On Manifolds (2000-2009)

"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)

"Manifolds are a type of topological spaces we are interested in. They correspond well to the spaces we are most familiar with, the Euclidean spaces. Intuitively, a manifold is a topological space that locally looks like Rn. In other words, each point admits a coordinate system, consisting of coordinate functions on the points of the neighborhood, determining the topology of the neighborhood." (Afra J Zomorodian, "Topology for Computing", 2005)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)

"One could also question whether we are looking for a single overarching mathematical structure or a combination of different complementary points of view. Does a fundamental theory of Nature have a global definition, or do we have to work with a series of local definitions, like the charts and maps of a manifold, that describe physics in various 'duality frames'. At present string theory is very much formulated in the last kind of way." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"Roughly speaking, a manifold is essentially a space that is locally similar to the Euclidean space. This resemblance permits differentiation to be defined. On a manifold, we do not distinguish between two different local coordinate systems. Thus, the concepts considered are just those independent of the coordinates chosen. This makes more sense if we consider the situation from the physics point of view. In this interpretation, the systems of coordinates are systems of reference." (Ovidiu Calin & Der-Chen Chang,  "Geometric Mechanics on Riemannian Manifolds : Applications to partial differential equations", 2005)

"Quantum physics, in particular particle and string theory, has proven to be a remarkable fruitful source of inspiration for new topological invariants of knots and manifolds. With hindsight this should perhaps not come as a complete surprise. Roughly one can say that quantum theory takes a geometric object (a manifold, a knot, a map) and associates to it a (complex) number, that represents the probability amplitude for a certain physical process represented by the object." (Robbert Dijkgraaf, "Mathematical Structures", 2005)

"A manifold is an abstract mathematical space, which locally (i.e., in a close–up view) resembles the spaces described by Euclidean geometry, but which globally (i.e., when viewed as a whole) may have a more complicated structure." (Vladimir G Ivancevic & Tijana T Ivancevic, "Applied Differential Geometry: A Modern Introduction", 2007)

"The idea of Morse theory is that the topology/geometry of a manifold can be understood by examining the smooth functions (and their singularities) on that manifold." (Steven G Krantz, "The Proof is in the Pudding", 2007)