K. Costello | Searcharxiv

arXiv2010

Ground-based multisite observations of two transits of HD 80606b

We present ground-based optical observations of the September 2009 and January 2010 transits of HD 80606b. Based on 3 partial light curves of the September 2009 event, we derive a midtransit time of T_c [HJD] = 2455099.196 +- 0.026, which is about 1 sigma away from the previously predicted time. We observed the January 2010 event from 9 different locations, with most phases of the transit being observed by at least 3 different teams. We determine a midtransit time of T_c [HJD] = 2455210.6502 +- 0.0064, which is within 1.3 sigma of the time derived from a Spitzer observation of the same event.

astro-ph.EP↗

arXiv2003

Hilbert schemes, Hecke algebras and the Calogero-Sutherland system

We describe the ring structure of the cohomology of the Hilbert scheme of points for a smooth surface X. When the canonical class K_X = 0, this was done by Lehn and Sorger, extending earlier work when X = C^2. Their approach does not generalize. Instead we recast this problem as a question of finding integrals of motion for a Hamiltonian which describes ``intersection with the boundary''. To do so, we use the identification of the cohomology of the Hilbert schemes with a Fock space modelled on the lattice H(X). With this identification, Lehn computed the operator of intersection with the boundary. It is essentially the Calogero-Sutherland Hamiltonian. We then solve the problem of finding integrals of motion by using the Dunkl-Cherednik operators to find an explicit commuting family of differential operators; these operators represent cup product on the Hilbert scheme. We provide two characterizations of the Hilbert scheme multiplication operators; the first as an algebra of operators that can be inductively built from functions and the CS Hamiltonian, and the second in terms of the centralizer of the CS Hamiltonian inside an appropriate ring of differential operators.

math.AG↗