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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