Georgia Institute of Technology College of Sciences

For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the exact threshold for the appearance of Hamilton $\ell$-cycles in an Erd\H{o}s--R\'enyi random hypergraph, confirming a conjecture of Narayanan and Schacht. The main difficulty is that the second moment is not tight for these structures. I’ll discuss how a variant of small subgraph conditioning and a subsampling procedure overcome this difficulty.

The Guderley problem describes the behavior of a strong self-similar shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock's strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. 

In this talk, I will present recent work on the rigorous construction of the self-similar converging and diverging shock solutions for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

TBA