TRINO

Speaker: Gus Schrader (Northwestern University)

Title: Quantization of Lagrangian submanifolds of cluster varieties and enumerative geometry

When: Tuesday February 17 @14h00

Where: SISSA, Room 133

Abstract: As discovered by Gekhtman-Shapiro-Vainshtein, a cluster variety comes with a canonical (pre-)symplectic form. Fock and Goncharov observed that this form is actually the image under the regulator map of a class in the algebraic K-group K_2 of the function field of the variety, and introduced the notion of "K2-Lagrangians": algebraic Lagrangian subvarieties on which the K2 symplectic form restricts to zero. I'll describe the quantization of such a Lagrangian in the moduli space of SL(2,C)-local systems on a sphere with m punctures, and explain how this quantization leads to a formula for Ekholm and Shende's enumerative invariant of holomorphic maps from open Riemann surfaces into a class of (real) Lagrangians in Kahler C^3. Based on joint work with Mingyuan Hu, Linhui Shen and Eric Zaslow.

Speaker: Stefano Canino (Università di Trento)

Title: Isotropic rank of harmonic polynomials

When: Friday March 13 @11h00

Where: SISSA, Room 133

Abstract: The Waring problem consists in computing the Waring rank of a homogeneous polynomial, i.e. the minimum number of powers of linear forms of which it is the sum. It is an NP-problem and we have only partial solutions. In particular, we can determine the rank of general forms, quadrics, monomials, binary forms, ternary cubics and a few other examples. In this talk we restrict to the class of harmonic polynomials, i.e. polynomials which are in the kernel of the Laplace operator, and we define the isotropic rank of a harmonic polynomial as the minimum number of powers of isotropic linear forms of which it is the sum. For the study of this rank, we introduce harmonic apolarity theory and the needed geometrical dictionary, and we use them to recover all the analogous results known for the Waring problem. In particular, we determine the isotropic rank for general harmonic forms, harmonic quadrics, harmonic monomials and harmonic ternary forms. This is a joint work with C. Flavi.