Related papers: Singularity, complexity, and quasi--integrability …
For an $n$-qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such…
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
A paper is devoted to study of local behavior of so-called $Q$-mappings including qua\-si\-con\-for\-mal mappings and mappings with bounded distortion. It is showed that, such mappings have removable isolated singularities whenever the grow…
Let $X$ and $Y$ be finite complexes. When $Y$ is a nilpotent space, it has a rationalization $Y \to Y_{(0)}$ which is well-understood. Early on it was found that the induced map $[X,Y] \to [X,Y_{(0)}]$ on sets of mapping classes is…
We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic…
In this paper I survey some recent results on finite determination, convergence, and approximation of formal mappings between real submanifolds in complex spaces. A number of conjectures are also given.
Spaces of quasi-invariant measures supplied with different topologies are studied. Their embeddings, projective decompositions, conditions for their metrizability are investigated. Theorems about convergence of nets of quasi-invariant…
In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary…
We study the Hamiltonian dynamics and spectral theory of spin-oscillators. Because of their rich structure, spin-oscillators display fairly general properties of integrable systems with two degrees of freedom. Spin-oscillators have…
A simple and algorithmic description of matrix shape invariant potentials is presented. The complete lists of generic matrix superpotentials of dimension $2\times2$ and of special superpotentials of dimension $3\times3$ are given…
In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion.…
We study the symmetry properties of autonomous integrating factors from an algebraic point of view. The symmetries are delineated for the resulting integrals treated as equations and symmetries of the integrals treated as functions or…
Near-integrability is usually associated with smooth small perturbations of smooth integrable systems. Studying integrable mechanical Hamiltonian flows with impacts that respect the symmetries of the integrable structure provides an…
The schematic finite spaces are those finite ringed spaces where a theory of quasi-coherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show…
This article has two purposes. The first is to give an expository account of the integrable systems approach to harmonic maps from surfaces to Lie groups and symmetric spaces, focusing on spectral curves for harmonic 2-tori. The most…
Interconnected dynamic systems are a pervasive component of our modern infrastructures. The complexity of such systems can be staggering, which motivates simplified representations for their manipulation and analysis. This work introduces…
We consider the general model for dynamical systems defined on a simplicial complex. We describe the conjugacy classes of these systems and show how symmetries in a given simplicial complex manifest in the dynamics defined thereon,…
We investigate the combinatorial and dynamical properties of so-called nearly Euclidean Thurston maps, or NET maps. These maps are perturbations of many-to-one folding maps of an affine two-sphere to itself. The close relationship between…
The relation between integrable systems and algebraic geometry is known since the XIXth century. The modern approach is to represent an integrable system as a Lax equation with spectral parameter. In this approach, the integrals of the…
This small text was written for the AMS Notices. It is a survey of integrality properties of complex local systems, where I tried to single out one example which is not entirely explicit in the literature. The focus is on the obstruction it…