# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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.

### Regular Poisson manifolds and their foliations

A regular Poisson manifold can be described as a foliated space supporting a tangentially symplectic form. As was discussed in a previous talk, such forms are solutions of a certain partial differential relation that can be shown to satisfy the h-principle when some natural condition on the underlying foliated space, namely that of being uniformly open is fulfilled. Examples of foliations will be presented in this talk that are not induced by any Poisson structure although the basic obstructions vanish. In particular, these foliated spaces are not uniformly open.

### Non formal deformation quantizations and Lie group actions

I will present a few aspects of convergent star product theory involving star representation techniques. The main goal is to define a class of non-commutative differential manifolds via explicit universal deformation formulae for (non-Abelian) group actions on Fréchet algebras. In the examples considered here, these formulae will be of two types: oscillatory integral and analytic. The first kind may be interpreted as finite dimensional versions of Cattaneo-Felder's formulae for Kontsevich star products on symplectic manifolds in presence of curvature. The second kind is based on deformations of Gelfand-Shilov's spaces of type S. Analytic continuations of those yield non-commutative (Fréchet) function algebras containing observables with exponential growth, unlike in the (topological) ${C}^{*}$ framework. The methods used to produce these formulae comes from symmetric spaces and harmonic analysis on Lie groups.

### Noncommutative geometry and number theory I

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