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We start with an epistemological introduction on the evolution of the concepts of space and time and more generally of physical concepts in the context of the relation between mathematics and physics from the point of view of deformation theory. The concepts of relativity, including anti de Sitter space-time, and of quantization, are important paradigms; we briefly present these and some consequences. The importance of symmetries and of space-time in fundamental physical theories is stressed. The last section deals with "composite elementary particles" in anti de Sitter space-time and ends with speculative ideas around possible quantized anti de Sitter structures in some parts of the universe. [ABSTRACT FROM AUTHOR]

Ideas from deformation quantization applied to algebra with one generator lead to the construction of non-linear flat connection, whose parallel sections have algebraic significance. The moduli space of parallel sections is studied as an example of bundle-like objects with discordant (sogo) transition functions, which suggests a method to treat families of meromorphic functions with smoothly varying branch points. [ABSTRACT FROM AUTHOR]

There exist foliated T2-bundles on close surfaces whose foliation W*-algebras are isomorphic to the hyperfinite factor of type IIIλ. We introduce a secondary invariant called a K-set for such foliations. The K-set can detect the value λ determined from Connes' S-set and its behavior for such foliations are quite similar to that of the S-set. This suggests that the K-set can be considered as a geometric counterpart of the S-set. [ABSTRACT FROM AUTHOR]

We describe irreducible representations, coherent states and star-products for algebras of integrals of motions (symmetries) of two-dimensional resonance oscillators. We demonstrate how the quantum geometry (quantum Kähler form, metric, quantum Ricci form, quantum reproducing measure) arises in this problem. We specifically study the distinction between the isotropic resonance 1 : 1 and the general l : m resonance for arbitrary coprime l,m. A quantum gyron is a dynamical system in the resonance algebra. We derive its Hamiltonian in irreducible representations and calculate the semiclassical asymptotics of the gyron spectrum via the quantum geometrical objects. [ABSTRACT FROM AUTHOR]

We present here classes of parabolic geometries arising naturally from Se-ashi's principle to form good classes of linear differential equations of finite type, which generalize the cases of second and third order ODE for scalar functions. We will explicitly describe the symbols of these differential equations. The model equations of these classes admit nonlinear contact transformations and their symmetry algebras become finite dimensional and simple. [ABSTRACT FROM AUTHOR]

The notion of μ-regulated algebras is given in a slightly revised version. Notions of their localizations, limit localizations and intertwiners among them are defined. A family of limit localizations parameterized by Σ is viewed as the space of deformations of expressions of the algebra obtained by a limit localization. The space has a flat connection defined through infinitesimal intertwiners. Several examples of localized algebra with finite generator system are given. In concrete calculation, there appear several strange phenomena such as an element having two different inverses, or an expression of some elements having sign ambiguity. However, such phenomena are related with Jacobi's theta functions, where Jacobi's theta function is redefined as the bilateral power series in a ✻-algebra. [ABSTRACT FROM AUTHOR]

Given a 2-vector field and a closed 1-form on a manifold, we consider the set of cotangent vectors which annihilate the deformed Schouten bracket of the 2-vector field by the closed 1-form. We show that if the space of cotangent vectors forms a vector bundle, it carries a structure of a Lie algebroid. We treat this theorem in the category of Lie algebroids. As a special case, this result contains the well-known fact that the 1-jet bundle of functions of a contact manifold has a Lie algebroid structure. [ABSTRACT FROM AUTHOR]

It is proven that a local Lie algebra in the sense of A. A. Kirillov determines the base manifold up to a diffeomorphism provided the anchor map is nowhere-vanishing. In particular, the Lie algebras of nowhere-vanishing Poisson or Jacobi brackets determine manifolds. This result has been proven for different types of differentiability: smooth, real-analytic, and holomorphic. [ABSTRACT FROM AUTHOR]

First the fundamental results of analytic curves with singularities which correspond to those in the classical theory of curves in Euclidean space are described. And then the geometry of surfaces with singularities are studied. [ABSTRACT FROM AUTHOR]

In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold (M, ω) of dimension 2n, we define induction as a way to construct a symplectic manifold (P, μ) of dimension 2n + 2. Given any symplectic manifold (M, ω) of dimension 2n and given a symplectic connection ▽ on (M, ω), we define induction as a way to construct a symplectic manifold (P, μ) of dimension 2n+2 and an induced connection ▽P which is a Ricci-flat symplectic connection on (P, μ). [ABSTRACT FROM AUTHOR]

A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating a complex Lie algebroid should be a holomorphic stack. [ABSTRACT FROM AUTHOR]

Let (M, g) be a compact connected locally conformally flat semi-symmetric space of dimension 3 and with principal Ricci curvatures ρ1 = ρ2 ≠ ρ3 = 0. Then M is a Seifert fibre space. Moreover, in case the holonomy group is discrete, M is commensurable to a Kleinian manifold. If the holonomy group is indiscrete, (M, ḡ) is a hyperbolic surface bundle over a circle and (M, g) has negative scalar curvature. Here ḡ denotes a metric induced from the flat conformal structure. [ABSTRACT FROM AUTHOR]

We study Möbius applicable surfaces, i.e., conformally immersed surfaces in Möbius 3-space which admit deformations preserving the Möbius metric. We show new characterizations of Willmore surfaces, Bonnet surfaces and harmonic inverse mean curvature surfaces in terms of Möbius or similarity invariants. [ABSTRACT FROM AUTHOR]

In the present article, we are concerned with the automorphisms of a contact Weyl manifold, and we introduce an infinite-dimensional Lie group structure for the automorphism group. [ABSTRACT FROM AUTHOR]

In this paper we will show that a Möbius structure or a conformal structure of a manifold induces a projective structure of a regular curve on the manifold, and that for a regular curve on the sphere, the curve has no self-intersection if the projective developing map of the curve is injective. [ABSTRACT FROM AUTHOR]

This is a survey of some topics where global and stochastic analysis play a role. An introduction to analysis on Banach spaces with Gaussian measure leads to an analysis of the geometry of stochastic differential equations, stochastic flows, and their associated connections, with reference to some related topological vanishing theorems. Following that, there is a description of the construction of Sobolev calculi over path and loop spaces with diffusion measures, and also of a possible L2 de Rham and Hodge-Kodaira theory on path spaces. Diffeomorphism groups and diffusion measures on their path spaces are central to much of the discussion. Knowledge of stochastic analysis is not assumed. [ABSTRACT FROM AUTHOR]