Related papers: Combination Theorems in Groups, Geometry and Dynam…
Some intensive observables of the electronic ground state in condensed matter have a geometrical or even topological nature. In this Review I present the geometrical observables whose expression is known in a full many-body framework,…
This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key…
After work of W. P. Thurston, C. Bavard and \'E. Ghys constructed particular hyperbolic polyhedra from spaces of deformations of Euclidean polygons. We present this construction as a straightforward consequence of the theory of…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of…
We prove the existence of continuous boundary extensions (Cannon-Thurston maps) for the inclusion of a vertex space into a tree of (strongly) relatively hyperbolic spaces satisfying the qi-embedded condition. This implies the same result…
This note is an attempt to give an answer for the following old I.M. Gelfand's question: why some important problems of integral geometry (e.g., the Radon transform and others) are related to harmonic analysis on groups, but for other quite…
We survey some of the connections linking complex dynamics to other fields of mathematics and science. We hope to show that complex dynamics is not just interesting on its own but also has value as an applicable theory.
The neat formulation that describes the gauge interactions associated with internal symmetries is extended to the case of a simple, yet non-trivial, symmetry group structure which mixes gravity and electromagnetism by associating a gauge…
The paper is mainly devoted to systematic developments and applications of geometric aspects of second-order variational analysis that are revolved around the concept of parabolic regularity of sets. This concept has been known in…
The requirement of diffeomorphism symmetry for the target space can lead to anomalous commutators for the energy-momentum tensor for sigma models and for fluid dynamics, if certain topological terms are added to the action. We analyze…
This is an invited survey article on higher gauge theory for the Encyclopedia of Mathematical Physics, 2nd edition. In particular, we provide a lightning introduction to higher structures and to the construction of the kinematical data of…
We introduce the notions of geometric height and graded (geometric) relative hyperbolicity in this paper. We use these to characterize quasiconvexity in hyperbolic groups, relative quasiconvexity in relatively hyperbolic groups, and convex…
We introduce a mixture of generalized hyperbolic distributions as an alternative to the ubiquitous mixture of Gaussian distributions as well as their near relatives of which the mixture of multivariate t and skew-t distributions are…
We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the…
Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…
We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic…
The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the…
The use of double groupoids and their associated double Lie algebroids and characteristic distributions is proposed for the description and analysis of continuous media that carry two different constitutive or geometric structures. Various…
In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…