Dyson's invitation: Opportunities in juxtaposition of incompatibles

Question

"Up to now, my examples of missed opportunities have been mathematical discoveries which actually occurred, although they could have occurred a long time earlier. In such cases one can be sure that an opportunity existed, but it existed only in the past. I now come to the more difficult task of identifying missed opportunities that are still open. Here one can no longer be sure that the opportunity is real, but if it is real then it has the virtue of existing in the present. The past opportunities which I discussed have one important feature in common. In every case there was an empirical finding that two disparate or incompatible mathematical concepts were juxtaposed in the description of a single situation. Taking the four examples in turn, the pairs of disparate concepts were respectively: modular functions and Lie algebras, field equations and particle dynamics, Lorentz invariance and Galilean invariance, quaternion algebra and Grassmann algebra. In each case the opportunity offered to the pure mathematician was to create a wider conceptual framework within which the pair of disparate elements would find a harmonious coexistence. I take this to be my methodological principle in looking for opportunities that are still open. I look for situations in which the juxtaposition of a pair of incompatible concepts is acknowledged but unexplained." - from Missed Opportunities by Freeman Dyson.

Dyson made a name for himself by showing the compatibility of the Schwinger, Feynman, and Tomonaga approaches to QED. He presents two unresolved juxtapositions of incompatible mathematical elements still under intense investigation:

A) General relativity and quantum mechanics

B) Feynman's sum over histories and existing theories of normed vector spaces.

Can you pose other Dyson pairs of incompatible mathematical structures (perhaps not as grand as Dyson's) that might present some interesting opportunities for integration under a yet-to-be-discovered (or invented) overarching theory? That is, can you extend Dyson's list?

Answer 1

My favourite "juxtaposition of a pair of incompatible concepts that is acknowledged but unexplained": the Riemann hypothesis and spectral theory, as phrased in the Hilbert-Pólya conjecture that the imaginary parts of the zeroes of the Riemann zeta function correspond to eigenvalues of an unbounded self-adjoint operator.

This is actually a topic on which Dyson himself has worked: When confronted with the statistical distribution of the zeroes on the critical line, which exhibited an anti-bunching or repulsion effect, Dyson noted that this seemed to coincide with that of the eigenvalues of a random Hermitian matrix, drawn from the Gaussian unitary ensemble.

The topic is nicely explained in The spectrum of Riemannium:

In the 1970's the mathematician Hugh Montgomery found that the statistical distribution of the Riemann zeroes on the critical line has a certain property that they tend not to cluster too closely together, but to repel. During a visit to the Princeton Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices. Dyson realized that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random and large Hermitian matrix. These distributions are of importance in physics, because for example the energy levels of an atomic nucleus satisfy such statistics.

In a posterior development that has given substantive force to this approach to the Riemann hypothesis through functional analysis and operator theory, the mathematician Alain Connes has formulated a “trace formula” using his non-commutative geometry framework that is actually equivalent to a certain generalized Riemann hypothesis. However, the mysterious operator believed to provide the Riemann zeta zeroes remains hidden yet. Even worst, we don’t even know on which space the Riemann operator is acting on.

This graph illustrates the highly suggestive juxtaposition of Riemann zeroes and spectral statistics.