16/06/2005, 11:00 — 12:00 — Room P12, Mathematics Building
William Goldman, University of Maryland
Projective geometry on manifolds (IV)
Poisson geometry on Fricke spaces
- Symplectic geometry.
- Fricke's theorem on rank two free groups.
- Fenchel-Nielsen coordinates and their generalizations.
- Action of the mapping class group.
- Penner-Fock coordinates.
16/06/2005, 09:30 — 10:30 — Room P12, Mathematics Building
William Goldman, University of Maryland
Projective Geometry on Manifolds (III)
Deformation spaces: geometries on the space of geometries.
- Markings.
- Definition of the deformation space.
- The Ehresmann-Thurston holonomy theorem.
- Representation spaces.
15/06/2005, 15:30 — 16:30 — Room P12, Mathematics Building
Jonathan Weitsman, University of California at Santa Cruz
Measures on Banach manifolds and supersymmetric quantum field theory (II)
15/06/2005, 14:00 — 15:00 — Room P12, Mathematics Building
Jonathan Weitsman, University of California at Santa Cruz
Measures on Banach manifolds and supersymmetric quantum field theory (I)
14/06/2005, 15:30 — 16:30 — Room P12, Mathematics Building
William Goldman, University of Maryland
Projective Geometry on Manifolds (II)
Affine structures
- Affine structures on the \(2\)-torus.
- Benzecri's theorems.
- Euler class of a surface group representations.
- The Milnor-Wood inequality.
- Flat bundles and geometric structures.
14/06/2005, 14:00 — 15:00 — Room P12, Mathematics Building
William Goldman, University of Maryland
Projective geometry on manifolds (I)
Geometric structures on topological spaces
- Klein geometries.
- Geometric atlases for Ehresmann structures.
- Fundamental domains and identification spaces.
- Developing the universal covering space.
- Representions of the fundamental group.
- The hierarchy of Klein geometries.
- Pathological developing maps.
18/06/2004, 11:30 — 12:30 — Room P3.10, Mathematics Building
Nikolay Mishachev, Lipetsk Technical University, Russia
Introduction to the \(h\)-principle (IV)
Wrinkling. Application of the wrinkling theory: new proofs of Thurston theorem on foliations and new proof of Eliashberg's theorem on mappings with prescribed singularities.
See also
http://math.stanford.edu/~nmish
18/06/2004, 09:30 — 10:30 — Room P3.10, Mathematics Building
Ryszard Nest, University of Copenhagen
Formal deformations and index theorems (IV)
Index theorems.
See also
https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/cuntz.ps
17/06/2004, 11:30 — 12:30 — Room P3.10, Mathematics Building
Nikolay Mishachev, Lipetsk Technical University, Russia
Introduction to the \(h\)-principle (III)
Convex integration.
See also
http://math.stanford.edu/~nmish
17/06/2004, 09:30 — 10:30 — Room P3.10, Mathematics Building
Ryszard Nest, University of Copenhagen
Formal deformations and index theorems (III)
Index of PDO's, relation to K-theory and cyclic cohomology.
See also
https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture2.ps
http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/ch6.ps
16/06/2004, 11:30 — 12:30 — Room P3.10, Mathematics Building
Nikolay Mishachev, Lipetsk Technical University, Russia
Introduction to the \(h\)-principle (II)
Holonomic approximation - II
See also
http://math.stanford.edu/~nmish
16/06/2004, 09:30 — 10:30 — Room P3.10, Mathematics Building
Ryszard Nest, University of Copenhagen
Formal deformations and index theorems (II)
Calculus of pseudodifferential operators and formal deformations
See also
https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture2.ps
15/06/2004, 11:30 — 12:30 — Room P3.10, Mathematics Building
Nikolay Mishachev, Lipetsk Technical University, Russia
Introduction to the \(h\)-principle (I)
Holonomic approximation - I
See also
http://math.stanford.edu/~nmish/
15/06/2004, 09:30 — 10:30 — Room P3.10, Mathematics Building
Ryszard Nest, University of Copenhagen
Formal deformations and index theorems (I)
Formal deformations of symplectic manifolds, structure and classification
See also
https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/Modules.ps
http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture1_1.ps
04/06/2003, 11:00 — 12:00 — Room P3.10, Mathematics Building
Constantin Teleman, Cambridge University
Twisted K-Theory and Applications (IV)
Twisted K-theory and the moduli of holomorphic G-bundles on a Riemann surface
The Frobenius algebra structure and relation to the index theory for the moduli of G-bundles on Riemann surfaces. The moduli space of flat G-bundles and the stack of all holomorphic G-bundles. (*Time permitting: higher twistings and general index formulas).
