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What are the recent and new applications of Mathematics in other Sciences ?

Let me try to be more precise about the question:

  • By "recent" I mean the last 15 years.
  • By "new" I want to exclude the standard answers like cryptography or finance
  • By "applications" I mean a mathematical concept (or even a trick) successfully used in another field (preferably not Theoretical Physics) to solve a problem or to shed a new light on a phenomenon.
  • I would prefer to see recent applications of modern Mathematics, but new applications of classical results should be considered as a valid answer too.
  • the answer should not simply be "XXX was successfully applied to YYY". it should contain a short explanation of the mathematical concept involve, and a description of the problem/phenomenon it solved/enlightened.

The typical example I have in mind does not strictly answer the question (first it is an application of Theoretical Physics to condensed matter, and then I am not giving the required details): it seems that some 2d quantum field theories which had a priori no physical meaning were successfully used to understand properties of graphene (I think this was really unexpected: 2d conformal field theory was considered as a toy model to approach the understanding of more relevant field theories in higher dimension).


v2: I hope this version is better than the previous one.

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From his web page, the OP knows a good deal more about this than the people he is asking. –  Will Jagy Apr 24 '11 at 22:19
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I cast a vote against closing - although I am instinctively not keen on the question, it seems a reasonable one with decent criteria for what would constitute a good answer. (It would help to give some idea of what counts as "recent", though) –  Yemon Choi Apr 24 '11 at 22:32
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@Ryan, the example Damien mentions is actually a very dramatic and unusual case; not just what you might typically see picking up a random physics or chemistry journal. –  Scott Morrison Apr 24 '11 at 22:41
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I also vote against closing (even though somehow the current phrasing was initially off-puting). Remember the new procedure; if you want to vote to close, you instead need to write a comment cancelling one of Yemon's, André's or my "anti-votes". –  Scott Morrison Apr 24 '11 at 22:43
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The question is undoubtedly interesting, but too vague, and I don't think that the OP has made the necessary effort to clarify the question. Usually, vague questions do not generate very good answers. –  Thierry Zell Apr 24 '11 at 23:16
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12 Answers

up vote 11 down vote accepted

(Disclaimer: I am writing based on what I remember from one seminar that I heard given by Alain goriely, and so claim responsibilities for all inaccuracies!)

Basic differential geometry has been applied to the problem of protein folding and dynamics of protein in an interesting way (and this is fairly new work from what I gathered). Here I give one simple example:

A little bit of remedial biology in case we forgot: a protein molecule is formed by one or more polypeptide chains. Each chain is made of building blocks called amino acids joined together by peptide bonds. As its name suggests, a polypeptide chain is just a long string of amino acids chained end to end. What determines the shape of protein molecules is the individual amino acids. Roughly speaking, each building block (amino acid) is formed by a backbone (something common to all amino acids), together with one or more things than hangs off the backbone. The backbone gives the initial chain like structure of the polypeptide chain. The interaction between the things hanging off the backbone, and between the things and the surrounding environment, is what drives the dynamical folding of the protein giving its final shape.

For traditional protein dynamics, or for traditional storage of protein structures, what they do is they take the numerically computed (or experimentally observed) protein structure, and define a map $\pi$. The map $\pi(n)$ roughly gives the (relative) spatial position of the $n$th amino acid in the chain. Separately there is also a map $\nu(n)$ which gives the orientation of the amino acid, and what is hanging off the backbone there.

For some less precise dynamical computations, going through the whole list of all positions can be computationally intensive (a protein can have upwards of tens or hundreds of thousands of amino acids), without being particularly accurate. On the other hand, the basic larger-scale structure of polypeptide chains are fairly well known (classical in the biology literature), and includes things like alpha-helices, beta-pleats, and turns. These three most common structures are all well-approximated by constant torsion and constant curvature space-curves. Therefore, a computationally less demanding way of storing the approximate structure of the protein backbone would be to decompose the folded structure into its "secondary structures", approximate each of those by these space-curves (each can be parametrized completely by the torsion, curvature, total length, starting position, and starting direction). This allows better memory use and faster computations for certain numerical simulations of protein dynamics.

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This is a remarkable breakthrough and I wish I could find some good review articles on the state of the art.The current research literature is very disorganized and difficult to ferret out by the usual search engine means.Any suggestions? –  Andrew L Apr 25 '11 at 3:11
    
UPDATE:I DID find this article at the archive-but as far as I know,it's the only one I could find so far:arxiv.org/PS_cache/arxiv/pdf/0809/0809.2079v1.pdf –  Andrew L Apr 25 '11 at 3:22
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FWIW I believe linking-to-the-arXiv orthodoxy is to link to the main page of the article and not the PS_cache copy of a particular version of it. arxiv.org/abs/0809.2079 –  anon Apr 25 '11 at 7:49
    
Unfortunately, my shallow knowledge on the subject all came from attending one seminar. You can try e-mailing professor Goriely if you want to learn more. –  Willie Wong Apr 25 '11 at 11:41
    
Thanks a lot for this very nice answer. –  DamienC Apr 26 '11 at 7:25
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The new area of discrete differential geometry is solving problems in computer graphics, such as creating more lifelike hair in animation. I became aware of this from an article in the New York Times (see http://www.nytimes.com/2010/12/30/movies/30animate.html). An excerpt: "... the images on screen are not the result of a patchwork of technical tricks, but of precise mathematical equations based on the way the world actually looks and operates — in a word, physics. They use what is known as discrete differential geometry, a field so new that the first textbook on the subject was published only two years ago. [...] The uses of discrete differential geometry go far beyond animation. Johns Hopkins Medical Center, for instance, is using Mr. Grinspun’s computer simulations to predict how needles move through human flesh, so that doctors can train to do laparoscopic surgery on virtual bodies instead of the real thing."

