Related papers: The Lorentzian distance formula in noncommutative …
Lorentzian polynomials are a fascinating class of real polynomials with many applications. Their definition is specific to the nonnegative orthant. Following recent work, we examine Lorentzian polynomials on proper convex cones. For a…
We introduce an estimator for distances in a compact Riemannian manifold based on graph Laplacian estimates of the Laplace-Beltrami operator. We upper bound the error in the estimate of manifold distances, or more precisely an estimate of a…
Noncommutative geometry, in its many incarnations, appears at the crossroad of various researches in theoretical and mathematical physics: from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and…
This contribution is an introduction to the metric aspect of noncommutative geometry, with emphasize on the Moyal plane. Starting by questioning "how to define a standard meter in a space whose coordinates no longer commute?", we list…
We define a Lorentzian distance function on the group of contactomorphisms of a closed contact manifold. This distance function is continuous with respect to the Hofer norm on the group of contactomorphisms defined by Shelukhin and finite…
Application of the noncommutative geometry to several physical models is considered.
We present an abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, and an alternative approach to Lorentzian length spaces that does not use auxiliary ``positive signature'' metrics or other unobserved fields. We begin…
To appear in Encyclopedia of Mathematical Physics, J.-P. Fran\c{c}oise, G. Naber and T.S. Tsou, eds., Elsevier, 2006. The article surveys the modern developments of noncommutative geometry in string theory.
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…
We prove a splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature. Additionally, we extend the first variation formula to spaces with any timelike curvature bound, either from above or below, and…
In this paper, we study a two-dimensional Lorentzian problem on the anti-de Sitter plane. Using methods of geometric control theory and differential geometry, it was possible to construct an orthonormal frame, calculate extremal…
I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…
A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played…
An introduction is given to some selected aspects of noncommutative geometry. Simple examples in this context are provided by finite sets and lattices. As an application, it is explained how the nonlinear Toda lattice and a discrete time…
A characterization of the proximal normal cone is obtained and a separation theorem for convex subsets of Riemannian manifolds is established. Moreover, the convexity of the distance function $d_S$ for a convex subset $S$ in the cases where…
We study a relative trace formula for a compact Riemann surface with respect to a closed geodesic $C$. This can be expressed as a relation between the period spectrum and the ortholength spectrum of $C$. This provides a new proof of…
Contribution to the Proceedings in honour of C.N. Yang, editor S-T Yau. To
The well known relation between extended supersymmetry and complex geometry in the non-linear sigma-models is reviewed, and some recent developments related to the introduction of the non-anti-commutativity, in the context of the…
Our aim is to prove the integration formula on the noncommutative (Moyal) plane in terms of singular traces {\it a la} Connes.
The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the…