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Partial differential equations have been used to establish fundamental results in mathematics such as the uniformization theorem, Hodge-deRham theory, the Nash embedding theorem, the Calabi-Yau theorem, the positive mass theorem, the Yamabe theorem, Donaldson's theory of smooth 4-manifolds, nonlinear stability of the Minkowski space-time, the Riemannian Penrose inequality, the Poincaré conjecture in 3D, and the differentiable sphere theorem. These examples all come from geometry and topology, and I was trying to find similar examples in other branches of mathematics without luck. I can sort of imagine why geometry and topology maybe amenable to PDE but this does not mean PDE cannot find applications in other branches. I asked probabilists and was told that most of the examples they think of seem to be the other way around, i.e., using probability theory to say something about PDE. Can you provide an example, or give a reason why such examples must be confined to geometry and topology.

The reason I am asking this question is that majority of "pure math" students don't seem to like PDE courses, thinking it as an "applied" subject so it has nothing to do with them. My impression is that for instance students in algebraic or differential geometry somehow get their "own version" of PDE theory from specialized books in their subject, specifically tailored for the problem at hand. It would be much easier and methodical if the student had taken a general PDE course before. So I thought this kind of list maybe helpful in convincing the beginning student to take PDE classes. As the list stands now, we have enough for geometry/topology and perhaps mathematical physics students, but it would be great for instance to have something for probability, number theory, analysis, and algebra students.

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Does "geometry" include algebraic geometry? My understanding is that there are applications there (see… ). – Qiaochu Yuan Aug 31 '11 at 17:48
Yes it includes. Thanks for the link! – timur Aug 31 '11 at 17:55
Community Wiki? – Willie Wong Aug 31 '11 at 19:57
See the MO question – Denis Serre Sep 1 '11 at 5:35
An interesting game for when people who are bored: much like the Erdos numbers, you can play PDE numbers for the AMS MSC numbers. A connetion between two MSC numbers is given by a paper that has both. The goal is to find the minimum number of papers required to link a given MSC code to 35XXX. – Willie Wong Sep 1 '11 at 14:18

As alluded-to by Qiaochu Y. above, and as I can personally attest, PDE arise in the modern theory of automorphic forms. Superficially/historically, this might be viewed as a formal generalization of "holomorphic" to "eigenfunction for Laplace-Beltrami operator". Indeed, already c. 1947, Maass showed that real quadratic fields' grossencharacter L-functions arose as Mellin transforms of "waveforms", Laplace-Beltrami eigenfunctions on $\Gamma\backslash H$, a complementary result to his advisor Hecke's result that $L$-functions for complex quadratic extensions of $\mathbb Q$ arose from holomorphic modular forms.

The spectral theory of automorphic forms, from Avakumovic, Roelcke, and Selberg c. 1956, in effect decomposes $L^2(\Gamma\backslash H)$ with respect to the invariant Laplacian, descended from the Casimir operator on the group $SL_2(\mathbb R)$, which (anticipating theorems of Harish-Chandra) almost exactly corresponds to decomposition into irreducible unitary representations.

The Selberg trace formula, and Langlands' and Arthur's, as well as Jacquet's "relative" trace formula, do afford an interpretation as spectral decompositions of various integral operators, rather than differential operators. Nevertheless, or "however", some aspects of the situation that are clumsy, because of their "extreme" features, but interesting for applications for the same reason, from that viewpoint are amenable to thinking about solutions of (invariant) inhomogeneous PDEs with distributional "targets". A typical scenario is a "Helmholtz" equation (a wave equation Fourier-transformed in the time parameter), $(\Delta-\lambda)u=f$. Among other cases of interest, the case that $f$ is an (automorphic) delta is very useful in various number-theoretic applications, such as proving "subconvex" bounds: Anton Good sketched this application already in 1983 (and Diaconu and I treated $GL_2$ over number fields recently... implicitly using this idea, although reference to classical special functions gave a shorter argument for the official version).

Philosophizing a bit, such experiences, and continuing ones of a related sort, indicate to me that geometrically meaningful, that is, group-invariant, "PDE" are a natural/obvious extension of "calculus"... so that, in particular, their natural solutions in Sobolev spaces (etc) are "natural objects", whether or not they are classical special functions, or entirely elementary.

