I haven’t had a long post for a while, but there is lots of great math on the internet. Here are some of the things I’m trying to find time to read.
Integrable systems, toric geometry and Okounkov bodies The answer to a question which many people have asked: What’s the relationship between integrable systems and torus actions? Allen Knutson has been telling people roughly what the picture should be for a while, but the details seemed very hairy; now they are all resolved. That gets us half way to the question I want to know the answer to: “What is the relation between these integrable systems and cluster algebras?”
A closed formula for the decomposition of tensor products of Specht modules for the symmetric group A positive formula for stable Kronecker coefficients! (Corollary 4.08) And a proof which relies on the sort of planar diagram philosophy that people on this blog love. Congratulations to Bowman, de Visscher and Orellana.
Causal diagrams and causal models Not new, but new to me. I had always learned that, if and are correlated, then the only way to tell which one causes which (or whether they are both caused by something else) was by a randomized trial. Not true! You can make this determination in a purely observational manner, by seeing how and both correlate with . Apparently, this was known since the late 80’s, but my stats course never covered it. Makes me want to go back and work in algebraic statistics.
And I’ll take the opportunity to plug a paper of my own which has been a long time coming. Schubert problems with respect to osculating flags of stable rational curves The Shapiro-Shapiro conjecture, now proved by Mukhin, Tarasov and Varchenko shows that any distinct points on give Schubert problems whose solutions are, astonishingly all real. What happens when the points collide? And what does this have to do with the work of Henriques and Kamnitzer?