Abstract:
We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.

Abstract:
A homeomorphism of a compact metric space is {\em tight} provided every non-degenerate compact connected (not necessarily invariant) subset carries positive entropy. It is shown that every $C^{1+\alpha}$ diffeomorphism of a closed surface factors to a tight homeomorphism of a generalized cactoid (roughly, a surface with nodes) by a semi-conjugacy whose fibers carry zero entropy.

Abstract:
Giving coding region structural features a role in the hypomethylation of specific genes, the occurrence of G+C content, CpG islands, repeat and retrotransposable elements in demethylated genes related to cancer has been evaluated. A comparative analysis among different cancer types has also been performed. In this work, the inter-cancer coding region features comparative analysis carried out, show insights into what structural trends/patterns are present in the studied cancers.

Abstract:
Let $M$ be a closed oriented surface endowed with a Riemannian metric $g$. We consider the flow $\phi$ determined by the motion of a particle under the influence of a magnetic field $\Omega$ and a thermostat with external field ${\bf e}$. We show that if $\phi$ is Anosov, then it has weak stable and unstable foliations of class $C^{1,1}$ if and only if the external field ${\bf e}$ has a global potential $U$, $g_{1}:=e^{-2U}g$ has constant curvature and $e^{-U}\Omega$ is a constant multiple of the area form of $g_1$. We also give necessary and sufficient conditions for just one of the weak foliations to be of class $C^{1,1}$ and we show that the {\it combined} effect of a thermostat and a magnetic field can produce an Anosov flow with a weak stable foliation of class $C^{\infty}$ and a weak unstable foliation which is {\it not} $C^{1,1}$. Finally we study Anosov thermostats depending quadratically on the velocity and we characterize those with smooth weak foliations. In particular, we show that quasi-fuchsian flows as defined by Ghys in \cite{Ghy1} can arise in this fashion.

Abstract:
Let $M$ be a closed orientable Riemannian surface. Consider an SO(3)-connection $A$ and a Higgs field $\Phi:M\to so(3)$. The pair $(A,\Phi)$ naturally induces a cocycle over the geodesic flow of $M$. We classify (up to gauge transformations) cohomologically trivial pairs $(A,\Phi)$ with finite Fourier series in terms of a suitable B\"acklund transformation. In particular, if $M$ is negatively curved we obtain a full classification of SO(3)-transparent pairs.

Abstract:
Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent SU(2)-connection associated to each meromorphic function on $M$.

Abstract:
We find obstructions to the existence of Einstein metrics of non-negative sectional curvature on a smooth closed simply connected manifold of any dimension. The results are achieved by combining the classical Morse theory of the loop space with a new upper bound for the topological entropy of the geodesic flow in terms of the curvature tensor.

Abstract:
We prove exponential growth rate of contractible closed geodesics for an arbitrary bumpy metric on manifolds of the form X#Y, where the fundamental group of X has a subgroup of finite index at least 3 and Y is simply connected and not a homotopy sphere.

Abstract:
Let $M$ be a closed orientable surface of negative curvature. A connection is said to be transparent if its parallel transport along closed geodesics is the identity. We describe all transparent SU(2)-connections and we show that they can be built up from suitable B\"acklund transformations.