We consider a variational problem in a planar convex domain, motivated by statistical mechanics of crystal growth in a saturated solution. The minimizers are constructed explicitly and are completely characterized.
We study the irreducible components of the moduli space of instanton sheaves on P^3, that is, µ-semistable rank 2 torsion-free sheaves E with c_1(E)= c_3(E)=0 satisfying h^1(E(−2))= h^2(E(−2))=0. In particular, we classify all instanton sheaves with c_2(E) ≤4, describing all the irreducible components of their moduli space. A key ingredient for our argument is the study of the moduli space T(d) of stable sheaves on P^3 with Hilbert polynomial P(t) = d · t, which contains, as an open subset, the moduli space of rank 0 instanton sheaves of multiplicity d; we describe all the irreducible components of T(d) for d ≤ 4.
We consider two nested billiards in ℝd, d≥3, with C2-smooth strictly convex boundaries. We prove that if the corresponding actions by reflections on the space of oriented lines commute, then the billiards are confocal ellipsoids. This together with the previous analogous result of the author in two dimensions solves completely the Commuting Billiard Conjecture due to Sergei Tabachnikov. The main result is deduced from the classical theorem due to Marcel Berger saying that in higher dimensions only quadrics may have caustics. We also prove versions of Berger's theorem and the main result for billiards in spaces of constant curvature: space forms.
Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson-Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.
For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure.