253,122 research outputs found
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Lectures on Kähler Geometry
These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different aspects of Kähler geometry
Characteristics, Bicharacteristics, and Geometric Singularities of Solutions of PDEs
Many physical systems are described by partial differential equations (PDEs).
Determinism then requires the Cauchy problem to be well-posed. Even when the
Cauchy problem is well-posed for generic Cauchy data, there may exist
characteristic Cauchy data. Characteristics of PDEs play an important role both
in Mathematics and in Physics. I will review the theory of characteristics and
bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e.,
those aspects which are invariant under general changes of coordinates. After a
basically analytic introduction, I will pass to a modern, geometric point of
view, presenting characteristics within the jet space approach to PDEs. In
particular, I will discuss the relationship between characteristics and
singularities of solutions and observe that: "wave-fronts are characteristic
surfaces and propagate along bicharacteristics". This remark may be understood
as a mathematical formulation of the wave/particle duality in optics and/or
quantum mechanics. The content of the paper reflects the three hour minicourse
that I gave at the XXII International Fall Workshop on Geometry and Physics,
September 2-5, 2013, Evora, Portugal.Comment: 26 pages, short elementary review submitted for publication on the
Proceedings of XXII IFWG
Continuous and discrete aspects of Lagrangian field theories with nonholonomic constraints
This dissertation is a contribution to the differential-geometric treatment of classical field theories. In particular, I study both discrete and continuous aspects of classical field theories, in particular those with nonholonomic constraints. After some introductory chapters dealing with the geometric structures inherent in field theories and the discretization of field theories, the first part of the thesis is concerned with discrete field theories taking values in Lie groupoids. It is shown that many previously known discrete field theories are particular instances of Lie groupoid field theories, and the geometry of Lie groupoids is used to construct a unifying framework for this class. In two further chapters, the effect of symmetry upon this setup is described, with particular attention to the case of Euler-Poincaré reduction, which can be rephrased using concepts of discrete differential geometry. In the second part of the thesis, nonholonomic constraints for field theories are described. A number of differential-geometric results that characterize the nature of nonholonomic constraints are derived: in particular, a version of the De Donder-Weyl equation suitable for constrained field theories is discussed and a so-called momentum lemma is derived (describing the influence of symmetry upon the nonholonomic framework). In the last chapter, a physical example of a nonholonomic field theory is given, based on the theory of Cosserat media. This example is treated using the theory of the preceding chapters. Furthermore, a geometric numerical integration scheme is derived and used to give a quantitative insight into the dynamics
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