# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

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### Remarks about diffusion mediated transport

Typical flow regimes in aerodynamics and fluid dynamics involve large Reynolds numbers. There are important issues regarding, for example, the relationship between kinetic and potential energy or turbulent behaviour. Here we shall discuss diffusion mediated transport, a property of systems with quite small Reynolds numbers, about 0.05. This is the environment of the living cell.

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 and involve subtle interactions on varying scales. Earlier, we were interested in the design of material microstructure, typically in order to optimize the performance of devices that do work by changing their microstructure. In such gadgets, like shape memory or magnetostrictive, energy transduction is very close to equilibrium in order to minimize the energy budget - think about remote controls. The chemical mechanical transduction in motor proteins is, by contrast, quite distant from equilibrium. These systems function in a dynamically metastable range.

We give a general dissipation principle and illustrate how it may be used to describe transport, for example in flashing rachet and conventional kinesin type motors. We introduce new methods based on the Monge-Kantorovich problem and Wasserstein metric to explore this. The equations we obtain are analogous to the ones already formulated by Astumian, and Oster, Ermentrout, and Peskin, and by Adjari and Prost and their collaborators. What is necessary for transport? What is the role of diffusion? What is the role of other elements of the system and how can dissipation be exploited to understand this? How successful are we? The opportunity to discover the interplay between chemistry and mechanics and to elaborate the implications of metastability could not offer a more exciting venue.

We are reporting here on joint work with Michel Chipot, Jean Dolbeault, Stuart Hastings, and Michal Kowalczyk.

### Some remarks on evolutions

We present some result concerning evolutions of Noetherian local rings of characteristic zero using Andre-Quillen homology of commutative rings.

### Residue Theorems and Lie Algebroid Invariants

The recent work of Harvey and Lawson has resulted in a family of residue theorems relating the singular sets (viewed as rectifiable currents) of a vector bundle map to the Chern-Weil characteristic forms of the bundles. Many of the classical singularity theorems (e.g., the Argument Principle, the Poincare-Hopf Index Theorem, and the Riemann-Roch Theorem) are special cases of these more general formulae.

In this talk, I will review briefly the techniques used by Harvey and Lawson and give an account of current work to extend these to the study of Lie algebroids.

### Poincaré recurrence, geodesic flows, hyperbolic geometry

Today, we are arguably in possession of a satisfactory approach to a quantitative description of recurrence. As one knows well: Poincaré only established that orbits return infinitely often, not how often or when they return.

This will be a colloquium type exposition of recent results that shed new light on the subject, also discussing joint work with Benoît Saussol, Christian Wolf, and Gianluigi Del Magno. The talk will include the view of the (maybe classical) hyperbolic geometer and briefly some consequences for number theory. No prerequisites from dynamical systems or ergodic theory will be necessary.

### Optimal strategies of time averaged optimization of cyclic motion

We consider dynamic inequalities with locally bounded derivatives on the circle and optimize the time averaged profit along admissible motions which are defined for all nonegative time. We describe the nature of optimal strategies, generic swichings between them when the problem depends additionally from a one dimensional parameter, and also the respective singularities of the averaged profit like a function of the parameter.

### Bound States in Curved Quantum Layers

We consider a nonrelativistic quantum particle constrained to a curved hard-wall layer of constant width built over a noncompact surface embedded in ${R}^{3}$. Under the assumption that the surface curvatures vanish at infinity, we find sufficient conditions which guarantee the existence of geometrically induced bound states.

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