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With this question I'm looking for suggestions about how and where to start learning differential topology starting from a good base of linear algebra and geometry, also consider that this know-how is supposed to be used in a programming context and for reading and writing algorithms.

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I don't think I have ever seen a programming problem that required good knowledge of differential topology--even problems that directly address topology! Why don't you tell us a little more about your "programming context" and the kinds of algorithms you are interested in. When you do that, please tell us how this question is related to GIS so that it won't be closed as off-topic. –  whuber Mar 6 '13 at 16:27
    
@whuber as far as I know anyone who wants to process and generate 3D meshes needs a know how about this, 3D world in general requires this as a good starting point. –  user2128456 Mar 6 '13 at 17:29
    
Generating and understanding 3D meshes has almost nothing to do with differential topology (which is the study of smooth real manifolds, which--among other properties--have no "kinks," edges, or cusps, which are characteristic of meshes and triangulations in general). It sounds like you ought to be more interested in basic topological concepts including point-set topology and the elements of algebraic topology, especially simplicial homology, as well as the combinatorial underpinnings. –  whuber Mar 6 '13 at 17:37
    
@whuber I'd think algorithms for processing Lidar point clouds would be a very good thing to know. Any idea what sort of math this would involve? –  Kirk Kuykendall Mar 7 '13 at 15:34
    
@Kirk It depends on the algorithm, but generally it helps to know some linear algebra, statistics, and foundations of computer science (analysis of algorithms). The skills in analytical geometry and visualization normally learned in an advanced calculus course would also be helpful. –  whuber Mar 7 '13 at 15:42
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closed as unclear what you're asking by whuber Jul 11 '13 at 2:41

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Here is a place to start, I believe it is an area called mesh physics. http://en.wikipedia.org/wiki/Topology_optimization

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This does not appear to be relevant to GIS, nor to differential topology. Could you please explain the connection? –  whuber Mar 7 '13 at 15:43
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