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The presence of fruitful mathematical themes suggests the unity of mathematics. What I mean by a mathematical theme here is a basic idea or guiding principle that motivates or directs the central questions of a subject. A few classic examples are "representation", "classification" and perhaps "duality". Another example that seems (to me) currently more active is "rigidity", by which I mean the exploration of conditions under which weak equivalence of a pair of objects implies stronger equivalence.

In the interest of seeing the general direction of contemporary mathematics as the resultant of such themes, I ask:

Question: What are the major mathematical themes driving mathematical exploration now?

A good answer ought to not only include the theme in question, but at least two specific areas of mathematics in which strong currents of research are driven by the theme. For example, the "rigidity" theme above is strongly driving the theory of finite von Neumann algebras (as can be seen for example in Popa's deformation/rigidity theory), but appears also in ergodic theory, geometric group theory and differential geometry. Of course, rigidity questions make sense in any area in which there is a heierarchy of equivalences of various strengths, but certain areas (due to the suitability of available techniques) are more strongly driven by efforts to address such questions than other areas are. It would be nice to have a sense of which areas are driven by which themes and (perhaps) why. Arguably, it is the state of the art of techniques in an area that drive the themes, but also those techniques were probably developed because their associated theme was natural.

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Categorization and "homotopization" seems like a pretty big one if MO is any indication. –  Steve Huntsman Aug 8 '13 at 16:10
@Steve: It would be nice to have a brief story of these themes you mention as an answer, I think. –  Jon Bannon Aug 8 '13 at 16:15
Categorification came up in the discussion that led me to ask this question. This might also be nice to hear about, and is perhaps what Steve meant in connection with homotopization? –  Jon Bannon Aug 8 '13 at 16:18
I don't mind myself, but maybe use a text editor at home and then sleep on it? About being OCD with editing, I think you're probably in fine company around these parts. –  Todd Trimble Aug 8 '13 at 17:44
@Jon: I was merely referring to the belief which I have heard expressed--but do not understand at all--that homotopy theory is foundational to mathematics as a whole. A prominent example along these lines would seem to be the homotopy type theory effort that Voevodsky et al. have been publicizing. –  Steve Huntsman Aug 8 '13 at 18:58

3 Answers 3

Stability. I interpret this in a very general sense. If A implies B, does a small perturbation of A implies a small perturbation of B? This "theme" is omnipresent. I omit the discussion of the classical notion of Lyapunov's stability...

And I give only two examples, as required.

I. A dynamical system is called "structurally stable" or "robust", if a small perturbation of dynamics (in an appropriate function space) leads to the "same behavior", for example the preturbed system is topologically conjugate to the unperturbed one.

See, for example, MR0925417 Andronov, A. A.; Vitt, A. A.; Khaĭkin, S. È. Theory of oscillators, Dover Publications, Inc., New York, 1987.

For a more recent example of the same, see MR0732343 Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.

II. Liouville's theorem says that conformal maps in Euclidean spaces of dimension $n\geq 3$ are "trivial", that is they are restrictions of M\"obius transformations. What if a map is "close to conformal"? There is a nice precise definition of this: quasiconformal with small dilatation.
See the wonderful book of Reshetnyak,

MR1326375 Reshetnyak, Yu. G. Stability theorems in geometry and analysis, Kluwer Academic Publishers Group, Dordrecht, 1994.

III. An example of unsolved problem (due to Fedya Nazarov). A classical theorem of Rado says that if $f$ is a continuous function in a region in the complex plane, and $f$ is analytic on the set $\{ z:f(z)\neq 0\}$ then $f$ is analytic everywhere. What if $f$ is known to be analytic on the set $\{ z:|f(z)|>\epsilon\}$. Is it globally close to an analytic function in some sense? Give a quantitative estimate in terms of $\epsilon$.

Everyone can add her favorite examples of stability.

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Is it meant to say $|f(z)| > \varepsilon$ in that second to last paragraph? –  Sam Hopkins Aug 8 '13 at 22:03
Sam: Yes, thanks. –  Alexandre Eremenko Aug 9 '13 at 1:06

Probabilistic methods have become an important tool in many areas. The idea is to show the existence of structures satisfying certain properties by defining an appropriate random model of the structure, and show that the property holds with high probability. Examples include:

  1. Random regular graphs are expanders
  2. Gromov's notion of random groups, which have been used to construct hyperbolic groups with various properties, including property T, the Haagerup property, surface subgroups, and groups which do not embed uniformly in Hilbert space
  3. Ranks of elliptic curves
  4. Constructions of closed surfaces in hyperbolic 3-manifolds by gluing "random" pairs of pants together, where the pants are shown to satisfy certain geometric constraints with high probability and uniformly distributed by ergodic theoretic methods.
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The study of random objects is of great interest in itself, not only as a tool for existence proofs. Random matrices, random polynomials, random Taylor and Fourier series are all interesting and some of these objects are relevant for physics. –  Alexandre Eremenko Aug 9 '13 at 15:14
@Alexandre: I was thinking the same thing. What came to mind for me is Voiculescu's free probability providing a solid tie between freeness and large random matrices. It's interesting that physics (I'm probably oversimplifying) seems to support the other "dichotomy" answer as well, since Wigner seemed to consider random matrices as occurring as high energy "random blocks" occurring in a block decomposition around the lower energy "stable" states. –  Jon Bannon Aug 9 '13 at 15:22
Mumford's "Age of stochasticity" is relevant here:stat.uchicago.edu/~lekheng/courses/191f09/mumford-AMS.pdf –  Piyush Grover Aug 9 '13 at 15:46
@AlexandreEremenko and Jon Bannon: Of course, the study of probability distributions is a long tradition in mathematics, so certainly not modern (although the specific distributions of random objects being studied in contemporary mathematics has changed as you point out; I would add e.g. SLE). The point of my answer is that probability is being used as a tool to prove existence results where the answer says nothing about the probability distribution (1 and 4) or gives one an idea of how common certain structures are (2 and 3), which I think is a modern trend. –  Ian Agol Aug 9 '13 at 16:16
Thanks, Ian, it was clear from your answer what you meant and I think your answer is very solid just as it stands. Alexandre's comment suggests another answer to this question distinct from yours may be "exploration of random structures". If this is understood to mean "study of probability distributions" then I agree, it is certainly not a modern theme. –  Jon Bannon Aug 9 '13 at 16:23

The dichotomy between structure and randomness is one such theme. Tao's paper focuses on additive number theory, where the idea is that almost all sets are either highly structured (e.g., contain arithmetic progressions) or similar to a random set. But similar themes appear in computational complexity theory; low computational complexity is associated with structure, and high computational complexity is associated with randomness. Razborov and Rudich's result on natural proofs can loosely be thought of as an argument that certain kinds of simple proofs of P≠NP are highly unlikely because they would imply the existence of a lot more structure in randomness than most people expect there to be.

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Though of a different flavor, this theme also appears with great prominence in set theory, specially on structures of size near the size of the reals. Typically, we can classify structures under a strong forcing axiom, such as $\mathsf{PFA}$, while such a classification is impossible in the presence of $\mathsf{CH}$. Among the many examples of this phenomenon, two prominent ones are the search for a basis for the uncountable linear orders (Justin Moore, Wacław Sierpiński), and the study of automatic continuity of homomorphisms of Banach algebras (Dales, Esterle, Woodin, Todorcevic). –  Andres Caicedo Aug 9 '13 at 15:23

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