Cursos de Verão em Geometria 

15/06/2005, 15:30 — 16:30 — Sala P12, Pavilhão de Matemática
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 — Sala P12, Pavilhão de Matemática
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 — Sala P12, Pavilhão de Matemática
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 — Sala P12, Pavilhão de Matemática
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 — Sala P3.10, Pavilhão de Matemática
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.

Ver também

http://math.stanford.edu/~nmish

18/06/2004, 09:30 — 10:30 — Sala P3.10, Pavilhão de Matemática
Ryszard Nest, University of Copenhagen

Formal deformations and index theorems (IV)

Index theorems.

Ver também

https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/cuntz.ps

17/06/2004, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Nikolay Mishachev, Lipetsk Technical University, Russia

Introduction to the \(h\)-principle (III)

Convex integration.

Ver também

http://math.stanford.edu/~nmish

17/06/2004, 09:30 — 10:30 — Sala P3.10, Pavilhão de Matemática
Ryszard Nest, University of Copenhagen

Formal deformations and index theorems (III)

Index of PDO's, relation to K-theory and cyclic cohomology.

Ver também

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 — Sala P3.10, Pavilhão de Matemática
Nikolay Mishachev, Lipetsk Technical University, Russia

Introduction to the \(h\)-principle (II)

Holonomic approximation - II

Ver também

http://math.stanford.edu/~nmish

16/06/2004, 09:30 — 10:30 — Sala P3.10, Pavilhão de Matemática
Ryszard Nest, University of Copenhagen

Formal deformations and index theorems (II)

Calculus of pseudodifferential operators and formal deformations

Ver também

https://www.math.tecnico.ulisboa.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture2.ps

15/06/2004, 11:30 — 12:30 — Sala P3.10, Pavilhão de Matemática
Nikolay Mishachev, Lipetsk Technical University, Russia

Introduction to the \(h\)-principle (I)

Holonomic approximation - I

Ver também

http://math.stanford.edu/~nmish/

15/06/2004, 09:30 — 10:30 — Sala P3.10, Pavilhão de Matemática
Ryszard Nest, University of Copenhagen

Formal deformations and index theorems (I)

Formal deformations of symplectic manifolds, structure and classification

Ver também

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 — Sala P3.10, Pavilhão de Matemática
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 — Sala P3.10, Pavilhão de Matemática
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 — Sala P3.10, Pavilhão de Matemática
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 — Sala P3.10, Pavilhão de Matemática
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 — Anfiteatro Ea2, Torre Norte (Electricidade)
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.

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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 — Anfiteatro Ea2, Torre Norte (Electricidade)
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.

10/07/2002, 14:30 — 15:30 — Anfiteatro Ea2, Torre Norte (Electricidade)
Viktor Ginzburg, Santa Cruz

Periodic Orbits and Symplectic Topology (III)

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.

10/07/2002, 11:30 — 12:30 — Anfiteatro Ea2, Torre Norte (Electricidade)
Ieke Moerdijk, Utrecht

Non-abelian Cohomology and Gerbes (III)

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.

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Gerbes and non-abelian cohomology in degree 2.

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).