The geometry of convex cones associated with the Lyapunov inequality and the common Lyapunov function problem.
The geometry of dented pentagram maps
We propose a new family of natural generalizations of the pentagram map from 2D to higher dimensions and prove their integrability on generic twisted and closed polygons. In dimension there are such generalizations called dented pentagram maps, and we describe their geometry, continuous limit, and Lax representations with a spectral parameter. We prove algebraic-geometric integrability of the dented pentagram maps in the 3D case and compare the dimensions of invariant tori for the dented maps...
The geometry of laminations
A lamination is a continuum which locally is the product of a Cantor set and an arc. We investigate the topological structure and embedding properties of laminations. We prove that a nondegenerate lamination cannot be tree-like and that a planar lamination has at least four complementary domains. Furthermore, a lamination in the plane can be obtained by a lakes of Wada construction.
The Geometry of Model Spaces for Probability-Preserving Actions of Sofic Groups
Bowen’s notion of sofic entropy is a powerful invariant for classifying probability-preserving actions of sofic groups. It can be defined in terms of the covering numbers of certain metric spaces associated to such an action, the ‘model spaces’. The metric geometry of these model spaces can exhibit various interesting features, some of which provide other invariants of the action. This paper explores an approximate connectedness property of the model spaces, and uses it give a new proof that certain...
The geometry of nondegeneracy conditions in completely integrable systems (corrected version of fascicule 4, volume XIV, 2005, p. 705-719)
Nondegeneracy conditions need to be imposed in K.A.M. theorems to insure that the set of diophantine tori has a large measure. Although they are usually expressed in action coordinates, it is possible to give a geometrical formulation using the notion of regular completely integrable systems defined by a fibration of a symplectic manifold by lagrangian tori together with a Hamiltonian function constant on the fibers. In this paper, we give a geometrical definition of different nondegeneracy conditions,...
The geometry of nondegeneracy conditions in completely integrable systems
The geometry of some natural conjugacies in dynamics.
The growth rate and dimension theory of beta-expansions
In a recent paper of Feng and Sidorov they show that for β ∈ (1,(1+√5)/2) the set of β-expansions grows exponentially for every x ∈ (0,1/(β-1)). In this paper we study this growth rate further. We also consider the set of β-expansions from a dimension theory perspective.
The hamiltonian formalism in higher order variational problems
The Hausdorff dimension of systems of linear forms
The Hirota method and soliton solutions to the multidimensional nonlinear Schrödinger equation.
The homeomorphism group of the hairy arc
The homotopy class of non-singular Morse-Smale vector fields on 3-manifolds.
The Hörmander and Maslov classes and Fomenko's conjecture.
The index of a vector field tangent to a hypersurface and the signature of the relative jacobian determinant
Given a real analytic vector field tangent to a hypersurface with an algebraically isolated singularity we introduce a relative Jacobian determinant in the finite dimensional algebra associated with the singularity of the vector field on . We show that the relative Jacobian generates a 1-dimensional non-zero minimal ideal. With its help we introduce a non-degenerate bilinear pairing, and its signature measures the size of this point with sign. The signature satisfies a law of conservation of...
The Intersection of Four Quadrics in IP6, Abelian Surfaces and their Moduli.
The Jeffrey-Kirwan localization theorem and residue operations in equivariant cohomology.
The Kolmogorov-Arnold-Moser theorem.
The Kowalevski's top and the method of Syzygies
A new algorithm for finding separation coordinates is tested on the example of Kowalev ski’s top.
The K-property of N billiard balls I.