(One can't help but note that there is an understandable, if unfortunate, human tendency to declare and understand "turf", so that one chooses one's own, and stays away from others'. Similarly, "experts" on subject X do not favor outsiders' appropriating bits of it "for applications", as though anything other than a life-long dedication could penetrate the mysteries... One may read about medieval European "guilds" and their protection of their "secrets".)

As a methodological philosophizing: my own experience tells me that means of description are useful. That is, structural, meaningful characterization of objects is good. Saying that something is a solution of a natural (group-invariant?...) PDE is a strong, meaningful constraint. Ergo, helpful/good.

The small rant at the end: the usual style of seemingly-turf-respecting narrowness is not so good for genuine progress, nor even for individual understanding.

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Monge-Amp`ere equations appear not only in geometry, but also in economics (though I cannot comment on their importance in that area due to lack of my education in economics), namely in the so called Monge-Kantorovich problem. By the way, Leonid Kantorovich was a mathematician and economist who received a Nobel prize in economics.

The problem is as follows. Let $\mu_1,\mu_2$ be two probability measures in $\mathbb{R}^n$. We are looking for a measurable map $f\colon \mathbb{R}^n\rightarrow\mathbb{R}^n$ such that $f_*(\mu_1)=\mu_2$ (where $f_*$ is the usual push-forward on measures), and $f$ minimizes certain cost functional.

Brenier has shown existence of such a map (called now the Brenier map) under appropriate conditions on the measures and the cost functional; he reduced the problem to solvability of certain Monge-Amp`ere equation. Other people proved some regularity of the solution.

The Brenier map was applied further by F. Barthe in a completely different area: to prove a new functional inequality called the inverse Brascamp-Lieb inequality (see "On a reverse form of the Brascamp-Lieb inequality", Invent. Math. 134 (1998), no. 2, 335–-361). He also obtained with his method a new proof of the known Brascamp-Lieb inequality. Moreover in the same paper, Barthe deduced from his functional inequality a new isoperimetric property of simplex and parallelotop: simplex is the ONLY convex body with minimal volume ratio, while parallelotope is the ONLY centrally symmetric convex body with minimal volume ratio. (Previously K. Ball has shown these minimality properties of simplex and parallelotop without proving the uniqueness, using a different technique.) Remind that volume ratio of a convex body is, by definition, the ratio of its volume to the volume of ellipsoid of maximal volume contained in it.

Later on other authors applied the Brenier map to obtain sharp constants in some other functional inequalities.

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Parabolic PDE and their generalizations provide analytic constructions of Markov processes. See, for example, K. Taira, Semigroups, Boundary Value Problems and Markov Processes, Springer, 2004, or N. Jacob, Pseudo-differential operators and Markov processes, Vols 1-3, Imperial College Press, London, 2001-2005.

Some elliptic equations appear in non-commutative harmonic analysis and representation theory (with further number-theoretic applications). See S. Lang. $SL_2(\mathbb R)$, Addison-Wesley, 1975.

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I'm assuming that non-mathematical subjects, like physics, don't count --- there the heat, wave, Schrödinger, KdV, water wave equation, Navier-Stokes, Helmholtz, ..., equations are all fairly important objects. In fact most of the PDE I could name would be related to physics in some way. I would say that most PDE are in this direction.

In some sense, the entire field of complex analysis comes down to genuinely understanding solutions to one PDE; complex analysis, I think you'd agree, is a pretty big field, with plenty of applications of its own.

A number of tools have been produced by PDE which are of universal appeal in analysis. For example, the Fourier transform, which has a broad range of applications in analysis, not to mention generalizations, e.g. the Gelfand map, was developed as a tool to solve the wave equation. Another is the convolution (which I'm assuming is also from PDE) and along with it a variety of dense functions, nice partitions of unity, and so on, along with notions of convergence which are also very useful in a variety of contexts. Things like the Poisson kernel and the Hilbert transform have become prototypical examples in integral operators.

