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[This lecture is also doubling as this week's "open problem of the week", as it (eventually) discusses the soliton resolution conjecture.]

In this third lecture, I will talk about how the dichotomy between structure and randomness pervades the study of two different types of partial differential equations (PDEs):

  • Parabolic PDE, such as the heat equation u_t = \Delta u, which turn out to play an important role in the modern study of geometric topology; and
  • Hamiltonian PDE, such as the Schrödinger equation u_t = i \Delta u, which are heuristically related (via Liouville’s theorem) to measure-preserving actions of the real line (or time axis) {\Bbb R}, somewhat in analogy to how combinatorial number theory and graph theory were related to measure-preserving actions of {\Bbb Z} and S_\infty respectively, as discussed in the previous lecture.

(In physics, one would also insert some physical constants, such as Planck’s constant \hbar, but for the discussion here it is convenient to normalise away all of these constants.)

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