20/04/2021, 17:00 — 18:00 Europe/Lisbon —
Tommaso Pacini, Università degli Studi di Torino
Minimal Lagrangian submanifolds, totally real geometry and the anti-canonical line bundle.
The category of totally real (TR) submanifolds was traditionally of interest mainly to complex analysts. We will present a survey of recent work towards a "TR geometry" and explain its relevance to the study of minimal Lagrangians and of convexity properties of the volume functional.
27/04/2021, 17:00 — 18:00 Europe/Lisbon —
Laura Schaposnik, University of Illinois at Chicago
On generalized hyperpolygons
In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.
04/05/2021, 17:00 — 18:00 Europe/Lisbon —
Yu-Shen Lin, Boston University
Correspondence theorem between holomorphic discs and tropical discs on (Log)-Calabi-Yau Surfaces
Tropical geometry is a useful tool to study the Gromov-Witten type invariants, which count the number of holomorphic curves with incidence conditions. On the other hand, holomorphic discs with boundaries on the Lagrangian fibration of a Calabi-Yau manifold play an important role in the quantum correction of the mirror complex structure. In this talk, I will introduce a version of open Gromov-Witten invariants counting such discs and the corresponding tropical geometry on (log) Calabi-Yau surfaces. Using Lagrangian Floer theory, we will establish the equivalence between the open Gromov-Witten invariants with weighted count of tropical discs. In particular, the correspondence theorem implies the folklore conjecture that certain open Gromov-Witten invariants coincide with the log Gromov-Witten invariants with maximal tangency for the projective plane.