Abstracts for Conference on "Integrability, Geometry and Moduli", July 29 - August 2, 2019

We study genus zero open-closed orbifold Gromov–Witten invariants counting holomorphic disks in a symplectic toric Calabi–Yau 3-orbifolds with boundary in Lagrangians of Aganagic–Vafa type. I will describe an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel–Jacobi maps of the mirror curve, based on joint work with Bohan Fang and Hsian-Hua Tseng. This generalizes the open mirror theorem for symplectic toric Calabi–Yau 3-manifolds conjectured by Aganagic–Vafa, Aganagic–Klemm–Vafa and proved in joint work with Fang.

Using the ideas of motivic integration I will explain how to compute the "number" of semistable Higgs bundles  (maybe with parabolic structure) of fixed rank and degree on a smooth complex curve. Based on that result I am going to discuss a similar problem in the case of bundles with connections. If time permits, I will explain open questions, including a conjectural relation to Satake correspondence for affine Grassmannians over 2-dimensional local rings. This is a joint project with Roman Fedorov and Alexander Soibelman.

Representations of the celebrated Heisenberg commutation relations and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis).  In this talk I explain how Heisenberg relations bridge the noncommutative geometry and signal analysis.  After providing a brief comparative dictionary of the two languages, I will show e.g.