Let \(n\) be a large positive integer. Recently I've been looking for a family of curves \(f_n: \mathscr{C}_n\to \mathbb{P}^1\) with the following properties:
I'm currently at SWAG, the awesomely named Summer Workshop in Algebraic Geometry at Georgia.
Jonathan Wise gave a really amazing talk today about logarithmic geometry, which got me very excited about the possibility of using log geometry to compute various structures on fundamental groups of degenerating curves (e.g. Hodge structures, Galois actions, etc). It was also by far the best introduction to the subject I've seen. Yu-jong Tzeng gave a nice introduction to algebraic cobordism, and sketched a proof of Goettsche's formula for the Euler characteristic of the Hilbert scheme of points of a surface, which was quite elegant. Renzo Cavalieri gave a beautiful overview of Pixton's work on the double ramification cycle(s); I'm really interested in understanding where the products over graphs in Pixton's formulas "come from" geometrically, but so far I've been unable to figure out a rigorous statement. Finally, Dmitry Zakharov spoke about his recent work with Clader, Janda, and Zvonkine on applying Pixton's work to prove various classical tautological relations. Overall a really great conference so far.
Here's a question Dmitry asked at dinner, which I think is pretty interesting. Given that we expect the motion of the 10+ objects in the solar system to be chaotic, can we explain the fact that it has been relatively stable over the past 4 or so billion years? How long can we expect this stability to continue. Dimitry reassures me that numerical evidence indicates we have at least a few million years, but I think the Bayesian argument that we have at least a billion is kind of weak. I'd be very interested to hear an argument from physical (rather than probabilistic) principles analyzing the long-term stability of the solar system.
I'm thinking of starting a blog. Let's see how this goes.