The authors analyse two topological invariants of an embedding of an arrangement of rational plane curves in the projective complex plane, namely, the cohomology ring of the complement and the characteristic varieties. Their main result states that the cohomology ring of the complement to a rational arrangement is generated by logarithmic 1 and 2-forms and its structure depends on a finite number of invariants of the curve (its combinatorial type).

Preliminaries Ring structure of $H^\bullet(\mathbb{P}^2\backslash \mathcal{R}; \mathbb{C})$ Generalized Aomoto complexes Characteristic varieties, local systems and rational arrangements Examples Bibliography.

We prove that the solutions of a cohomological equation of complex dimension one and in the analytic category have a monogenic dependence on the parameter. This cohomological equation is the standard linearized conjugacy equation for germs of holomorphic maps in a neighborhood of a fixed point.

The goal of this work is to describe the dynamics of generic homeomorphisms of certain compact metric spaces $X$. Here ``generic'' is used in the topological sense -- a property of homeomorphisms on $X$ is generic if the set of homeomorphisms with the property contains a residual subset (in the sense of Baire category) of the space of all homeomorphisms on $X$. The spaces $X$ we consider are those with enough local homogeneity to allow certain localized perturbations of homeomorphisms; for example, any compact manifold is such a space. We show that the dynamics of a generic homeomorphism is quite complicated, with a number of distinct dynamical behaviors coexisting (some resemble subshifts of finite type, others, which we call `generalized adding machines', appear strictly periodic when viewed to any finite precision, but are not actually periodic). Such a homeomorphism has infinitely many, intricately nested attractors and repellors, and uncountably many distinct dynamically-connected components of the chain recurrent set. We single out several types of these ``chain components'', and show that each type occurs densely (in an appropriate sense) in the chain recurrent set. We also identify one type that occurs generically in the chain recurrent set. We also show that, at least for $X$ a manifold, the chain recurrent set of a generic homeomorphism is a Cantor set, so its complement is open and dense. Somewhat surprisingly, there is a residual subset of $X$ consisting of points whose limit sets are chain components of a type other than the type of chain components that are residual in the space of all chain components. In fact, for each generic homeomorphism on $X$ there is a residual subset of points of $X$ satisfying a stability condition stronger than Lyapunov stability.

Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.

Complex symplectic spaces are non-trivial generalizations of the real symplectic spaces of classical analytical dynamics. This title presents a self-contained investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionality.

Investigates the anisotropic Hardy spaces associated with very general discrete groups of dilations. This book includes the classical isotropic Hardy space theory of Fefferman and Stein and parabolic Hardy space theory of Calderon and Torchinsky.

Deals with weighted projective lines, a class of non-commutative curves modelled by Geigle and Lenzing on a graded commutative sheaf theory. They play an important role in representation theory of finite-dimensional algebras; the complexity of the classification of coherent sheaves largely depends on the genus of these curves.

We develop the basic theory of root systems $R$ in a real vector space $X$ which are defined in analogy to the usual finite root systems, except that finiteness is replaced by local finiteness: the intersection of $R$ with every finite-dimensional subspace of $X$ is finite. The main topics are Weyl groups, parabolic subsets and positive systems, weights, and gradings.

Proves two equalities of local Kloosterman integrals on $\mathrm{GSp}\left(4\right)$, the group of $4$ by $4$ symplectic similitude matrices. This book conjectures that both of Jacquet's relative trace formulas for the central critical values of the $L$-functions for $\mathrm{g1}\left(2\right)$ in [{J1}] and [{J2}].