Here is a link to a book: http://www.amazon.com/Discrete-Differential-Geometry-Graduate-Mathematics/dp/0821847007.

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http://en.wikipedia.org/wiki/Compressed_sensing

The AMS article is pretty good: http://www.ams.org/happening-series/hap7-pixel.pdf

Basically, some new math results show how to combine a lot of 1-pixel sensor readings into a complete picture of something that would normally be done with a sensor array in a camera.

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Agreed - the AMS article is indeed VERY good. –  DamienC May 4 '11 at 10:23
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Space-time codes (or why your router has two antennas): http://en.wikipedia.org/wiki/Space_time_code

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Thanks a lot for your answer - it would be nice to explain a bit more what are the math involved. –  DamienC May 4 '11 at 10:20
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How about applications of discrete complex analysis to statistical physics? There was a surge of work this past decade on the subject, such as proofs of conformal invariance of 2-D models (Ising, Potts, Spinglass, O(n),etc.). Before, there were mainly unrigorous physics arguments to prove the various facts involved, such as the value of the Honeycomb Constant. The machinery of SLE and discrete complex analysis has been extremely insightful in the proofs involved. Much of the methodology is based on the foundational work done by Onsager and Baxter decades before.

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There are applications of homology theory both to Topological Data Analysis and other parts of Applied Topology (work of Gunnar Carlson and his coworkers in Stanford), via the CHOMP project to the structure of materials, dynamics of Evolution of Pattern Complexity during Phase Separation, etc. (The Stanford webpage mentions many more applications and they really merit a mention but I leave the 'reader' the joy of browsing around the links there.) There is also work by Robert Ghrist again on Applied Topology.

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The work of Ghrist might deserve its own separate answer, as some of it seems to tackle very different types of applications. –  Thierry Zell Apr 25 '11 at 15:46
    
You are probably right, so why not add a summary as a separate answer. –  Tim Porter Apr 26 '11 at 7:03
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(From SIAM web site.) The behavior of a dynamical system is determined by its "skeleton," which consists of the different attractors (steady states, periodic solutions, or more complicated sets), as well as saddle-type objects with their global stable and unstable manifolds. Global manifolds are complicated objects that must be found numerically. They are hypersurfaces consisting of infinitely many trajectories that end up (or come from) a saddle-type object. All other trajectories qualitatively follow the dynamics given by the "nearest" global manifolds. This feature was recently utilized in the Genesis Mission, which sent a spacecraft to a saddle-type periodic orbit around the Lagrange point between the earth and the sun to collect solar dust particles. The spacecraft traveled on global manifolds to its destination and back to earth virtually without any fuel.

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Applications of non-Archimedean analysis (a p-adic analog of the symmetric stable process and the corresponding heat-like equation) to dynamics of complex systems like proteins or spin glasses. See http://www.worldscinet.com/brl/03/0303/S1793048008000836.html

or http://iopscience.iop.org/0305-4470/35/2/301

Applications of fractional calculus (a branch of classical analysis dealing with fractional derivatives and integrals) to diffusion phenomena in disordered systems. See a survey paper with an interesting title "The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics":

http://iopscience.iop.org/0305-4470/37/31/R01

Mathematically, this is (in the simplest case) a kind of a theory of parabolic equations with a fractional time derivative.

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Statistics applied to microarray data in biology. And Bernd Sturmfels and his students have been applying algebraic geometry to this. He wrote a book titled Algebraic statistics for computational biology. Biology is a field that will explode in coming decades. Advances in that field will probably capture the public imagination the way physics did in the 20th century. The next Einstein could be a biologist.

(Does any of the empirical data used in any research papers published in that field have any validity? That's something that could bear inspection. People put vast numbers of large data sets on the internet, and others base research papers on them. But before they do that they transform raw numbers by, for example, raising everything to the power 3/2. If you later ask them specifically how they transformed it, they may be blindsided by the question and think you're making a strange unusual request that they'd never have expected in a million years. People don't normally ask such weird things. At least such has been my impression.)

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Gröbner bases give a nice way to solve many problems in computer vision and robotics. Often these problems involve solving systems of polynomial equations eg. those describing the projection of a point in 3D on the 2D plane of a camera sensor or those describing how the position of a robot component depends on the details of a sequence of articulated joints. Here's a paper that uses such methods to investigate the minimum amount of information to infer properties of cameras, among other things.

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Pseudodifferential (PDO) and Fourier integral operators (FIO) and also Wick and antiWick operators have applications in signal processing, especially in analysis of non stationnary signals like speech signals. Examples of such time frequency representation are spectrograms, scalograms which use wavelet theory and similar tools like Gaborets, chirplets, ridglets,...etc. This tools are also used to solve PDEquations as they serve to almost diagonalization of a large class of PDO and FIO.

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