# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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.

### An old algorithm in a new era: MacMahon's partition analysis

In the late 1890's, Major Percy Alexander MacMahon developed his Partition Analysis with the object of proving his conjectures on plane partitions. While writing at length on this topic and clearly indicating its power, MacMahon eventually admitted that he was unable to apply his new tool to solve any of his conjectures on plane partitions.

In this talk, we shall begin by examining MacMahon's method from a purely elementary point of view and shall illustrate its power by considering questions of general interest such as: How many triangles are there with integer sides and perimeter n? Our object will be to illustrate the broad utility of a 19th century idea that is only now coming into its own in the era of computer algebra.

### Optimal control of a chemotaxis system

Chemotaxis is the process by which cells aggregate under the force of a chemical attractant. The cell and chemoattractant concentrations are governed by a coupled system of parabolic partial differential equations. We investigate the optimal control of the proportion of cells being generated. The optimality system contains forward and backward reaction-diffusion and convection-diffusion equations. A numerical scheme is discussed that addresses the special needs of this coupled system.

This is joint work with K. Renee Fister, Murray State University.

### Local symbols on algebraic curves

In 1968 John Tate gave a definition of the residues of differentials on curves in terms of traces of certain linear operators on infinite-dimensional vector spaces. The aim of the talk is to give a new definition of two local symbols on algebraic curves (the Tame Symbol and the Hilbert's Norm Residue Symbol) from the commutator of a central extension of groups. This definition, which involves topics of Steinberg symbols, is valid for curves over a perfect field. Moreover, when the curve is complete, analogously to Tate's construction the reciprocity laws of both symbols can be deduced from the finiteness of the cohomology groups ${H}^{0}\left(C,{O}_{C}\right)$ and ${H}^{1}\left(C,{O}_{C}\right)$.

### Métodos Variacionais em Equações Diferenciais

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