Axiomatic homotopy theory for operads

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.Comment: 29 pages, revised for publicatio

Leibniz homology of Lie algebras as functor homology

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated to the Lie operad.Comment: 26 page

Character formulas for the operad of two compatible brackets and for the bihamiltonian operad

We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.Comment: 24 pages, accepted by Functional Analysis and its Applications, a few typos correcte

Dissipative motion in galaxies with non-axisymmetric potentials

Due to the clumpy nature of the self gravitating gas composing the interstellar medium, it is not clear whether galactic gas dynamics can be discussed in terms of standard hydrodynamics. Nevertheless, it is clear that certain generic properties related to orbital structure in a given potential and the effect of dissipation can be used to qualitatively understand gas motion in galaxies. The effect of dissipation is examined in triaxial galaxy potentials with and without rotating time dependent components. In the former case, dissipative trajectories settle around closed loop orbits when these exist. When they do not, e.g., inside a constant density core, then the only attractor is the centre and this leads to mass inflow. This provides a self regulating mechanism for accession of material towards the centre --- since the formation of a central masses destroys the central density core and eventually stops the accession. In the case when a rotating bar is present, there are usually several types of attractors, including those on which long lived chaotic motion can occur (strange attractors). Motion on these is erratic with large radial and vertical oscillations.Comment: Contibution to the conference on ``Astrophysical Fluids: From Atomic Nuclei to Stars and Galaxies'' (in honour of Giora Shaviv's 60 th birthday). To appear in Physics Report

Physics, Topology, Logic and Computation: A Rosetta Stone

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.Comment: 73 pages, 8 encapsulated postscript figure