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(This question might not be appropriate for this site. If so, I apologize in advance. I would have posted to mathstack, but I'm looking for advice from active researchers.)

I am an undergrad in math and physics getting ready to apply to grad school. So far, I know that I enjoy PDEs (and most kinds of analysis, generally), mathematical physics, and especially dynamical systems. But this is way too broad for choosing a Ph.D. program. My background in these subjects is somewhere between the undergraduate and graduate level, but certainly not 'up to date' or research-level.

I want to get a clearer picture of these fields as they exist today. If I'm going to try to contribute to these topics in the next 5-10 years, I'd like to know what I'll be getting into.

If you work in these fields, or you have colleagues who do, how would you describe the current state of affairs?

Specifically:

  • What are people trying to accomplish?
  • How do those things fit into a larger picture?
  • What are the obstructions?
  • What are some of the major recent advances?
  • What would you like to see happen over the next few decades?

I'm looking for technical descriptions, preferably with references to actual papers. Pretend you were describing your work to a fellow mathematician from an entirely different specialty. Don't worry if it's over my head (I'm sure it will be); the point is to get a taste, to help narrow my interests, and to have a guide to come back to over the years.

Thanks!

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This is a good question, but should be made Community Wiki. I see that there is already 1 vote to close. To users with closing power: I ask that you keep the question open for at least a few days to collect a few good answers for @AJGibson. If having a big list is annoying at that point, then we can close it. –  Tom LaGatta Mar 20 '13 at 10:01
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For PDEs, I'd recommend Klainerman's exposition for the Princeton Companion to Mathematics. An extended version may be found on his homepage: web.math.princeton.edu/~seri/homepage/papers/gws-2006-3.pdf –  Timothy Chow Mar 21 '13 at 15:17
    
I also think this is a very good question, and is suitable for this website, but I would also vote to make it a CW. –  William Mar 23 '13 at 0:33
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10 Answers 10

I am afraid that the task you describe as "obtaining a clear picture" of PDE, Mathematical physics and dynamical systems is impossible. I doubt that there are people who have a "clear picture" of all these three fields. Each of them is enormous. My advise is to find an adviser who works in the broad area of analysis, and to rely on his directions for the study of some specific area. Of course you can read about the broader area, and perhaps after 40 years of work and reading you will obtain some general picture:-)

For a general introduction to dynamical systems, I recommend the book of Katok and Hasselblatt, Introduction to modern theory of dynamical systems, or another book of the same authors, A first course in dynamics, with a panorama of recent developments.

I don't think there exists a modern survey of the whole mathematical physics. But Reed and Simon 4 volume course, Methods of Mathematical Physics gives an introduction to some parts of it.

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I did my graduate work in PDEs, and I never felt I had an overview of what was known and what was open. Neither did any of the PDE researchers I talked to. So I asked them "How do you know whether what you're working on is new?" Basically they had no answer. In retrospect I'd say if you work on something obscure and unimportant, it stands a good chance of being original, or at least publishable, which of course is a weaker criterion. –  John D. Cook Mar 28 '13 at 2:30
    
John, what you are saying looks very plausible. Even if we consider only LINEAR PDE, where there is a 4-volume Hormander book, there is still an enormous amount of research on the topics not covered in this book. I doubt that a comprehensive survey of such an area as Linear PDE covering all important topics is possible. –  Alexandre Eremenko Mar 28 '13 at 12:43
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There is lots of work going on in mathematical relativity (and more generally in non-linear hyperbolic PDEs) on trying to establish global non-linear stability around interesting exact solutions. The case of Minkowski space was treated in the seminal work of Christodoulou and Klainerman. The main challenge at present is tackling the Kerr family of black hole solutions. The solution of the non-linear problem is not yet in sight, but several linear problems are being studied as toy models.

Just to clarify, the problem of global non-linear stability consists of showing that a small perturbation of the initial data for the background solution of interest produces an exact solution that is both qualitatively and quantitatively close to the background solution everywhere and for all time.

