Connected components of definable groups and o-minimality I

We give examples of groups G such that G^00 is different from G^000. We also prove that for groups G definable in an o-minimal structure, G has a "bounded orbit" iff G is definably amenable. These results answer questions of Gismatullin, Newelski, Petrykovski. The examples also give new non G-compact first order theories.Comment: 26 pages. This paper corrects the paper "Groups definable in o-minimal structures: structure theorem, G^000, definable amenability, and bounded orbits" by the first author which was posted in December (1012.4540v1) and later withdraw

Toy amphiphiles on the computer: What can we learn from generic models?

Generic coarse-grained models are designed such that they are (i) simple and (ii) computationally efficient. They do not aim at representing particular materials, but classes of materials, hence they can offer insight into universal properties of these classes. Here we review generic models for amphiphilic molecules and discuss applications in studies of self-assembling nanostructures and the local structure of bilayer membranes, i.e. their phases and their interactions with nanosized inclusions. Special attention is given to the comparison of simulations with elastic continuum models, which are, in some sense, generic models on a higher coarse-graining level. In many cases, it is possible to bridge quantitatively between generic particle models and continuum models, hence multiscale modeling works on principle. On the other side, generic simulations can help to interpret experiments by providing information that is not accessible otherwise.Comment: Invited feature article, to appear in Macromolecular Rapid Communication

Generic Automorphisms and Green Fields

We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories having the definable multiplicity property. We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap in the construction of the bad fiel