A graph (or more general relational structure) is homogeneous if any isomorphism between finite induced subgraphs extends to an isomorphism of the graph. Surprisingly, in view of the rigidity of almost all finite graphs, a countably infinite random graph is homogeneous with probability 1: indeed, it is isomorphic to a particular graph, the Rado graph, with probability 1. All countable homogeneous graphs are known; the others are not "random" in any reasonable sense, though they are all residual in the sense of Baire category (another interpretation of the phrase "almost all". Recently, Jaroslav Nesetril and I have investigated the analogue of homogeneity for graphs and posets, where "isomorphism" is replaced by "homomorphism" (or "monomorphism" in the definition. The talk will also touch on other kinds of structure.
We tend to approach mathematics of the past from our perspective today: understanding of earlier developments is shaped by what happened later. Nevertheless, a better historical understanding of a given subject will result from a comparison with what came before, from a study of the origins and background of the development under consideration. The subject of mathematical existence came to the forefront of various branches of analysis in the nineteenth century. The recurring interest in existence questions after 1830 represented a new theoretical tendency in mathematics, one that was not present in the writings of eighteenth-century masters of analysis.
In the calculus of variations there was the well-known use of variational arguments such as Dirichlet's principle to establish the existence of solutions of boundary-value problems defined by partial differential equations. Existence questions also came up in other parts of this subject: in the theory of the second variation, in Weierstrasss field theory and in the study of constrained optimization. The lecture will examine the history of some technical results that involved existence assumptions or raised questions concerning the existence of mathematical objects. Included in this survey will be the work of Adolph Mayer, Edmund Husserl, Weierstrass, Adolf Kneser, Hilbert, Hadamard, and Oskar Bolza.
In this talk I will propose a set of mechanistic rules that can be used to understand the process of territorial pattern formation through interactions with scent marks. The models are described as systems of partial differential equations, coupled to ordinary differential equations. Under realistic assumptions the resulting territorial patterns include spontaneous formation of `buffer zones' between territories which act as refuges for prey such as deer. This result is supported by detailed radiotracking studies. In some cases, energy methods can be applied to the system, and the lowest energy solution corresponds to a spatial territory.
The model will also be analysed using game theory, where the objective of each pack is to maximize its fitness by increasing intake of prey (deer) and by decreasing interactions with hostile neighboring packs. Predictions will compared with radio tracking data for coyotes and wolves, including some new data from Yellowstone, where topography and local prey density can be shown to affect movement behavior.
In turbulence, the most interesting stationary and attracting solutions are finite flux Kolmogorov spectra which describe an energy density distribution that allows for a constant flux of energy from large scale sources to small scale sinks. The manner in which such spectra are realized can be surprising. I will discuss what happens for wave turbulence situations and then speculate on what may very well be generic behavior. What is most interesting is the introduction of the notion of entropy, more accurately entropy production, which I will show plays a meaningful role even in nonisolated systems. I will try to make the ideas accessible to an audience with a broad background.
The classical method of moving frames was developed by Elie Cartan into a powerful tool for studying the geometry of curves and surfaces under certain geometrical transformation groups. In this talk, I will discuss a new foundation for moving frame theory based on equivariant maps. The method is completely algorithmic, and can be readily applied to completely general Lie group and even infinite-dimensional pseudo-group actions. The resulting theory and applications are remarkably wide-ranging, including geometry, classical invariant theory, differential equations, the calculus of variations, symmetry and object recognition in computer vision, and the design of symmetry-preserving numerical algorithms.
Recent and continuing studies on branching tube flows will be described, the motivation coming from applications to the cardiovascular system, lung airways and cerebral arteriovenous malformations. The work is based partly on modelling for increased flow rates, partly on direct numerical simulations and partly on the various comparisons possible. Small differentials in pressure acting across a multiple branching may be considered first, followed by substantial pressure differentials in a side branching, in a multiple branching or in a basic three-dimensional branching. All of these cases include a comparison of results between the modelling and the direct simulations. Wall shear, pressure variation, influence lengths, and separation or its suppression will be examined, showing in particular sudden spatial adjustment of the pressure between mother and daughter tubes, nonunique flow patterns and an almost linear increase of flow rate with increasing number of daughters, depending on the specific conditions. The extension to large networks of vessels will also be addressed. The agreement between modelling and direct simulations is found to be generally close at moderate flow rates, suggesting their combined use in the biomedical applications.
Groups and semigroups gradings on rings and algebras are studied very intensively last years. One of the basic problems of the structure theory of graded algebras is to describe all finite dimensional simple objects. In particular, it is very important to describe all possible gradings on finite dimensional simple algebras. For example, in case of associative algebras over an algebraically closed field any simple finite dimensional algebra is a matrix algebra. So, one of the first step is the classification of all group gradings on matrix algebras. The results concerning the classification of all group gradinds on matrix algebras, on finite dimensional simple Lie and Jordan algebras will be presented in the talk.