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What Igor describes is one facet of a general theme in evolutionary PDEs nowadays. A lot of energy is now focussed on PDEs modelling attractive self-interaction (such as gravity). For these "focussing" type equations, it makes sense to ask (a) do there exist stationary solutions, and if so, are they unique? (b) are these stationary e solutions stable? (c) do these stationary solutions describe asymptotic behaviour? In the context of relativity, (a) is the "black hole uniqueness theorems", (b) is "stability of Kerr", and (c) is "final state conjecture". –  Willie Wong Mar 20 '13 at 15:41
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In the context of, say, nonlinear waves with power nonlinearities, (a) is the characterisation of ground states (b) is stability of ground states (see e.g. Nakanishi and Schlag's monograph Invariant Manifolds and ...; Walter Strauss's review of which is great for your purpose ams.org/journals/bull/2013-50-02/S0273-0979-2012-01389-7/… ) and (c) is the "soliton resolution conjecture". (Incidentally, for equations which do not admit bound states/stationary solutions, the analogue of "soliton resolution conjecture" is just the "global wellposedness/scattering".) –  Willie Wong Mar 20 '13 at 15:48
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Another major research program in this subject (which, for many equations, is actually more or less complete) is the task of establishing wellposedness for subcritical or critical equations (with the latter requiring either a defocusing equation or for the data to be smaller than the ground state in some sense, the latter assertion being known nowadays as the "threshold conjecture"). See e.g. web.math.princeton.edu/~seri/homepage/papers/telaviv.pdf –  Terry Tao Mar 20 '13 at 16:33
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Dynamical systems is a huge field, with at least 3 (or more) subdisciplines which often interact with each other, but also have self-contained advances. Ergodic theory, topological dynamical systems, and smooth (differentiable) dynamical systems.

As far as the topological dynamical systems and hydrodynamics are concerned, a list of 8 unsolved problems for 21st century was presented around 2001 by HK Moffatt of Cambridge. Here's the link to that file: http://www.igf.fuw.edu.pl/KB/HKM/PDF/HKM_128_s.pdf

The gist of the first few questions is as follows: It was established by V. Arnold and others that given a laminar (time-invariant) velocity field in a 3-d fluid volume, the resulting fluid-particle trajectories can show chaotic behavior (e.g. see ABC flow on wikipedia). While this much is known, not much is known about the measure of the phase-space on which such chaotic trajectories exist. Further questions arise when instead of particle trajectories themselves, we are talking about scalars that may or may not be passive (e.g. dyes, temperature).

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For a list of 15 open problems in mathematical physics (in 2000) see Simon's Problems, http://mathworld.wolfram.com/SimonsProblems.html.

Some of these problems are solved, as mentioned in the link.

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I would make this a comment, but it seems to be too long for that. As I have suggested making the question a CW, I would like this answer to be treated as an extensive comment expressing personal opinion.

Great answers have already been provided, so I just want to make it clear that the theory of dynamical systems in general, and what the reader seems to be looking for, are somewhat different things. First, the theory of dynamical systems is incredibly vast (see the book series suggestions by Ian Morris in one of the answers). There are a few reasons for this vastness, the most obvious of which are probably the following.

1) Dynamical systems is a field that draws from many different fields (topology, geometry, complex, real and functional analysis, algebra and number theory, and more). To understand why this is so, consider:

2) A dynamical system, in its broadest sense, is an action of a semigroup or a group on a set. For example, consider $f: \mathbb{R}\rightarrow\mathbb{R}$ given by, say, $f(x) = x/2$. Taking the semigroup $G = \mathbb{Z}_{\geq 0}$ of non-negative integers, we can define an action on $\mathbb{R}$ as follows: $k\in G$ acts on $x$ via $kx = f^k(x)$ where $k$ denotes $k$-fold composition of $f$ with itself. Since $f$ is invertible, we can consider the action of the group $\mathbb{Z}$ on $\mathbb{R}$ via $kx = f^k(x)$, where if $k < 0$, then $f^k$ denotes $k$-fold composition of the inverse of $f$ with itself. The interesting questions begin to appear when the set on which the dynamics is defined (action of a group) carries more than just a set structure; say a topological space, or some kind of an algebraic structure, such as a module or a ring or a field, or a smooth structure, such a smooth manifold, or an analytic structure, such as a complex-analytic manifold, or a measurable. In this case the actions that are most interesting to consider are those that preserve the structure (for example, $f$ above is a homeomorphism, an (analytic) diffeomorphism and an isomorphism when $\mathbb{R}$ is viewed as a topological space with the usual Euclidean topology, an (analytic) manifold, or an algebraic field, respectively. Then one beigns to raise the following questions: what is the structure (in terms of the structure carried by the space) of orbits? For example, are there dense orbits? Periodic ones? How many periodic orbits of a given period are there? How does the number of periodic orbits grow as a function of the period?

The modern theory of dynamical systems can thus be roughly broken into areas according to the "structure" of the space on which the dynamics is considered: topological dynamics, algebraic/arithmetic dynamics, smooth dynamics, holomorphic dynamics and ergodic theory (considered, respectively, on topological spaces, algebraic spaces such as number fields, smooth manifolds, analytic manifolds where most of the time the action is given by a rational--i.e. quotient of polynomials--map, and measured spaces where the action is typically measure-preserving and satisfies a few other technical conditions). Of course, all these may intersect. The modern theory is concerned with the following (related) problems:

(1) Investigation of specific dynamical systems as a source of interesting examples, or looking for examples of certain dynamical phenomena;

(2) Classification: classifying dynamical systems up to a certain quite natural equivalence (here is an example for topological systems: http://en.wikipedia.org/wiki/Topological_conjugacy). This program is huge and the problems here are very difficult;

(3) Related to (2), search for invariants (properties that are preserved by the equivalence from (2)). There are two sets of invariants: complete and incomplete. A complete invariant is a property such that if two dynamical systems have this property, they they are necessarily equivalent, and an incomplete invariant is a property that is shared by two equivalent dynamical systems, but is not a complete invariant.

