###
14/06/2005, 15:30 — 16:30 — Room P12, Mathematics Building

William Goldman, *University of Maryland*

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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*

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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*

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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*

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Formal deformations and index theorems - IV

Index theorems.

#### See also

http://www.math.ist.utl.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*

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Formal deformations and index theorems - III

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

#### See also

http://www.math.ist.utl.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

http://www.math.ist.utl.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

http://www.math.ist.utl.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*

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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*

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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*

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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*

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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, *Non-abelian Cohomology and Gerbes*

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Non-abelian Cohomology and Gerbes

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.

#### Lecture 4

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*

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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 — Amphitheatre Ea2, North Tower, IST

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 — Amphitheatre Ea2, North Tower, IST

Ieke Moerdijk, *Utrecht*

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```###
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.

#### Lecture 3

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).

###
10/07/2002, 09:30 — 10:30 — Amphitheatre Ea2, North Tower, IST

Alexander Varchenko, *North Carolina*

```
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Arrangements, Hypergeometric Functions, and KZ-Type Equations - IV

Conformal blocks in conformal field theory and their
quantizations are solutions of Knizhnik-Zamolodchikov differential
and difference equations. The geometric version of these objects is
multidimensional hypergeometric functions and hypergeometric
equaitons. The interrelations of these subjects, the theory of
solvable models in statistical mechanics, and representation theory
will be discussed.

The course will be elementary and accessible to graduate
students and advanced undergraduate students.

###
09/07/2002, 16:30 — 17:30 — Amphitheatre Ea2, North Tower, IST

Viktor Ginzburg, *Santa Cruz*

```
```###
Periodic Orbits and Symplectic Topology

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.