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Conference programme


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Conference schedule

Monday
4 September
Tuesday
5 September
Wednesday
6 September
Thursday
7 September
Friday
8 September
9:30-10:30
Registration and coffee
R. Dijkgraaf
M. Atiyah
N. Manton
H. B. Lawson
10:30-11:00
Conference opening
Coffee break
Coffee break
Coffee break
Coffee break
11:00-12:00
R. Bryant
S. Ramanan
D. Freed
J. E. Andersen
G. Segal
12:15-13:15
P. Boalch
W. Goldman
S.-T. Yau
J. Hurtubise
Video
(12:00-12:30)
Lunch buffet
(starts at 13:30)
Free afternoon
Session on open problems
(12:40-14:10)
16:00-17:00
M. Gualtieri
A. I. Bobenko
T. Hausel
End of Conference
17:00-17:30
Coffee break
Coffee break
Coffee break
17:30-18:30
S. Tolman
C. LeBrun
M. Roček
18:30-19:30
19:30-21:00
Round table* (19:30-21:00)
Social Dinner
(starts at 20:00)

*: Note that the round table will take place at the Residencia de Estudiantes.

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Registration

The standard place and time for registration will be Monday, 4th September, 9:30-10:30, in the venue of the Conference. The registration fee covers the lunch buffet on the Monday (the first day of the Conference), as well as coffee breaks and refreshments. The (optional) Social Dinner will be held on Wednesday, 6th September. The price is 50 euros per person. Please note the following important points: Please note that credit cards and cheques cannot be accepted.

Conference opening

The Conference will be opened by:

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Lectures

Monday, 4 September

11:00-12:00, Mon 4 Sep: Robert Bryant (Duke University)
Some nonembedding and nonextension results in special holonomy
In the analytic category, the conditions for realizing a given SU(3)-structure on a 6-manifold as a hypersurface in a G2-manifold or a given G2-structure on a 7-manifold as a hypersurface in a Spin(7)-manifold are well-known. It is easy to prove their sufficiency using Cartan-Kähler theory. However, in the smooth category, these conditions fail to be sufficient. In this talk, I will discuss some examples of this failure and their geometric meaning.
12:15-13:15, Mon 4 Sep: Philip Boalch (ENS, Paris)
Painlevé, Klein and the icosahedron
We will discuss "algebraic Painlevé transcendents": the sixth Painlevé equation was discovered by R. Fuchs around 1905 and recent interest stems largely from its link to 2d QFT, Frobenius manifolds and Einstein metrics (and I guess the fact that it is a dimensional reduction of the anti-self-dual Yang-Mills equations). In spite of the fact that generic solutions are known to be `new transcendental functions' we will describe many explicit solutions, some of their underlying geometry and the method used to construct them.
16:00-17:00, Mon 4 Sep: Marco Gualtieri (MIT)
Generalized Complex Geometry
We will take a hike through generalized complex geometry; hopefully there will be a good view by the end. I will concentrate on the implications of having an extended symmetry group consisting of B-field transformations.
17:30-18:30, Mon 4 Sep: Sue Tolman (University of Illinois at Urbana-Champaign)
Bi-Hermitian Structures and Generalized Kähler reduction
Based on joint work with Yi Lin. We define reduction in generalized complex and generalized Kähler geometry. Using this, we give explicit constructions of bi-Hermitian structures on a variety of spaces, including projective spaces (or their blow-up at arbitrarily many points), Hirzebruch surfaces, and complex Grassmannians.

