Reeb dynamics in dimension 3 and broken book decompositions

Vincent Colin
Université de Nantes
September 11, 2020
In a joint work with Pierre Dehornoy and Ana Rechtman, we prove that on a closed 3-manifold, every nondegenerate Reeb vector field is supported by a broken book decomposition. From this property, we deduce that in dimension 3 every nondegenerate Reeb vector field has either 2 or infinitely periodic orbits and that on a closed 3-manifold that is not graphed, every nondegenerate Reeb vector field has positive topological entropy.

An asymptotic version of the prime power conjecture for perfect difference sets

Sarah Peluse
Institute for Advanced Study and Princeton University; Veblen Research Instructor, School of Mathematics
September 10, 2020
A subset D of a finite cyclic group Z/mZ is called a "perfect difference set" if every nonzero element of Z/mZ can be written uniquely as the difference of two elements of D. If such a set exists, then a simple counting argument shows that m=n2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in Z/(n2+n+1)Z whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists.

Physics and Astrophysics With Gravitational-Wave Observations of Black Hole Binaries

Emanuele Berti
Johns Hopkins University
September 10, 2020
The observation of compact binary mergers by the LIGO/Virgo collaboration marked the dawn of a new era in astronomy. LISA will expand this vision by opening a new observational window at low frequencies. The gravitational radiation emitted by compact binary systems in these two frequency windows encodes important information on their astrophysical formation mechanism. Furthermore, compact objects - whether in isolation or in binaries - are excellent astrophysical laboratories to probe our understanding of high-energy physics and strong-field gravity.