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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.)