• 5/16/2024: On the Donaldson-Scaduto conjecture
• 3/16/2023: Towards a Monopole Fueter Floer Homology

May 16, 2024
TITLE: On the Donaldson-Scaduto conjecture

ABSTRACT: Motivated by G2-manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed 3-sphere with three asymptotically cylindrical ends in the G2-manifold X×ℝ3, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold X×C, where X is an A2-type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge-Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical U(1)-invariant special Lagrangians in X×C, where X arises from the Gibbons-Hawking construction. This talk is based on a joint work with Yang Li.

Slides of Lecture

March 16, 2023
TITLE: Towards a Monopole Fueter Floer Homology 

ABSTRACT: Monopoles appear as the dimensional reduction of instantons to 3-manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higher-dimensional spaces. The most interesting examples appear on Calabi-Yau 3-folds and G2-manifolds. Monopoles, conjecturally, can be used to define invariants of 3-manifolds, Calabi-Yau 3-folds, and G2-manifolds. These monopole invariants, conjecturally, are related to certain counts of calibrated submanifolds, similar to the Taubes’ theorem, which relates the Seiberg-Witten and Gromov invariants of symplectic 4-manifolds.  
Motivated by this conjecture, we propose a Floer theory for 3-manifolds, generated by Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3.  A major difficulty in defining these homology groups is related to the non-compactness problems. We prove partial results in this direction, examining the different sources of non-compactness, and proving some of them, in fact, do not occur.