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.
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.
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.
Surveys of young stellar objects (YSOs) populating nearby molecular clouds are bringing us closer to an integrated picture of star and cluster formation, one that incorporates processes spanning many orders of magnitude in size, from accretion on stellar scales to the formation of clusters and associations on molecular cloud scales. Following a spectacular 15 years of infrared astronomy with Spitzer and Herschel, we now have a nearly compete census of the dusty YSOs (those with disks or infalling envelopes) in the clouds within 500 pc of the Sun.