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This was originally a question posted here on MathSE. But I'll ask again here to see if I can get some different answers.

I'm looking for current areas of research which apply techniques from differential geometry to biological processes. I'm (scantly) aware of a handful of applications to cell science and microbiology, but I've heard almost nothing of differential geometric methods in say ecology or evolution. Any sort of reference (textbook, paper, researcher, etc) would be appreciated.

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Not exactly what you asked, but I would like to leave it here nevertheless, since it is a very beautiful application of differential geometry in cell biology.

The Wilmore energy describes physically the bending energy of a compact and oriented surface embedded (or even immersed) into $\mathbb R^3$. This is used in biology (aka Canham-Helfrich energy) to explain the different shapes of blood cells, since cells are trying to minimize their bending energy. See also this talk.

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Infinite-dimensional geometry in shape analysis, which has many applications across biology and medicine. I would suggest to look at this list of examples to get started.

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I have seen some applications of differential geometry in model/parameter identification (see this method paper). From a quick search on Google Scholar, I only found a handful of papers on this topic. Usually, differential algebra and computational means (MCMC) are the preferred techniques, but I think it is due to a lack of development of this application. I have only recently started looking into this application, but the issue of model/parameter identification is directly related to personalized medicine, ecological models, and evolutionary models, etc.(see this review paper).

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Bill Allard described a method to compute the shape of the cornea from keratometer measurements in

W. Allard, The reconstruction of surfaces in $\mathbb{R}^3$ by reflection. J. Geom. Anal. 8 (1998), no. 5, 709–722.

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