# Fall 2018 Activity

This was my first ever semester at the University of Rochester. I had only one class to teach, and thankfully, it was one I’d done before: MTH 201, Introduction to Probability. I spent most of September getting settled, and buying books and computers for my office.

I visited Ido Ben-Ari at the University of Connecticut on Sep 8 2017. I spoke about my joint work with Jon Chaika (University of Utah) that I’m quite excited about. It’s a generalized description of the collective behavior of random, semi-infinite walks on the lattice. The two major assumptions on the walks are that they have to coalesce if they meet. Under some very mild assumptions, we prove a striking dichotomy theorem: either all the walks coalesce, or they form biinfinite trajectories with an asymptotic velocity that foliate the space. We built this theory to understand the behavior of geodesics in first- and last-passage percolation. We use a lot of tools from ergodic theory, and I learned a lot from Jon. We also construct a bunch of examples. Our main example is a first-passage model with non-crossing geodesics that do have asymptotic direction.

It was almost by accident that I met Jon. I had this general framework that I thought would be a useful way to understand the behavior of generalized ‘Busemann functions’ in first- and last-passage percolation. Generalized Busemann functions appear as minimizers in the newly discovered variational formulas for first- and last-passage percolation (Georgiou, Rassoul-Agha, Seppalainen, Comm. Math. Phys 2016, and Krishnan, Comm. Pure Appl. Math., 2016). Luckily, the first ergodic theorist I randomly approached at Utah was an amazing mathematician. There is a typo-ridden draft on the arXiv, but we have a new version that we’re almost ready to send out for review.

I spent October putting together my NSF grant. It was on the properties of the Busemann functions I mentioned. It was my first time writing a grant about this project, and it took a while to put together.

I spent a part of November working on a project with an undergraduate student that I advised at the University of Utah. We spent some time refining our writeup and sending it out to people. Scott came up with the idea for the project: it’s on a connection between colored non-crossing partitions (see Marberg, Eric (2013) ‘Crossings and nestings…’, MR3139391), longest increasing subsequences constrained to be smaller than a certain number, and certain special Kostka numbers. Kostka numbers appear in the theory of symmetric functions in a seemingly unrelated context. We worked on cleaning up and writing Scott’s bijection between Kostka numbers and longest increasing subsequences constrained to be smaller than a certain number. I’m mainly interested in the combinatorics because of the connection between longest increasing subsequences and several classical models in probability.

We wrote to a few experts about the results, and it seems that some parts of it ought to be well-known. However, some of our generalizations appear to be new. The most interesting thing for us, at least, it to connect the Kostka numbers to colored non-crossing partitions. We’ve made some progress, but we’re not really close to anything. So its not quite ready to publish, but it’s certainly a very good start for Scott. He’s just applied to grad schools this year. He won the prestigious Churchill scholarship to spend a year at Cambridge. He intends to learn more about analytic and algebraic combinatorics there.

I also attended the North East Probability Seminar at Columbia in mid-November. I really enjoyed Hugo Duminil-Copins talk, and want to look into this paper of Schramm that he mentioned. It has a cool new inequality in it. I also had a chance to speak to LP Arguin, and we’re collaborating on a project in log-correlated fields and spin glasses.

Jeremy Quastel and I discovered a small hiccup in a paper that is under review. We were using a theorem from Borodin, Corwin, Ferrari and Veto 2013 that isn’t quite right. We spent a bit of time fixing it, and incorportating the referees suggestions. We hope to be done in the next few weeks.