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.
Best Answer
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.