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I am a second year undergraduate college student interested in applied math program. I hear a lot that general topology(e.g. the first half of Munkres' book Topology) is very useful, but is it really helpful for people like me who are into "applied" math, rather than pure math??

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  • $\begingroup$ Why not just learn it? It doesn't take too long to learn provided your analysis is good (which seeing as you're "applied", it should be). Maybe a couple months of self study. $\endgroup$ – user223391 May 20 '15 at 4:45
  • $\begingroup$ Depends... on what you are applying math for. Market researchers, for example, employ topologists to find the connected components of their demographical base. $\endgroup$ – David Wheeler May 20 '15 at 4:45
  • $\begingroup$ @DavidWheeler That sounds more like graph theory than topology. $\endgroup$ – augurar May 20 '15 at 4:49
  • $\begingroup$ Its not essential, insofar as there's an applied mathematician at my university who doesn't know any general topology (he's an expert in complex analysis, partial differential equations etc.) However, I tried to sit through one of his courses and just couldn't do it, it was so non-rigorous and logically just full of holes. I really do recommend doing some pure mathematics, if only so that you have the tools to teach the material properly! $\endgroup$ – goblin May 20 '15 at 4:57
  • $\begingroup$ en.wikipedia.org/wiki/Topological_fluid_dynamics ;). Any tool can be useful in the right hands. $\endgroup$ – N. G. M. May 20 '15 at 6:47
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For example in data analysis one could use topological approach. And they actually do. Look the cloud of data (very big data) may have some topological characteristics: cycles, wholes, components, and so on.

Traditional statistical tools are not robust enough to deal with certain high dimensional and noisy data. We need additional methods that we can use to modify and preprocess the data to a form which is more suitable for statistical tools. One very promising direction is to use topology. It was Gunnar Carlsson and Herbert Edelsbrunner who first realize the potential of topological tools to obtain new qualitative information about large dimensional data sets. For example the topological data analysis was essential to identify a subgroup of breast cancers with a unique mutational profile and excellent survival, information invisible to classical methods.

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