(4) Study of invariant objects. As an example, let us return back to the example $f$ acting on $\mathbb{R}$ from above. Notice that $f(0) = 0$. This in particular shows that the set $\{0\}$ is invariant under $f$. Also notice that for each $x\in\mathbb{R}$, $f^n(x)\rightarrow 0$ as $n\rightarrow\infty$. In this case the point $0$ (or the set $\{0\}$ is called an attractor. In general, given a dynamical system and a subset of the space on which the system acts which is invariant, one wants to study the structure of this set in terms of the structure of the ambiant space (e.g. the topology of invariant sets in topological dynamics) as well as the dynamics of the dynamical system when restricted to the invariant set. Notice that this also includes dimension theory (that has been mentioned above).

I presume the dynamical systems that you were curious about are those that arise as ordinary and partial differential equations, in which case one is working with an action of a continuous group on some space in the smooth (or analytic) category (manifolds). Certainly the field here is huge (in fact, systematic development of dynamical systems started as study of certain differential equations which model planetary orbits). Some of the biggest questions in this area are the $N$-body problems (http://en.wikipedia.org/wiki/N-body_problem), the Navier-Stokes equations (http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations) and in general, as has already been mentioned above, different types of ODEs and PDEs that arise as models in biology, physics, computer science, etc (in physics in particular, wave equations are abundant (http://en.wikipedia.org/wiki/Wave_equation), such as the Schroedinger equation from quantum mechanics).

Also arising in quantum and statistical mechanics are dynamical systems on infinite-dimensional spaces (Hilbert spaces). In this case the dynamics may be given by the action of a unitary operator on a Hilbert space, such as the solution of the Schrodinger equation. Techniques here are usually principally different than those in the finite dimensional case (e.g. on a finite dimensional manifold).

In general, the field of mathematical physics is huge, and dynamical systems are certainly not foreign to mathematical physics. Some of the areas of math physics where dynamical systems arise naturally are: statistical mechanics (here measurable dynamics, ergodic theory are prevalent); classical mechanics and general relativity (Newtonian dynamics modeled by a system of ODEs and systems of PDEs from relativity); fluid mechanics (the main problem here is the Navier-Stokes equations) which studies dynamics of compressible and incompressible fluids (or gasses) in a given environment, modeled by (a system of) PDEs; quantum mechanics and spectral theory (infinite dimensional dynamics given by unitary operators acting on Hilbert spaces); wave mechanics (wave equations, given by (a system of) PDEs).

I hope this helps.

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You could get a good overview of current research in dynamical systems via the book series Handbook of Dynamical Systems, which is a collection of surveys of the various areas of contemporary research in that field. Each survey is of the order of 40 pages long. This book series might also help to illustrate the sheer vastness of dynamical systems as a research topic: its four volumes total 4071 pages.

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I would not recommend that series to someone who only wants to get an overview of the field, especially barely at the graduate level. I would say that series is more of a reference for researchers or graduate students who have already chosen to work in the area. –  William Mar 23 '13 at 0:39
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Try the Nonlinearity open problems volume (2008, but most are still relevant).

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Thats an excellent list of problems! –  Piyush Grover Mar 20 '13 at 16:33
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Beside the book of Katok and Hasselblatt, mentioned above, i would recommend Dynamics in One Complex Variable of John Milnor (if your are interested in holomorpic dynamics) and the book Dimension Theory in Dynamical Systems: Contemporary Views and Applications of Yakov Pesin (if your are inetersted in dimensional theoretical aspect of dynamics). More from a physical view point we have the chaos book http://chaosbook.org/ for free. I wish You to have much fun with math.

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At least in a more applied side of PDE, one of the 'hot' topics seems to be adding nonlocal interactions/terms to classical PDEs. This arises in trying to model pattern formation in biology, social interactions, swarming of birds etc. There are some very interesting nontrivial analytic questions. Have a look at some of the papers of Razvan Fetecau:

http://people.math.sfu.ca/~van/publicationssp.html

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Percolation is a major outstanding research area in both theoretical and applied probability. I recommend reading the recent survey Percolation Since St. Flour by Geoffrey Grimmett and Harry Kesten (July 2012) which gives an up-to-date list of references.

Please feel free to send me an email, AJ, if you've got any specific questions.

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