Abstract:
We consider a magnetic flow without conjugate points on a closed manifold $M$ with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ \pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is closed. This result was proved in \cite{DP2} under the assumption that the flow of $\G$ is Anosov.

Abstract:
We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the "renewal flow", which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.

Abstract:
This paper is devoted to study thermodynamic formalism for suspension flows defined over countable alphabets. We are mostly interested in the regularity properties of the pressure function. We establish conditions for the pressure function to be real analytic or to exhibit a phase transition. We also construct an example of a potential for which the pressure has countably many phase transitions.

Abstract:
We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

Abstract:
For any toric automorphism with only real eigenvalues a Riemannian metric with an integrable geodesic flow on the suspension of this automorphism is constructed. A qualitative analysis of such a flow on a three-solvmanifold constructed by the authors in math.DG/9905078 is done. This flow is an example of the geodesic flow, which has vanishing Liouville entropy and, moreover, is integrable but has positive topological entropy. The authors also discuss some open problems on integrability of geodesic flows and related subjects.

Abstract:
The Jones-Witten invariants can be generalized for non-singular smooth vector fields with invariant probability measure on 3-manifolds, giving rise to new invariants of dynamical systems [22]. After a short survey of cohomological field theory for Yang-Mills fields, Donaldson-Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. These invariants have the information of the flows and they are interpreted as the intersection number of these flow orbits and constitute invariants of smooth four-manifolds admitting global flows. We study the case of Kahler manifolds by using the Witten's consideration of the strong coupling dynamics of N=1 supersymmetric Yang-Mills theories. The whole construction is performed by implementing the notion of higher dimensional asymptotic cycles a la Schwartzman [18]. In the process Seiberg-Witten invariants are also described within this context. Finally, we give an interpretation of our asymptotic observables of 4-manifolds in the context of string theory with flows.

Abstract:
In this paper, we compute universal minimal flows of groups of automorphisms of uncountable $\omega$-homogeneous graphs, $K_n$-free graphs, hypergraphs, partially ordered sets, and their extensions with an $\omega$-homogeneous ordering. We present an easy construction of such structures, expanding the jungle of extremely amenable groups.

Abstract:
In this article we study the mean return times to a given set for suspension flows. In the discrete time setting, this corresponds to the classical version of Kac's lemma \cite{K} that the mean of the first return time to a set with respect to the normalized probability measure is one. In the case of suspension flows we provide formulas to compute the mean return time. In particular, this varies linearly with continuous reparametrizatons of the flow and takes into account the mean escaping time from the original set.

Abstract:
The available expressions for the effective viscosity can not provide good predictions compared with the experiment data measured in the semi-dilute shear flows of fiber suspension with small aspect ratio. The departure of the theoretical prediction from the measured data increases with the decrease of the fiber aspect ratio. Therefore, by experiment for the fiber with 20 μm diameter, a new expression for the effective viscosity in the semi-dilute shear flows of fiber suspension with small aspect ratio is proposed, the relationship between the shear viscosity of fiber suspensions and the fiber concentration is given. The results show that the effective viscosity is not a linear function of the fiber concentration.

Abstract:
It is a well-known fact, that the greatest ambit for a topological group $G$ is the Samuel compactification of $G$ with respect to the right uniformity on $G.$ We apply the original destription by Samuel from 1948 to give a simple computation of the universal minimal flow for groups of automorphisms of uncountable structures using Fra\"iss\'e theory and Ramsey theory. This work generalizes some of the known results about countable structures.