Related papers: Helly's Theorem: New Variations and Applications
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
We review the special geometry of N = 2 supersymmetric vector and hypermultiplets with emphasis on recent developments and applications. A new formulation of the local c-map based on the Hesse potential and special real coordinates is…
Surveys of the local and distant Universe are the means to test and improve our models of galaxy formation. Substantial successes in the models are evident, while there is also considerable recent progress in identifying what remains to be…
The physical consistency of the match of piecewise-$C^0$ metrics is discussed. The mathematical theory of gravitational discontinuity hypersurfaces is generalized to cover the match of regularly discontinuous metrics. The mean-value…
Recent developments in Seiberg-Witten theory and relations with Complex Geometry.
This note is an expanded and updated version of our entry with the same title for the 2006 Encyclopedia of Mathematical Physics. We give a brief overview of graded Poisson algebras, their main properties and their main applications, in the…
Over the past three decades, black holes have played an important role in quantum gravity, mathematical physics, numerical relativity and gravitational wave phenomenology. However, conceptual settings and mathematical models used to discuss…
Black-hole perturbation theory is a useful tool to investigate issues in astrophysics, high-energy physics, and fundamental problems in gravity. It is often complementary to fully-fledged nonlinear evolutions and instrumental to interpret…
New results on the convexity of geodesic-length functions on Teichm\"{u}ller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A…
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)k\"ahler reduction; projective superspace; the generalized Legendre construction;…
We point out how some recent developments in the theory of constant scalar curvature K\"ahler metrics can be used to clarify the existence issue for such metrics in the special case of geometrically ruled complex surfaces.
A slight modification to one of Tarski's axioms of plane Euclidean geometry is proposed. This modification allows another of the axioms to be omitted from the set of axioms and proven as a theorem. This change to the system of axioms…
This paper is about Information Geometry, a relatively new subject within mathematical statistics that attempts to study the problem of inference by using tools from modern differential geometry. This paper provides an overview of some of…
Some mathematical theorems represent ideas that are discovered again and again in different forms. One such theorem is Hall's marriage theorem. This theorem is equivalent to several other theorems in combinatorics and optimization theory,…
This is a (mostly expository) paper on Reidemeister classes, twisted Burnside-Frobenius theory, congruences, R-infinity property and all that. It was written in 2005 and published in 2008. We post it as it was, only the bibliography data is…
Many moduli spaces are constructed as quotients of group actions; this paper surveys the classical theory, as well as recent progress and applications. We review geometric invariant theory for reductive groups and how it is used to…
The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and…
Survey article on the geometry of spherical varieties. Invited survey for Transformation Groups.
We give a brief introduction to some of the recent works on finding geometric structures on triangulated surfaces using variational principles.