Abstract:  On a closed orientable surface of genus $g$, a 1-system is a collection of pairwise non-homotopic simple closed curves that pairwise intersect at most once. Obtaining bounds on the maximum size of a 1-system has proved to be a surprisingly hard problem. Constructions with roughly $g^2$ curves have been known for the last few decades, but upper bounds are trickier: in 2012, Malestein-Rivin-Theran produced an upper bound that is exponential in $g$. Przytycki in 2014 improved this to a bound that is $O(g^3)$, and in 2018 Josh Greene achieved an upper bound that behaves like $g^2 \log(g)$.

Our main result is a quadratic-in-$g$ upper bound, resolving the problem up to explicit multiplicative constants. We achieve this by choosing an appropriate hyperbolic metric and paying careful attention to how certain polygons formed by curves in the 1-system distribute their area over the surface. This represents joint work with Jonah Gaster.

Organizational Meeting

This is the organizational meeting of the Algebra Seminar for the fall semester. Please come if you are interested.

Suppose that a continuous-time, stochastic diffusion (i.e., the Susceptible-Infected process) spreads on an unknown graph. We only observe the time at which the diffusion reaches each vertex, i.e., the set of infection times. What can be learned about the unknown graph from the infection times? While there is far too little information to learn individual edges in the graph, we show that certain high-level properties — such as the number of vertices of sufficiently high degree, or super-spreaders — can surprisingly be determined with certainty. To achieve this goal, we develop a suite of algorithms that can efficiently detect vertices of degree asymptotically greater than $\sqrt{n}$ from infection times, for a natural and general class of graphs with n vertices. To complement these results, we show that our algorithms are information-theoretically optimal: there exist graphs for which it is impossible to tell whether vertices of degree larger than $n^{1/2 - \epsilon}$ exist from vertices' infection times, for any $\epsilon > 0$. Finally, we discuss the broader implications of our ideas for change-point detection in non-stationary point processes. This talk is based on joint work with Anna Brandenberger (MIT) and Elchanan Mossel (MIT).

Join us for the customary first Math Club meeting with introductions, math puzzles/games, and of course, free pizza!

Several geometric questions about planar Brownian motion remain open. For instance, is the adjacency graph of the complementary connected components of planar Brownian motion connected? Does the intersection of two independent planar Brownian motions contain a non-singleton connected component? In this talk, I will discuss analogous questions for SLE and show that there is a change of behavior in the regime $\kappa \in (4,8)$. I will also explain an application to the conformal non-removability of SLE. The main intuition comes from Mandelbrot's fractal percolation. Based on joint work with Haoyu Liu (PKU).