Bob told me the other day that he discovered something in number theory that might have been known to number theorists for ages, namely most product of three primes has one very large prime factor and two smaller ones. His idea uses integration of asymptotic density of primes, given by the prime number theorem. He also gave some motivation for this result in terms of probabilistic number theory, namely trying to compute the moments of some arithmetic functions with Bernoulli random variables. Everything made sense to me so I was quite happy. Bob is a great conceptual lecturer.
Right now I am still stuck on my old Riemannian topology problem: given a manifold in R^n with sectional curvature bounded below by a positive constant, show that its intersection with any plane will be a submanifold. I still believe that convexity is the right approach to this problem. The original motivation was to show the intersection of the standard orthogonal group as embedded in R^n^2 intersected with any plane gives a submanifold. I also accidentally discovered that O(n) is contained in S^(n^2-1)(sqrt(n)), which somehow eluded me before because I was too focused on proving the bigger result. But this doesn’t help showing the intersection is a submanifold directly. Still unable to compute the curvature of O(n).
For probability, I decided to focus on Louis Arguin’s thesis on random probability cascade. He’s got some really solid proofs which clarify a lot of concepts for me. The environment of probability I course was too applied and too many nonmathematical geniuses were sitting in the room that I felt I learned very little.