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We give a new definition, based on considerations of well-posedness for a certain asymptotic initial value problem, of the phase space for the radiative degrees of freedom of the gravitational field in exact General Relativity. This space…
We review double field theory (DFT) with emphasis on the doubled spacetime and its generalized coordinate transformations, which unify diffeomorphisms and b-field gauge transformations. We illustrate how the composition of generalized…
This is a light survey article about the origins of contact and symplectic topology in dynamics and the more recent developments in the field. In lieu of formulas, numerous anecdotes are given.
Some boundedness properties of function spaces (considered as topological groups) are studied.
The flows of phase trajectories of cosmological models based on the vacuum classical Higgs field and their behavior on the Einstein-Higgs surface near singular points of a dynamical system are investigated by numerical simulation. The…
Various methods have been used to construct rational points and rational curves on rationally connected algebraic varieties. We survey recent advances in two of them, the descent and the fibration method, in a number-theoretical context…
Topological bases of behaviour of trajectories for autonomous differential systems of the second order on sphere are stated. Stereographic atlas of trajectories is constructed. Differential connections between trajectories of…
The Radon transform and its dual are central objects in geometric analysis on Riemannian symmetric spaces of the noncompact type. In this article we study algebraic versions of those transforms on inductive limits of symmetric spaces. In…
The Funk, cosine, and sine transforms on the unit sphere are indispensable tools in integral geometry. They are also known to be interesting objects in harmonic analysis. The aim of the paper is to extend basic facts about these transforms…
The analysis of domain wall dynamics is often simplified to one dimensional physics. For domain walls in thin films, more realistic approaches require the description as two dimensional objects. This includes the study of vortices and…
Recently proposed forms for gauge transformations with finite parameters in double field theory are discussed and problematic issues are identified. A new form for finite gauge transformations is derived that reveals the underlying gerbe…
Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with…
A new functional model for pairs of commuting isometries is described. Intertwining operators between such models are then studied in order to approach the classification of invariant subspaces of such pairs.
Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new…
We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and…
We survey recent developments on rationality problems for algebraic varieties, with a particular emphasis on cycle-theoretic and combinatorial methods and their applications to hypersurfaces.
The equation of motion for a point particle in the background field of double field theory is considered. We find that the motion is described by a geodesic flow in the doubled geometry. Inspired by analysis on the particle motion, we…
By using higher K-theory, we study deformation theory of K-theoretic cycles. As an application, we answer two questions posed by Mark Green and Philip Griffiths: (1). How to define tangent spaces to cycle class groups in general? (2).…
We review theoretical results obtained recently in the framework of statistical mechanics to study systems with long range forces. This fundamental and methodological study leads us to consider the different domains of applications in a…
In this work we explore the physics associated to Calabi-Yau (CY) n-folds that can be described as a fibration in more than one way. Beginning with F-theory vacua in various dimensions, we consider limits/dualities with M-theory, type IIA,…