# Summer Lectures in Geometry

## Past sessions

### 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

### 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

### 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.

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

### Formal deformations and index theorems - IV

Index theorems.

http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/cuntz.ps

### Introduction to the $$h$$-principle - III

Convex integration.

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

### Formal deformations and index theorems - III

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

http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture2.ps
http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/ch6.ps

### Introduction to the $$h$$-principle - II

Holonomic approximation - II

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

### Formal deformations and index theorems - II

Calculus of pseudodifferential operators and formal deformations

http://www.math.ist.utl.pt/~jmourao/ist-fcSCCAG/SummerLect04/lecture2.ps

### Introduction to the $$h$$-principle - I

Holonomic approximation - I

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

### Formal deformations and index theorems - I

Formal deformations of symplectic manifolds, structure and classification

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

### 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).

### 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.

### 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.

### 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.

### 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).

### 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.

### 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.

### 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).

### 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.

### 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.

Older session pages: Previous 5 Oldest

For detailed overviews of each course see http://camgsd.ist.utl.pt/encontros/slg/.