differential geometry applied to Biology 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.
 A: As you are aware, it is somewhat common to characterize molecular/cellular biology (e.g., modelling cell membranes, protein surfaces, organelle geometry, DNA folding, etc...) or physiology (e.g., here) with differential geometry, since one is describing a physically curved surface (i.e., the geometry is "real").
However, (differential) geometry in ecology and/or evolution is necessarily more abstract, since the models tend to be either probabilistic or based on differential equations (or both) of abstract variables. One issue is that not only would a geometric model need to be a good descriptor of the problem, but it would also have to be useful (e.g., provide some intuition/insight that the other description cannot). For instance, diffusion models are not uncommon in ecology, and they can be transformed into "simpler" diffusions on a Riemannian manifold, where the metric tensor is closely related to the diffusivity tensor of the Fokker-Planck equation (or diffusion coefficient of the equivalent Ito process it describes). However, I guess this kind of approach simply doesn't yield enough additional insight for it to be common. Maybe someone who knows the field better than I do can comment on why.
Nevertheless, I've seen a couple applications:


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*Work from Peter Antonelli applies differential geometric methods to ecology and evolution, and has quite a few papers in the area. He seems to heavily utilize Finsler manifolds and stochastic processes. For instance:


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*Antonelli et al, Gradient-Driven Dynamics on Finsler Manifolds: The Jacobi
Action-Metric Theorem and an Application in Ecology, 2014

*Antonelli and Zastawniak, Fundamentals of Finslerian diffusion with applications, 2012


*There is work on evolutionary modelling using differential geometry, by "smoothing" the discrete space of alleles (modelling evolution as continuous). There are also interesting links to information geometry (where one views a space of probability distributions as a Riemannian manifold) in this case. Check out these (and their citers & citations):


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*Shahshahani, A new mathematical framework for the study of linkage and selection, 1979

*Harper, Information Geomtry and Evolutionary Game Theory, 2009
A: The 32 citations in this paper may help:

Wei, Guo-Wei. "Mathematics at the eve of a historic transition in biology." Computational and Mathematical Biophysics 5, no. 1 (2017).
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