This pre-talk for the Geometric Analysis and Application Seminar on 10/13, provides background on the basic tools used in singularity analysis for mean curvature flow, and if time permitted also discusses Huisken’s classification of mean-convex shrinkers.

When does an algebraic differential equation have solutions which are themselves algebraic functions? This question goes back at least to Fuchs, in 1875. Its conjectural answer, due to Grothendieck and Katz in the linear case and Ekedahl–Shepherd-Barron–Taylor and Bost in general, situates it as one of number theory. I’ll explain these conjectures and some progress towards them, joint with Josh Lam, for isomonodromy differential equations. This class of differential equations includes, for example, the Painlevé VI equation. This work is in part an excuse to study “motivic” aspects of the space of all algebraic differential equations.

Given a discrete group of isometries of a real hyperbolic space, we can look at its limit set, i.e., the set of accumulation points of its orbits on the sphere at infinity. This is a compact metric space on which the group acts, and which enjoys many interesting geometric, topological, and dynamical features. For example, classical Kleinian groups provide many examples with limit sets that are quasicircles, Cantor sets, and Sierpinski carpets. In this talk, I will discuss how to construct examples whose limit sets are various low-dimensional trees of manifolds. In particular, we answer a question of M. Kapovich, by constructing convex cocompact groups of isometries of real hyperbolic spaces, whose limit sets are Čech cohomology 3-spheres not homeomorphic to S^3. These groups are right-angled Coxeter groups and our construction is flexible enough to produce infinitely many quasi-isometry classes. This is joint work with S. Douba, G.-S. Lee, and L. Marquis.