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Daniel Platt: Lectures

May 15, 2024
TITLE: New Spin(7)-instantons on compact manifolds


Slides of Lecture

September 13, 2023
TITLE: Approximations of harmonic 1-forms on real loci of Calabi-Yau 3-folds

ABSTRACT: Nowhere vanishing harmonic 1-forms on real loci of Calabi-Yau 3-folds are useful for the construction of G2-manifolds. If they have zeros, they are potentially useful for the construction of G2-manifolds with conical singularities, which are important in M-theory. Recent breakthroughs in complex geometry give explicit descriptions of some degenerating Calabi-Yau metrics. This has implications for nowhere vanishing 1-forms. I will explain one example, one conjectural example, one non-example, and one variety for which I know no theorem suggesting anything. The main novelty are some numerical results obtained by neural networks approximately solving the complex Monge–Ampère equation and the harmonic 1-form equation. This is joint work in progress with Michael Douglas and Yidi Qi.

September 11, 2022
TITLE: Associatives in the generalised Kummer construction

ABSTRACT: Associatives are a class of 3-dimensional submanifolds of G2-manifolds. They are examples of minimal surfaces and interesting in their own right, but there are also ideas to count them to define numerical invariants of G2-manifolds. For this, it is important to understand all possible degenerations of associatives in 1-parameter families, but all previously known families of associatives in 1-parameter families in compact G2-manifolds are roughly constant. I will explain some new associatives in the Generalised Kummer Construction whose volume tends to zero as the ambient G2-manifolds degenerate. The construction starts from a family of obstructed, non-rigid associatives, finitely many of which survive a perturbation of the ambient metric. I will explain the original family and the perturbation step. This is joint work with Shubham Dwivedi and Thomas Walpuski.

September 15, 2021
TITLE: New estimates for G2-structures on resolutions of orbifolds

ABSTRACT: Joyce and Karigiannis extended the generalised Kummer construction and constructed torsion-free G2-structures on resolutions of G2-orbifolds. In the talk I will explain a different analytic setup to study the same problem, using weighted Hölder norms, which gives improved control over the torsion-free G2-structure and a slightly simpler proof compared to the original proof. This has applications in G2-instantons, and potential applications in associative submanifolds and resolutions of singularities at different length scales. This is the content of arXiv:2011.00482.

Slides of Lecture