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Tuesday, 5 September

9:30-10:30, Tue 5 Sep: Robbert Dijkgraaf (University of Amsterdam)
Gauge Theories and Topological Strings
I will discuss the relation between Gromov-Witten invariants on Calabi-Yau threefolds with quantum gauge theories in various dimensions.
11:00-12:00, Tue 5 Sep: S. Ramanan (Chennai Mathematical Institute)
A survey of Hitchin pairs and involutions of their moduli
Hitchin, in a delightful paper, studied the space of representations of the fundamental group of a Riemann surface into split real forms of reductive algebraic groups and showed that they correspond to what he called Higgs pairs of a special type. He described particular components each of which is actually an affine space of half the dimension. These are connected components of the fixed points of an obvious involution on the moduli. My aim will be to discuss the fixed subsets of similar involutions that act and relate its components to representations of the fundamental group into other real forms.
12:15-13:15, Tue 5 Sep: William Goldman (University of Maryland)
Spaces of surface group representations
In the 1980's Nigel Hitchin introduced analytic techniques to investigate moduli spaces of representations of fundamental groups of closed Riemann surfaces in real semisimple Lie groups. In this talk I will discuss the historical context of this work, including several recent developments and generalizations.
16:00-17:00, Tue 5 Sep: Alexander I. Bobenko (Technische Universität Berlin)
Linear and nonlinear theories of discrete analytic functions
Based on joint work with Yu. Suris and Ch. Mercat. Two discretizations, linear and nonlinear, of basic notions of the complex analysis are considered. The linear theory is based on the discrete Cauchy-Riemann equations on an arbitrary rhombic tiling of a plane, the nonlinear one is based on the notion of circle patterns. We clarify the role of the rhombic condition in both theories: under this condition the corresponding equations are integrable. The linear theory is a linearization of the nonlinear one: the tangent space to a set of integrable circle patterns at an isoradial point consists of discrete holomorphic functions. We prove that Kenyon's discrete Green's function is an isomonodromic solution of the discrete Cauchy-Riemann equations, and that it is a tangent vector to the space of integrable circle pattarns along the family of isomonodromic discrete power functions.
17:30-18:30, Tue 5 Sep: Claude LeBrun (SUNY at Stony Brook)
Curvature Functionals, Optimal Metrics, and the Differential Topology of 4-Manifolds
This lecture will discuss the question of which smooth compact simply connected 4-manifolds admit Riemannian metrics that minimize the L2-norm of the curvature tensor. Metrics with this property are called OPTIMAL. Einstein metrics and scalar-flat anti-self-dual metrics provide us with two interesting classes of examples, and in the second case we now have a rather complete understanding of the problem of determining when such a metric exists. However, this circle of ideas also allows us to see that large classes of 4-manifolds simply do not admit ANY optimal metric. In fact, the difference between existence and non-existence turns out to delicately depend on one's choice of smooth structure; there are smooth 4-manifolds which carry optimal metrics, but which are homeomorphic to infinitely many distinct smooth 4-manifolds on which no optimal metric exists.

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Wednesday, 6 September

9:30-10:30, Wed 6 Sep: Sir Michael Atiyah (University of Edinburgh)
Nigel Hitchin and Geometry
Nigel Hitchin has introduced many original and significant ideas in geometry. These have been related to geometrical aspectsof theoretical physics and have been much used by the physics community. Often he has been well ahead of the mainstream and it has taken years for the importance of his ideas to be fully appreciated. I will give a rapid survey of his contributions.
11:00-12:00, Wed 6 Sep: Dan Freed (University of Texas at Austin)
Consistent orientation of moduli spaces
In joint work with Hopkins and Teleman we give a topological construction of a topological quantum field theory. This requires consistent K-theory orientations of moduli spaces of flat connections. An analogous problem was first considered by Donaldson using excision properties of elliptic operators. Our approach is topological, making use of bordism spaces introduced by Madsen and Tillmann.
12:15-13:15, Wed 6 Sep: Shing-Tung Yau (Harvard University)
Spacetime with torsion
In this talk, I will discuss the construction of complex manifolds with supersymmetries that satisfy the equations of Strominger that I had with Jun Li and Fu. They provide vacuum solution to Heterotic string theory. Its role in the subject of complex manifolds will be discussed as well.

