# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### 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 ${H}^{1}\left(C,{O}_{C}\right)$.

### Remarks about diffusion mediated transport

Diffusion mediated transport is implicated in the operation of many molecular level systems. These include some liquid crystal and lipid bilayer systems, and, especially, the motor proteins responsible for eukaryotic cellular traffic. All of these systems are extremely complex. In this expository talk we address the basic elements of transport in an environment of diffusion, concentrating on the flashing rachet. We illustrate how to characterize the properties of this process in terms of a Markov chain. It is of essential importance to understand what it means to determine a Markov chain in this manner, namely, the sense in which the discrete problem is an accurate description of the original continuous one. Here we are able to exploit some novel concepts about the dynamics of evolutionary systems including an appeal to the classical Monge transfer problem. This perspective illuminates the fundamental connection between the two problems and presents an opportunity for future applications.

This is joint work with Michal Kowalczyk.

### Singularly perturbed ordinary differential equations: an overview. I

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