartan gravitation
and beyond. What if the idea of connection had been defined in 1900? - Slides
Expand Abstract
The talk has two parts. The first sticks to the historical order of events, and
describes the E. Cartan realization of the possibility (now called Newton–Cartan’s theory) of reinterpreting Newton’s theory of gravity as curvature of a galilean space-time. Of course, the geometrical structure of such a galilean space-time is different from the relativistic one in Einstein’s theory; in particular the absolute time of Newtonian mechanics is preserved, yet space time has curvature, albeit not so rich as in Einstein’s theory of gravitation. This possibility has been developed since by several authors leading to a definition of ‘Classical SpaceTime’ which include as a particular case the reformulation of Newton gravitation by Cartan but allow some possibilities going beyond the Newton–Cartan theory.
Hence the idea of gravitation as curvature is not a specifically relativistic trait, pertaining exclusively to the Einstein’s theory, but includes naturally the classical Newton’s gravitation. This is not yet common knowledge, and misleading statements on this topic continue to be found. There are some reasons for this state of affairs, linked with the history of the subject, which are worth discussing from the perspective afforded by this reformulation.
The second part is an “What if” exercise. I will argue that by exploring a contrafactual question “What if the idea of connection had been defined in 1900?” one can build a plausible and conceptually sound uchronia, inverting the actual historical order of some events, where Special Relativity could have been discovered starting from (a Newton–Cartan style reformulation of) Newtonian gravitation, in sharp contrast with the actual historical order, which started from Maxwell’s electromagnetism. Again, and for reasons of internal consistency, later this would have led to the need to modify Newton’s gravitation, leading to the Einstein’s theory.
Carlos Simpson (CNRS, Univ. Côte d'Azur): Twistor space and parabolic structures - Slides
Expand Abstract
We’ll look at the interplay between parabolic structures and the twistor space construction. The basic idea is that the appearance of parabolic structures is related to the fact that local fundamental group elements at a puncture have weight −2. This is viewed most clearly in the case of rank 1 local systems, and we’ll then discuss the recent extension to the case of rank 2.
Richard Wentworth (Univ. Maryland):
Some remarks on Yang–Mills type equations in higher dimensions
Expand Abstract
The talk consists of two parts. First, I will discuss a variant of anti-self-dual connections in higher dimensions that is motivated by calibrated geometry and multipolarizations in Kaehler geometry. Such connections are shown to satisfy a version of Uhlenbeck weak compactness. In the case of Hermitian manifolds, we prove analogs of the Donaldson–Uhlenbeck–Yau and nonabelian Hodge theorems. Second, I describe an extension of the Donaldson–Uhlenbeck–Yau theorem to normal projective varieties. Taken together, the two parts give an analytic proof of Miyaoka’s version of the Bogomolov–Gieseker inequality, with a sharp result in the case of equality. This is joint work with Xuemiao Chen.
Video recordings
Video recordings are now available on the ICMAT YouTube channel.
Posters
David Alfaya (Univ. Pontificia Comillas): Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces - Poster
Expand Abstract
In this work we consider moduli spaces of flat Lie algebroid connections on a Riemann surface. These types of moduli spaces constitute a simultaneous generalization of several classes of moduli spaces which are broadly used in differential geometry, algebraic geometry, and mathematical physics, such as moduli spaces of Higgs bundles, twisted Higgs bundles, flat connections, and logarithmic or meromorphic connections.
For a general choice of a Lie algebroid, the geometry of theses spaces is mostly
unknown, but, for rank 1 Lie algebroids, we were able to obtain a lot of information about their geometry. We obtain some mild conditions which guarantee that these moduli spaces are smooth, nonempty and irreducible, we compute various higher homotopy groups and we prove several invariance results on their motives, including a “motivic nonabelian Hodge correspondence” between moduli spaces of twisted Higgs bundles and of flat Lie algebroid connections which allows us to compute explicitly the motives and E-polynomials (and, thus, Betti numbers) of these moduli spaces in rank 2 and 3. Along the way, we were able to verify computationally in low rank a conjectural formula by Mozgovoy on the motives of moduli spaces of Twisted Higgs bundles. Joint work with André Oliveira.
Erroxe Etxabarri-Alberdi (Loughborough Univ.): Fano 3-folds with 1-dimentional moduli - Poster
Expand Abstract
We give a brief introduction to K-stability, which allows the construction of moduli in Fano varieties. We present a collaboration project with Hamid Abban, Ivan Chelchov, Elena Denisova, Dongchen Jiao, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia and Theodoros Papazachariou where we study the Fano 3-folds with 1 dimensional moduli and where we will compactify the moduli for the ones containing K-stable object. We will also give an example (Family 3.12).
Natsuo Miyatake (IMI, Kyushu Univ.): Toda lattice with opposite sign on Riemannian manifolds and the variational method - Poster
Expand Abstract
The Hermitian–Einstein equation for diagonal metrics on cyclic Higgs bundles is
called the Toda lattice with opposite sign. We generalize it to a PDE on Riemannian manifolds and solve it when the manifold is compact using the variational method inspired by Kempf–Ness’s theorem for linear torus actions. We will also show the existence and the uniqueness of the time-global solution of the parabolic equations and the existence and the uniqueness of the solution to the Dirichlet problem.
Ronald A. Zúñiga-Rojas (Univ. Costa Rica): Stratifications on the Nilpotent Cone of the moduli space of Hitchin pairs - Poster
Expand Abstract
We consider the problem of finding the limit at infinity (corresponding to the downward Morse flow) of a Higgs bundle in the nilpotent cone under the natural ℂ∗-action on the moduli space. For general rank we provide an answer for Higgs bundles with regular nilpotent Higgs field, while in rank three we give the complete answer. Our results show that the limit can be described in terms of data defined by the Higgs field, via a filtration of the underlying vector bundle. This is joint work with Peter B. Gothen.
Social dinner
The (optional) social dinner will take place on Wednesday 14 September, 20:00, at Restaurante Panamera in Madrid. The price is €55 per person. To confirm your participation and payment, please contact the local organizers and specify whether you have any special diet requirement (such as vegetarian or vegan).
Address of Restaurante Panamera: Calle de Fernández de la Hoz, 57, Madrid