Using the techniques of arc presentations and open book foliations, we prove new lower bounds on the crossing number and arc index of a satellite knot in terms of it's companion, pattern, and wrapping number.
In this talk I will discuss homological and homotopical isoperimetric inequalities, including those of higher rank. I will emphasise what is known in the non-positive curvature case.
I will present my recent results on global regularity for a canonical shell model of the Navier-Stokes equations, covering the unforced equation and certain classes of forcing. I will discuss the distinction between the forced and unforced regularity problems in this model, with the corresponding Navier-Stokes questions as motivation.
The study of the maximum diameter of $d$-dimensional pseudomanifolds on $n$ vertices was initiated by Criado and Santos as an abstraction of the analogous problem for polytopes, in relation to the Polynomial Hirsch Conjecture.
A series of works by Santos, Criado, Newman and Bohman establish the asymptotics. Here we use a mixture of deterministic and random tools to
determine the exact value for every large enough $n$ when $d=2$, and for a positive fraction of $n$ when $d\geq 3$.
Our pseudomanifolds with maximum diameter crucially depend on a surprising connection to the cube of Euler trails in uniform hypergraphs. To this end, for every fixed uniformity $d$ and power $r \geq 2$, we show that satisfying the natural divisibility conditions implies the existence of the $r$th power of an Euler tour/trail in any large enough $d$-uniform hypergraph with large enough codegree. The talk represents joint work with Stefan Glock, Olaf Parczyk, and Silas Rathke.