# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### The dynamics of the Siegel Upper Half Plane

We will present the classification of the isometries of the hyperbolic plane, and the definition of some concepts like limis sets. We'll then generalize some of these definitions results for the action of the symplectic group on the Siegel upper half plane - we'll define both the group and the half plane, and present some basic results about them. We'll also mention a few different ways of compactifying the half plane.

### Modular invariants and subfactors

Braided subfactors provide a natural framework for understanding modular invariant partition functions in rational conformal field theory. Here we outline some of the mathematical structures in the underlying operator algebraic framework.

### Sobre hiperbolicidade uniforme em sistemas não uniformemente hiperbólicos

Apresentam-se condições que garantem que sistemas não uniformemente hiperbólicos passem a ser uniformemente hiperbólicos.

As condições envolvem a expansão não uniforme em conjuntos de probabilidade total e possibilitam provas bastante simples dos resultados.

### The duality of double structures

The notion of a double vector bundle is implicit in a traditional approach to connection theory, and in the formulation of theoretical mechanics developed by Tulczyjew and others. Double vector bundles have an unexpected 3-fold duality which is best understood in terms of triple structures and which underlies recent work on Lie bialgebroids and associated double structures.

### Integrable systems and geometry

Holomorphic functions can be useful when studying many natural nonlinear differential equations. Two examples are nonlinear magnetic monopoles and the nonlinear KdV equation that describes solitary water waves. The rich structure of complex analysis makes this a powerful tool whenever it can be applied. More specifically, a meromorphic function can be uniquely determined by very little information, say knowledge of its poles and residues. This can sometimes enable us to deduce many features, including explicit solutions, of a nonlinear differential equation.

This talk will give an introductory account of these ideas. It is hoped that the talk contains something for both novices and experts in the field.

### Spectral sequences, pairings and twisted products

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