PDE in general are rather hard, and so any particular PDE is likely to be rather narrow in scope. So many of the things of greatest interest to come out of it are tools to solve problems rather than necessarily specific solutions.

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A professor here at Cornell gave a talk several years ago where he began by briefly estimating that the majority of mathematics (as measured by number of articles) of the last 50-60 years relates to differential equations (he includes most of applied math, numerical analysis, much of analysis itself, and so on). I think by this measure, differential equations represents the greatest source of interesting problems in all of mathematics. It is probably rather difficult to find an area of mathematics which does not relate in some (potentially very loose) way to or use ideas from PDE. – Peter Luthy Sep 1 '11 at 1:12
Solving PDE is rather tricky business; even some of the simplest ones are so rich! The Cauchy-Riemann equations are innocent looking yet produce an incredible theory. Since there is much interest in solving these problems from an applied standpoint, people work hard and come up with ingenious ways to solve them, and in doing so produce a host of great abstract problems which don't relate that closely to the original problem anymore. – Peter Luthy Sep 1 '11 at 1:19

Klainerman writes in PDE as a Unified Subject

...the range of applications of specific PDE's is phenomenal, many of our basic equations being in fact at the heart of fully fledged fields of Mathematics or Physics such as Complex Analysis, Several Complex Variables, Minimal Surfaces, Harmonic Maps, Connections on Principal Bundles, Kahlerian and Einstein Geometry, Geometric Flows, Hydrodynamics, Elasticity, General Relativity, Electrodynamics, Nonrelativistic Quantum Mechanics, etc. Other important subjects of Mathematics, such as Harmonic Analysis, Probability Theory and various areas of Mathematical Physics are intimately tied to elliptic, parabolic, hyperbolic or Schrodinger type equations. Specific geometric equations such as Laplace-Beltrami and Dirac operators on manifolds, Hodge systems, Pseudoholomorphic curves, Yang-Mills and recently Seiberg-Witten, have proved to be extraordinarily useful in Topology and Symplectic Geometry. The theory of Integrable systems has turned out to have deep applications in Algebraic Geometry; the spectral theory Laplace-Beltrami operators as well as the scattering theory for wave equations are intimately tied to the study of automorphic forms in Number Theory. (p. 2)

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Much as I like seeing Klainerman quoted, once you remove geometry, topology, and obvious mathematical physics from that list, you are left with: harmonic analysis, probability theory, "integrable systems applied to algebraic geometry", and applications to automorphic forms. The last one is explained by Paul Garrett… . The first three are rather vague. – Willie Wong Sep 1 '11 at 14:11
And unfortunately, in that paper/talk, those items were not particularly explained. Because the goal was to show that PDE is a unified subject (as opposed to a bunch of people studying their own favourite equations), rather than showing that PDE is a subject that "unifies mathematics". – Willie Wong Sep 1 '11 at 14:12
Fair enough, but at least Klainerman's "...intimately tied to elliptic, parabolic, hyperbolic or Schrodinger type equations" prods us to look further, perhaps to 'Elliptic PDE's in probability and geometry'… – David Corfield Sep 1 '11 at 15:03

In principle a big set of functiontheory (i.e. the theory of complex differentiable functions) is the application of the results for harmonic functions:

  • mean value property
  • analyticity
  • Liouville's theorem
  • maximums principle
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Not sure if this counts, but aren't there a whole bunch of neat results in combinatorics on discrete analogs of PDE? I'm thinking, for example, of Stone's theorem for the discrete Schrodinger equation; the characterization of graph spectra; and how some solutions of the discrete Laplacian/discrete Helmholtz/discrete modified Helmholtz lead to special instances of ADE Dynkin diagrams.

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Probability: PDEs are all over the place in problems related to optimal filtering problems. For example, the Kushner-Stratanovich equation of nonlinear filtering. Several of the optimal filtering type results are approached by forming a cost functional, and reducing the problem to a Euler-Lagrange PDE.

My impression is that the Euler-Lagrange PDE is pervasive in lots of areas of math, although my exposure is mostly with Hamiltonian dynamics, dynamical systems and estimation theory.

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