“[Floer] homology theory depends only on the topology of your manifold. [This] is Floer’s incredible insight,” said Agustin Moreno of the Institute for Advanced Study.
Floer homology theory depends only on the topology of the manifold.
3 Matching Annotations
- Dec 2021
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www.quantamagazine.org www.quantamagazine.org
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- Jun 2021
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math.stackexchange.com math.stackexchange.com
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Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R
Dual spaces also appear in geometry as the natural setting for certain objects. For example, a differentiable function f:M→R where M is a smooth manifold is an object that produces, for any point p∈M and tangent vector v∈TpM, a number, the directional derivative, in a linear way. In other words, ==a differentiable function defines an element of the dual to the tangent space (the cotangent space) at each point of the manifold.==
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en.wikipedia.org en.wikipedia.org
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To put it succinctly, differential topology studies structures on manifolds that, in a sense, have no interesting local structure. Differential geometry studies structures on manifolds that do have an interesting local (or sometimes even infinitesimal) structure.
Differential topology take a more global view and studies structures on manifolds that have no interesting local structure while differential geometry studies structures on manifolds that have interesting local structures.
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