Georgia Institute of Technology College of Sciences

In this talk, I will present a classification result for time-periodic oscillation modes of capillary droplet under zero gravity, agitated at low Rayleigh modes (\leq14). To overcome the difficulties of nonlinear Fourier analysis on curved manifold, we develop a toolset paradifferential and para-Fourier integral operators on the sphere based on invariant Littlewood-Paley decomposition. In case of zonal symmetry, an effectively computable symbolic calculus is available. This paradifferential method balances the loss of regularity caused by small denominators, allowing a "fixed point" style proof instead of Nash-Moser/KAM iteration. Furthermore, the bifurcation analysis predicts observable frequency-amplitude relations, which were previously only partially addressed in physical references due to mishandled internal resonances. The bifurcation analysis is computer-assisted due to the hard task of finding all integral points on a given elliptic curve, together with heavy computations involving Legendre polynomials. The talk is based on a working paper co-authored with Haocheng Yang. 

TBA

We discuss the bow lemma or Schur comparison theorem, and in particular its higher dimensional analogs. We first investigate isoperimetry under upper and lower curvature bounds, including the Cartan-Hadamard conjecture. We then discuss Fenchel's theorem for CAT(0) spaces and its higher dimensional versions concerning tight (hyper)surfaces in Cartan-Hadamard manifolds. We also develop a rigidity mechanism for flat hypersurfaces in Alexandrov ambient spaces, which is applied to completely characterize surfaces which optimize total Gauss curvature in 3-folds with upper or lower curvature bounds.

Teams link: https://teams.microsoft.com/meet/2172100941812051?p=0YimD0FGzxQfer6TrD

Fix a graph or uniform hypergraph $F$ and consider the following simple 
process. Start with the complete (hyper)graph $K_n$ on $n$ vertices and 
repeatedly remove the edge set of a copy of $F$, chosen uniformly at 
random among all copies present at that time. Stop when no copy of $F$ 
is left. This is the $F$-removal process, and the key question is when 
it ends, that is, how many edges are left at the end.

This question has proven to be very hard. In 2015, Bohman, Frieze and 
Lubetzky famously proved that the triangle-removal process typically 
ends with $n^{3/2+o(1)}$ edges. Until now, the answer was not known for 
any other $F$.

We determine when the $F$-removal process ends for many choices of $F$. 
This includes cliques, where it confirms a well-known conjecture of 
Bennett and Bohman on the final number of edges. This is joint work with 
Kühn.