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To get to the point I'm looking for a book on Topology that covers specifically its uses in the medical field.

I've seen a lot of book requests in Topology, but they are all about learning topology or engineering based applications of topology, which are great topics, but not what I'm looking for.

I would love a book that specifically deals with Cancer, Medicine, Viruses, and/or Bacteria. Really anything that has a medical aspect to it. I've taken one topology course so even if there is no book that has what I'm looking for if you could offer a book that covers concepts/theorems that are used in applications in the medical field that would be appreciated too.

My professor recommended me Topology Now and I believe Topology and it's Applications, but they don't seem to cover anything medical related. I have not got my hands on a physical copy of either book so I'm just basing that previous sentence on synopses that I've read online.

Thanks for any thoughts, ideas, or recommendations you may have to offer! I'm also heading to the library later today to look, so I might have some specific titles to ask for recommendations of too. Thanks.

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    $\begingroup$ Do you have an example of general topology applied to the topics you list? $\endgroup$ – parsiad Apr 14 '16 at 16:38
  • $\begingroup$ Why do you think such applications would exist? The best I can come up with is knot theory, but that subject can get very technical and requires a nontrivial background. What do you have in mind? $\endgroup$ – anomaly Apr 14 '16 at 16:43
  • $\begingroup$ What spurred this conquest is the MAA conference in St. Joesph Missouri that I recently attended. One lecturer spoke on graphs imbedded in R3. She discussed, very briefly, on knot theoyry used in Chemistry and the idea of DNA and different materials being tied in knots. Which got me thinking are there any topological concepts that can be used in the medical field. $\endgroup$ – Fmonkey2001 Apr 14 '16 at 16:44
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    $\begingroup$ Also, I've been reading a little on Topological Data Analysis which seems to be being used in cancer research. systemsbiology.columbia.edu/… $\endgroup$ – Fmonkey2001 Apr 14 '16 at 16:46
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    $\begingroup$ There are, of course, many topology books on surgery theory :-P $\endgroup$ – Pedro Apr 14 '16 at 18:59
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If you've already studied some topology, you might consider Computational Topology: An Introduction by Edelsbrunner & Harer. It will help with understanding topological data analysis and has a few biological applications in its final chapter. See also Simplicial Models and Topological Inference in Biological Systems by Nanda & Sazdanovic. Here is the abstract from the latter:

This article is a user’s guide to algebraic topological methods for data analysis with a particular focus on applications to datasets arising in experimental biology. We begin with the combinatorics and geometry of simplicial complexes and outline the standard techniques for imposing filtered simplicial structures on a general class of datasets. From these structures, one computes topological statistics of the original data via the algebraic theory of (persistent) homology. These statistics are shown to be computable and robust measures of the shape underlying a dataset. Finally, we showcase some appealing instances of topology-driven inference in biological settings – from the detection of a new type of breast cancer to the analysis of various neural structures.

For applications to neuroscience, see Two’s company, three (or more) is a simplex: Algebraic-topological tools for understanding higher-order structure in neural data by Giusti, Ghrist and Bassett.

For the basics of knot theory, try either The Knot Book by Adams or the brief introduction in Chapter 12 of Introduction to Topology: Pure and Applied by Adams & Franzosa.

A useful resource is http://appliedtopology.org/. In fact, the latest entry at the time of writing is Funding opportunities for interdisciplinary research from the Center for Topology of Cancer Evolution and Heterogeneity.

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