May 15, 2024
TITLE: Infinite-Time Singularities of Lagrangian Mean Curvature Flow

ABSTRACT: Lagrangian mean curvature flow is the name given to the phenomenon that, in a Calabi-Yau manifold, the class of Lagrangian submanifolds is preserved under mean curvature flow. An influential conjecture of Thomas and Yau, refined since by Joyce, proposes to utilise the Lagrangian mean curvature flow to prove that certain Lagrangian submanifolds may be expressed as a connect sum of volume minimising ‘special Lagrangians’.

This talk is an exposition of recent joint work with Wei-Bo Su and Chung-Jun Tsai, in which we exhibit a Lagrangian mean curvature flow which exists for infinite time and converges to an immersed special Lagrangian. This demonstrates one mechanism by which the above decomposition into special Lagrangians may occur, and is also the first example of an infinite-time singularity of Lagrangian mean curvature flow. Our construction is a parabolic analogue of work of Dominic Joyce and Yng-Ing Lee on desingularisation of special Lagrangians with conical singularities, and is inspired by the work of Simon Brendle and Nikolaos Kapouleas on ancient solutions of the Ricci flow.

Slides of Lecture

May 13, 2024
TITLE: On the Moduli Space of G2 Manifolds

ABSTRACT: In his two seminal articles, Dominic Joyce not only constructed the first examples of closed manifolds with G2-holonomy metrics, but also proved that the moduli space of all G2-metrics on a closed manifold is itself a finite-dimensional manifold. The statement is, however, only a local one, and the global topological properties of these moduli spaces have remained quite mysterious ever since. Indeed, up to now, we only know that they may be disconnected by the work of Crowley, Goette, and Nordström; the question whether all path components are contractible or not has not been answered yet.

In this talk, I will outline a construction of a non-trivial element in the second homotopy group of the more accessible observer moduli space of G2 metrics on one of Joyce’s examples. If time permits, I will indicate why and how this non-trivial example might also descend to the (full) moduli space.

This talk is based on ongoing joint work with Sebastian Goette.

Slides of Lecture

May 14, 2024
TITLE: G2-instantons over generalised Kummer constructions via finite group actions

ABSTRACT: This talk explanes a method for producing new examples of G2-instantons over generalised Kummer constructions. This method is based on an extension of Walpuski’s original gluing theorem for G2-instantons over generalised Kummer constructions and deforms a connection that is (in a quantified sense) close to being an instanton. The novelty compared to previous constructions is that we use finite group actions to overcome possible obstructions. More precisely, we choose the (pre-glued) almost-instanton to be invariant under such a group action. In order for the linear equation inside the fix-point iteration of the gluing theorem to be solvable, it then suffices that the invariant part of the cokernel of the linearised instanton operator vanishes (instead of the full cokernel). This allows for more general conditions on the gluing data than in previous constructions.

Slides of Lecture

May 17, 2024
TITLE: The Moduli Space of Graphical Associative Submanifolds

ABSTRACT: In this talk, I discuss an infinite-dimensional Lagrange-multipliers problem that first appeared in Donaldson and Segal’s paper “Gauge Theory in Higher Dimensions II”. The longterm goal is to apply Floer theory to a functional whose critical points are generalizations of three (real) dimensional, special Lagrangian submanifolds. I will discuss a transversality theorem related to the moduli space of solutions to the Lagrange multiplers problem.

Slides of Lecture

May 17, 2024
TITLE: Geometric flows of G_2 and Spin(7)-structures

ABSTRACT: We will discuss a family of flows of G_2-structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G_2-structures. We will prove short-time existence and uniqueness of solutions to the flows and a priori estimates for some specific flows in the family. We will discuss analogous flows of Spin(7)-structures. This talk is based on arXiv:2311.05516 (joint work with P. Gianniotis and S. Karigiannis) and arXiv:2404.00870.

Slides of Lecture

May 16, 2024
TITLE: Stability of Cayley fibrations

ABSTRACT: Motivated by the SYZ conjecture, it is expected that G_2 and Spin(7)-manifolds admit calibrated fibrations as well. One potential way to construct examples is via gluing of complex fibrations, as in the program of Kovalev. For this to succeed we need the fibration property to be stable under deformation of the ambient Spin(7)-structure, with the main difficulty being the analysis of the singular fibres. In this talk I will present a stability result for fibrations with conically singular Cayleys modeled on the complex cone {x^2 + y^2 + z^2 = 0} in C^3.

Slides of Lecture

May 15, 2024
TITLE: Resolutions of Spin(7)-Orbifolds

ABSTRACT: In Joyce’s seminal work, he constructed the first examples of compact manifolds with exceptional holonomy by resolving flat orbifolds. Recently, Joyce and Karigiannis generalised these ideas in the G2 setting to orbifolds with Z2-singular strata. In this talk I will present a generalisation of these ideas to Spin(7) orbifolds and more general isotropy types. I will highlight the main aspects of the construction and the analytical difficulties.

Slides of Lecture

May 14, 2024
TITLE: Spectral theory of singular G2-instantons

ABSTRACT: G2-instantons on 7-dimensional manifolds generalize
both flat connections in dimension 3, and anti self-dual connections in dimension 4. Donaldson-Segal program expects a certain count of G2-instantons and other objects could yield a topological invariant for 7–manifolds, called the prospective G2–Casson invariant. Related to the compactification/boundary of the moduli space, Walpuski proposed to construct singular G2–instantons via gluing. The analytic part of this singular perturbation problem is expected to encounter indicial roots, that are essentially related to the spectrum of a certain Dirac operator on the standard 5-dimensional unit sphere.

In this talk, we report some work on the spectral theory, the consequent obstruction theory, and some expansions of harmonic sections related to these G2-instantons with 1-dimensional singularities. This is the preliminary of a joint project with Thomas Walpuski and Henrique Sá Earp.

Slides of Lecture

September 10, 2023
TITLE: G2 Manifolds from 4d N=1 Theories

ABSTRACT: We propose new G2-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N = 1 duality domain walls of 5d N = 1 theories. These domain walls interpolate between different extended Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the geometric realization of such a 5d superconformal field theory and its extended Coulomb branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its Kahler cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the Calabi-Yau three-fold over a real line, whilst varying its K¨ahler parameters as prescribed by the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through a canonical singularity at the locus of the domain wall. Due to the 4d N = 1 supersymmetry that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy G2. Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are known to admit torsion-free G2-holonomy metrics. We develop several generalizations to new 7-manifolds, which realize domain walls in 5d SQCD theories and walls between 5d theories which are UV-dual.

September 11, 2023
TITLE: On complete Calabi-Yau manifolds asymptotic to cones

ABSTRACT: We proved a “no semistability at infinity” result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in the 2-step degeneration theory developed by Donaldson-Sun. As a consequence, a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay is given. Moreover a byproduct of the proof is a polynomial convergence rate to the asymptotic cone for such manifolds. Joint work with Song Sun.