This seminar will not be held in the Spring 2018 Semester.
The Algebraic Geometry Seminar will be every Thursday from 3pm-4pm in CMSA Building, 20 Garden Street, Room G10.
The schedule will be updated as details are confirmed.
|09-14-17||Yu-Wei Fan (Harvard Math)||
Abstract: We will recall the notion of entropy of an autoequivalence on triangulated categories, and provide counterexamples of a conjecture by Kikuta-Takahashi.
|Shamil Shakirov, Harvard Math||
Undulation invariants of plane curves
Abstract: “One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has singularities or other distinctive features of interest). A classical example of such a problem, described by Cayley and Salmon in 1852, is to determine whether or not a given plane curve of degree r > 3 has undulation points — the points where the tangent line meets the curve with multiplicity four. Cayley proved that there exists an invariant of degree (r – 3)(3 r – 2) that vanishes if and only if the curve has undulation points. We construct this invariant explicitly for quartics (r=4) as the determinant of a 21 times 21 matrix with polynomial entries, and we conjecture a generalization for r = 5
|Alexander Moll, IHES||
Hilbert Schemes from Geometric Quantization of Dispersive Periodic Benjamin-Ono Waves
ABSTRACT: By Grojnowski and Nakajima, Fock spaces are cohomology rings of Hilbert scheme of points in the plane. On the other hand, by Pressley-Segal, Fock spaces are spaces of J-holomorphic functions on the loop space of the real line that appear in geometric quantization with respect to the Kähler structure determined by the Sobolev regularity s= -1/2 and the Hilbert transform J. First, we show that the classical periodic Benjamin-Ono equation is a Liouville integrable Hamiltonian system with respect to this Kähler structure. Second, we construct an integrable geometric quantization of this system in Fock space following Nazarov-Sklyanin and describe the spectrum explicitly after a non-trivial rewriting of our coefficients of dispersion \ebar = e_1 + e_2 and quantization \hbar = – e_1 e_2 that is invariant under e_2 <-> e_1. As a corollary of Lehn’s theorem, our construction gives explicit creation and annihilation operator formulas for multiplication by new explicit universal polynomials in the Chern classes of the tautological bundle in the equivariant cohomology of our Hilbert schemes, in particular identifying \ebar with the deformation parameter of the Maulik-Okounkov Yangian and \hbar with the handle-gluing element. Our key ingredient is a simple formula for the Lax operators as elliptic generalized Toeplitz operators on the circle together with the spectral theory of Boutet de Monvel and Guillemin. As time permits, we discuss the relation of dispersionless \ebar -> 0 and semi-classical \hbar \rightarrow 0 limits to Nekrasov’s BPS/CFT Correspondence.