You are currently browsing the tag archive for the ‘Ricci flow’ tag.

[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.)

### Recent Comments

 Lars Ericson on Finite time blowup for an aver… Sergey_Ershkov on Finite time blowup for an aver… arithmetica on Finite time blowup for an aver… Anonymous on Finite time blowup for an aver… Anonymous on An airport-inspired puzzle Anonymous on An airport-inspired puzzle Terence Tao on Gaps between primes April Jashmer Anting on Gaps between primes Anonymous on Finite time blowup for an aver… Terence Tao on Finite time blowup for an aver… What's the univ… on 254A, Lecture 8: The mean ergo… danield on Finite time blowup for an aver… Terence Tao on On the number of solutions to… Fred on Gaps between primes A Phrase Is Born on A nonstandard analysis proof o…