References
- Beaville, Laszlo: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994).
- Teleman: Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve. Invent. Math. 134 (1998).
03/06/2003, 11:00 — 12:00 — Room P3.10, Mathematics Building
Constantin Teleman, Cambridge University
Twisted K-Theory and Applications (III)
The Dirac-Ramond operator for a loop group
Kostant's "cubic" Dirac operator. The Dirac operator on a loop group. Construction of the twisted K-class for a positive energy representation of a loop group, by coupling the Dirac operator to a connection.
References
- Landweber: Multiplets of representations and Kostant's Dirac operator for equal rank loop groups. Duke Math. J. 110 (2001).
- Mickelsson: Gerbes, (twisted) K-theory, and the supersymmetric WZW model. hep-th/0206139.
02/06/2003, 16:30 — 17:30 — Room P3.10, Mathematics Building
Constantin Teleman, Cambridge University
Twisted K-Theory and Applications (II)
The Verlinde algebra as twisted K-theory
A refresher on loop groups and their positive-energy representations, the fusion product and the Verlinde algebra. Computation of the twisted \(K_G(G)\) in simple cases (\(S^1\), \(SU(2)\), \(SO(3)\)). Gradings and graded representations.
References
- Pressley, Segal: Loop Groups. Oxford University Press.
- Freed: The Verlinde algebra is twisted equivariant K-theory. Turkish J. Math. 25 (2001).
- Freed, Hopkins, Teleman: math.AT/0206257.
02/06/2003, 10:30 — 11:30 — Room P3.10, Mathematics Building
Constantin Teleman, Cambridge University
Twisted K-Theory and Applications (I)
K-Theory and its twisted versions: definitions and properties
Definition of K-theory of a space, using vector bundles and using families of bounded (Fredholm) operators. Group actions and equivariant K-theory. The Chern character. Twistings for K-theory and the twisted Chern character.
References
- Bouwknegt et. al: Twisted K-theory and K-theory of bundle gerbes. Comm. Math. Phys. 228 (2002).
- Freed: ICM Proceedings 2002.
- Freed, Hopkins, Teleman: math.AT/0206257.
11/07/2002, 14:30 — 15:30 — Amphitheatre Ea2, North Tower, IST
Ieke Moerdijk, Utrecht
Non-abelian Cohomology and Gerbes (IV)
The purpose of these lectures is to introduce the audience to the topological concepts of stack, gerbe and bundle gerbe, and the non-abelian cohomology classes to which they give rise.
Bundle gerbes and extensions of smooth groupoids.
References
- L. Breen, On the classification of 2-gerbes and 2-stacks, Asterisque 225, (1994).
- J. Giraud, Cohomologie nonabelienne, Springer-Verlag, 1971.
- I. Moerdijk, On the classification of regular Lie groupoids, preprint math.DG/0203099.
- M. Murray, Bundle gerbes, J. London Math. Soc. 54 (1996).
11/07/2002, 11:30 — 12:30 — Amphitheatre Ea2, North Tower, IST
Viktor Ginzburg, Santa Cruz
Periodic Orbits and Symplectic Topology (IV)
Introduction: Hamiltonian flows, the Arnold and Weinstein conjectures, examples: convex Hamiltonians and flows on twisted cotangent bundles, review of results.
Floer homology: review of Morse theory, Floer homology and symplectic homology, application: the Arnold conjecture.
Almost existence theorems for periodic orbits: symplectic capacities almost existence and the Hofer-Zehnder capacity, application: Viterbo's proof of Weinstein conjecture and almost existence in twisted cotangent bundles.
The Hamiltonian Seifert conjecture: the Seifert conjecture and counterexamples, Hamiltonian dynamical systems without periodic orbits.