Abstract:
In this paper we study a criterion for the viability of stochastic semilinear control systems on a real, separable Hilbert space. The necessary and sufficient conditions are given using the notion of stochastic quasi-tangency. As a consequence, we prove that approximate viability and the viability property coincide for stochastic linear control systems. We obtain Nagumo's stochastic theorem and we present a method allowing to provide explicit criteria for the viability of smooth sets. We analyze the conditions characterizing the viability of the unit ball. The paper generalizes recent results from the deterministic framework.

Abstract:
This article is part of my upcoming masters thesis which investigates the following open problem from the book, Free Lattices, by R.Freese, J.Jezek, and J.B. Nation published in 1995: "Which lattices (and in particular which countable lattices) are sublattices of a free lattice?" Despite partial progress over the decades, the problem is still unsolved. There is emphasis on the countable case because the current body of knowledge on sublattices of free lattices is most concentrated on when these sublattices are countably infinite. In this article, a simple sufficient condition for a \emph{finitely generated} lattice to not be embeddable in a free lattice is derived, which is easier to verify than checking if Jonsson's condition, L = D(L) = D^d(L), holds. It provides a systematic way of providing counterexamples to the false claim that: "all finitely generated semidistributive lattices satisfying Whitman's condition is a sublattice of a free lattice." To the best of the author's knowledge, this result is new. A corollary that is derived from this is a non-trivial property of free lattices which might not yet be known in the literature. Moreover, a problem concerning finitely generated semidistributive lattices will be posed.

Abstract:
In this paper we study the effect of a homoclinic tangency in the variation of the topological entropy. We prove that a diffeomorphism with a homoclinic tangency associated to a basic hyperbolic set with maximal entropy is a point of entropy variation in the $C^{\infty}$-topology. We also prove results about variation of entropy in other topologies and when the tangency does not correspond to a basic set with maximal entropy. We also show an example of discontinuity of the entropy among $C^{\infty}$ diffeomorphisms of three dimensional manifolds.

Abstract:
We study various classes of real hypersurfaces that are not embeddable into more special hypersurfaces in higher dimension, such as spheres, real algebraic compact strongly pseudoconvex hypersurfaces or compact pseudoconvex hypersurfaces of finite type. We conclude by stating some open problems.

Abstract:
We will generalize the Treibich-Verdier theory about elliptic solitons to a Hitchin system by constructing a particular ruled surface and we will propose a generalization of a tangency condition associated with elliptic solitons to a Hitchin system. In particular, we will calculate the dimension of the moduli space of Hitchin covers satisfying the tangency condition. With this new point of view, we will see a subtle relation between the characterizations of coverings and the singularities of divisors in a particular algebraic surface.

Abstract:
We give an account of some results, both old and new, about any $n\times n$ Markov matrix that is embeddable in a one-parameter Markov semigroup. These include the fact that its eigenvalues must lie in a certain region in the unit ball. We prove that a well-known procedure for approximating a non-embeddable Markov matrix by an embeddable one is optimal in a certain sense.

Abstract:
We classify five dimensional Anosov flows with smooth decomposition which are in addition transversely symplectic. Up to finite covers and a special time change, we find exectly the suspensions of symplectic hyperbolic automorphisms of four dimensional toris, and the geodesic flows of three dimensional hyperbolic manifolds.

Abstract:
We study and classify up to a linear conjugation germs of flows in the space of quadratic birational transformations of the complex projective space of dimension 3. As a consequence we show that every quadratic flow preserves a pencil of planes through a line.

Abstract:
We obtain an implicit equation for the correlation dimension of dynamical systems in terms of an integral over a propagator. We illustrate the utility of this approach by evaluating the correlation dimension for inertial particles suspended in a random flow. In the limit where the correlation time of the flow field approaches zero, taking the short-time limit of the propagator enables the correlation dimension to be determined from the solution of a partial differential equation. We develop the solution as a power series in a dimensionless parameter which represents the strength of inertial effects.

Abstract:
Some equivalent gradient and Harnack inequalities of a diffusion semigroup are presented for the curvature-dimension condition of the associated generator. As applications, the first eigenvalue, the log-Harnack inequality, the heat kernel estimates, and the HWI inequality are derived by using the curvature-dimension condition. The transportation inequality for diffusion semigroups is also investigated.