# Analysis, Geometry, and Dynamical Systems Seminar

## Past sessions

Newer session pages: Next 12 11 10 9 8 7 6 5 4 3 2 1 Newest

### Combining logic systems: Why, how, what for?

Motivated by applications in artificial intelligence and software engineering that require the joint use of different deduction formalisms, the interest in combination of logic systems has recently been growing, but the topic is also of interest on purely theoretical grounds. Several forms of combination have been studied, like product, fusion, temporalization, parameterization, synchronization and, more recently, fibring. In this guided tour of the issues raised by the combination of logics, we define fibring (the most general form of combination) in a very simple (yet useful) context, discuss some examples and establish some interesting transference results, namely preservation of strong completeness and nonpreservation of congruence. We end the tour with a brief reference to some open problems. The talk is based on a recent overview paper (together with C. Sernadas) available at http://www.cs.math.ist.utl.pt/ftp/pub/SernadasA/03-SS-fiblog22.pdf to appear in the CIM Bulletin.

### Hofer-Zehnder sensitive capacity of cotangent bundles and symplectic submanifolds

We will consider a modified Hofer-Zehnder capacity sensitive to the homotopy class of the periodic orbits and show that if a symplectic manifold admits a free Hamiltonian circle action then it has bounded Hofer-Zehnder (sensitive) capacity. We give two applications of this result. Firstly, we prove that every closed symplectic submanifold has a neighborhood with finite Hofer-Zehnder capacity. Secondly, consider a closed manifold with an effective circle action whose fixed point set has trivial normal bundle. Then, its standard cotangent bundle has bounded Hofer-Zehnder capacity.

### 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.

Older session pages: Previous 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 Oldest