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.
[Math] differential geometry applied to Biology
applicationsbiologydifferential-geometryreference-request
Related Solutions
If you find the question: "How many groups are there of order 35?" motivating, why don't you find the question: "How many differentiable manifolds of dimension 2 are there?", motivating as well?
There are millions of applications of manifolds in pure mathematics. Lie groups (continuous symmetries) are a beautiful example.
As an aside, your question got downvoted (not by me), because the tone is somewhat arrogant. For example your statement:
"It's a language to talk about differential equations, which are "naturally interesting". But honestly, I don't think I care about applications to differential equations, knowing that what it comes down to in the end is that the equations are rammed into a computer and brutally 'solved' by numerical methods anyway, no matter how fancy you formulate them."
is an incredibly ill informed view of the subject. The theory of differential equations is extremely rich (both from a pure and applied viewpoint).
Disclaimer: I hope it is OK to post "late" answers. The original question is quite vague, therefore, my answer will be quite general and lengthy.
General answer: There are so many books on Mathematical Biology now, that it is very hard to recommend just one book. The two volume book by James Murray, recommended in one of the answers, is not an exception, because it is primarily built on the works by Murray himself and his co-workers, and is not a textbook per se, however, it gives a very nice introduction to the mathematical models in biology, described by ODEs and PDEs. This is especially true for the population dynamics models, chemical kinetics and pattern formation.
Some people approach Mathematical Biology having originally biological background, and for them there are numerous textbooks, which provide basics of the required math. This site is devoted to mathematicians and people studying math, therefore I would like, in my answer, to point out mathematical books, mathematics of which was generated by biologically motivated problems.
Specifics:
Your area of expertise is Dynamical systems theory generally, ODEs in particular, including Stability theory. A recent book I would recommend is Dynamical Systems and Population Persistence by Hal Smith and Horst Thieme.
You are interested in Game theoretic models, including the analysis of ODEs. The point of entry, which also includes some classical works in mathematical genetics, is Evolutionary Games and Population Dynamics by Josef Hofbauer and Karl Sigmund. For people having necessary background a paper by the same authors is much shorter way to start research.
Your want to apply your knowledge of probability theory, especially of stochastic processes (SDEs and diffusion processes), to the real-world problems in mathematical genetics. This is an extremely mathematical area of research, good starting points are Probability Models for DNA Sequence Evolution and a now classic by Warren J. Ewens Mathematical Population Genetics: I. Theoretical Introduction.
Deterministic models of theoretical genetics are treated at length in a very mathematically involved book by Burger The Mathematical Theory of Selection, Recombination, and Mutation. The models include interesting integral equations of Quantitative Genetics with a lot of open mathematical problems.
(Might be close to the original question) There are a lot of mathematical models that study PDEs arising as a limit of some stochastic spatial processes for interacting particles, including populations of cells. This is a huge area of research, but one book, which may be interested to a mathematically mature person is Transport Equations in Biology by Benoît Perthame.
A huge number of models in Mathematical Biology (especially in Mathematical Epidemiology) deal with Random Graphs dynamics. A good book to start is Random Graphs Dynamics by Rick Durrett.
If you are an expert in optimization problems, Monte Carlo simulations, etc, then it would be interesting to look into Inferring Phylogenies by Joseph Felsenstein, or in the mathematical problems of the protein folding, which was also recommended in one of the answers. There is no unique mathematical book, which I consider a good introduction (one can always start with Physical Approaches to Biological Evolution by Mikhail V. Volkenstein and A. Beknazarov).
The list can be made much longer, but in my opinion, the mentioned books are a good start for a mathematically prepared person.
Edit:
- And, of course, for pure mathematicians, it may be very interesting to read the ideas, review papers, and an introduction to biology by Misha Gromov.
Best Answer
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:
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:
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):
Shahshahani, A new mathematical framework for the study of linkage and selection, 1979
Harper, Information Geomtry and Evolutionary Game Theory, 2009