In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below.IntroductionDefinition
A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood that is homeomorphic to Euclidean 3-space.Mathematical theory of 3-manifolds
The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very spe
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This course gives an introduction to the fundamental concepts and methods of the Digital Humanities, both from a theoretical and applied point of view. The course introduces the Digital Humanities circle of processing and interpretation, from data acquisition to new understandings.
We develop, analyze and implement numerical algorithms to solve optimization problems of the form: min f(x) where x is a point on a smooth manifold. To this end, we first study differential and Riemannian geometry (with a focus dictated by pragmatic concerns). We also discuss several applications.
In this seminar we will study the foundations of Knot theory, which studies entanglements of closed curves in three dimensional space.
What seems like a simple subject in topology turns out to be a rich theory with applications to 3-manifolds, representation theory, physics and much more.
In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analog
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure
In the mathematical field of geometric topology, the Poincaré conjecture (UKˈpwæ̃kæreɪ, USˌpwæ̃kɑːˈreɪ, pwɛ̃kaʁe) is a theorem about the characterization of the 3-sphere, which is the hypersphere th
In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrodinger equation. We also give a natural interpretation of the function. introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.