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Thursday, 7 September

9:30-10:30, Thu 7 Sep: Nicholas Manton (University of Cambridge)
Skyrmions and Nuclei
New solutions of the Skyrme field equations with baryon numbers 8, 12, 16, ..., which take into account the positive pion mass, have been discovered. These solutions are the Skyrme model analogue of alpha-particle cluster models of the atomic nuclei 8Be, 12C, 16O, ... There is also recent progress on calculating the spins and electromagnetic properties of quantized Skyrmions. (This is joint work with Richard Battye and Paul Sutcliffe hep-th/0605284, and with my students Olga Manko and Stephen Wood.) I will mention the relationship between all this and the work by Nigel Hitchin and others on SU(2) Monopoles.
11:00-12:00, Thu 7 Sep: Jørgen Ellegaard Andersen (Aarhus Universitet)
Toeplitz operators and Hitchin's projectively flat connection
We will discuss a number of applications of the theory of Toeplitz operators to the study of Hitchin's projectively flat connection and the associated projective representations of the mapping class groups. These are all of particular interest in the study of the associated TQFT's providing strong invariants of knots and 3-manifolds and they are closely related to Thurston's classification program of diffeomorphisms of surfaces.
12:15-13:15, Thu 7 Sep: Jacques Hurtubise (McGill University)
Moduli of calorons
Calorons (instantons on S1×R3), by work of Nye and Singer, can be seen to be equivalent to solutions to Nahm's equations on the circle, and by a twistor transform, to certain classes of vector bundles on an associated twistor space. In joint work with Benoit Charbonneau, we compute the moduli space of these objects. This talk is by way of being a coda to many beautiful ideas developed by Nigel Hitchin during the 1980s, and is meant to be a tribute to his role in developing them.
16:00-17:00, Thu 7 Sep: Tamás Hausel (University of Oxford & University of Texas at Austin)
S-duality in hyperkähler Hodge theory
In this talk I discuss how elementary mathematics (like enumerative combinatorics and representation theory of finite groups and quivers) leads to modular forms and Langlands duality by studying predictions of S-duality on the Hodge theory of hyperkähler moduli spaces of Yang-Mills instantons, magnetic monopoles and Hitchin pairs. The spaces and their hyperkähler metrics were constructed by Nigel Hitchin and his collaborators more than 20 years ago and their study has been a very active research area since then.
17:30-18:30, Thu 7 Sep: Martin Roček (SUNY at Stony Brook)
Hyperkähler cones and the c-map
The c-map relates special Kähler geometries to a class of quaternion-Kähler geometries. We give a simple construction of their Swann bundle or hyperkähler cone, and discuss some recent applications.

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Friday, 8 September

9:30-10:30, Fri 8 Sep: H. Blaine Lawson, Jr. (SUNY at Stony Brook)
Plurisubharmonicity and Convexity in Calibrated Geometry
I will introduce and discuss the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics on a calibrated manifold are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. I shall discuss questions of: pseudo-convexity and cores, boundary convexity, positive phi-currents, Duval-Sibony Duality, and boundaries of phi-submanifolds, all in the context of a general calibrated manifold (X,phi). Analogues of totally real submanifolds are introdued and used to construct enormous families of strictly phi-convex spaces with every topological type allowed by Morse Theory. Analogues of the Hodge Conjecture in calibrated geometry will be discussed. This is a report on joint work with Reese Harvey.
11:00-12:00, Fri 8 Sep: Graeme Segal (University of Oxford)
Quantum field theory and locality
A quantum field theory can be defined as a functor from a cobordism category to a linear category, but one would like to be able to say that it is a local structure on space-time. I shall discuss the motivation for this, and describe some different approaches to the question, and some partial results.

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Video

Monopoles in Motion: A study of the low-energy scattering of magnetic monopoles

By Sir Michael Atiyah (University of Edinburgh), Nigel Hitchin (University of Oxford), John Merlin (University of Vienna), David E. L. Pottinger and M. Williams Ricketts (IBM, Winchester). IBM UK Scientific Centre video and accompanying Pamphlet 207, 1989.

Session on open problems

The Conference will close with a session on open problems, on Friday, 8 September, 12:40-14:10. This session will be coordinated by:

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Round table

New interactions between geometry and physics

On Thursday, 7 September, 19:30-21:00, an informal round table on new interactions between geometry and physics will take place at the Residencia de Estudiantes. This is close to the venue of the Conference (3 minutes walking distance), also in the main campus of CSIC (a local map is available here). This event is organised by the Spanish Red Temática de Geometría y Física and the Residencia de Estudiantes. This is addressed to the general public and all participants in the Conference are encouraged to participate.

Download poster in jpg or pdf format.

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Lunch buffet

A lunch buffet will take place on Monday, 4 September, in the cloister of calle Serrano, 123 (123 Serrano Street). This is close to the venue of the Conference (3 minutes walking distance), also in the main campus of CSIC. A local map of the CSIC main campus is available here.

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Social dinner

The social dinner will take place on Wednesday, 6 September, 20:00, at Restaurante Jai-Alai. This restaurant is within 15 minutes walking distance from the venue of the Conference:

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Conference proceedings

An edited volume will be prepared based on the meeting. Its publication is currently under consideration at Oxford University Press.

OUP