SCSHGAP meeting, Durham NC, 1317 May 2024
Arrival date: Sunday, May 12.
Departure date: Friday, May 17.
All times are EDT. Organizers are Robert Bryant (Duke University) and Mark Haskins (Duke University).
Schedule:
MON 13 MAY 
TUES 14 MAY 
WED 15 MAY 
THUR 16 MAY 
FRI 17 MAY 

8:009:30 
BREAKFAST  BREAKFAST  BREAKFAST  BREAKFAST  BREAKFAST 
9:3010:30 
J. Lotay  G. Parker (Zoom) 
V. Majewski  G. Englebert (Zoom) 
S. Dwivedi 
11:0012:00 
A. Fino (Zoom) 
B. Acharya  D. Platt  S. Habibi Esfahani  E. Windes 
12:0014:00 
Lunch  Lunch  Lunch  Lunch  Lunch 
14:0015:00 
E. Svanes  D. Gutwein  R. Conlon  S.K. Chiu  Z. Liu (Zoom) 
15:0016:00 
Break  Break  Break  Break  Break 
16:0017:00 
T. Hertl  Y. Wang  A. Wood  Y. Li (Zoom) 

18:0020:30 
BANQUET 
Speakers:
The links will take you to abstracts, slides of lectures, and/or video recordings of the lectures (when available).
 Bobby Acharya (King’s College London and ICTP), Confinement from Special Holonomy Cones
 ShihKai Chiu (Vanderbilt), Special Lagrangians in K3fibered CalabiYau 3folds
 Ronan Conlon (University of Texas at Dallas), A family of Kahler flying wing steady Ricci solitons
 Shubham Dwivedi (Humboldt Universität zu Berlin), Geometric flows of G_2 and Spin(7)structures
 Gilles Engelbert (Oxford), Stability of Cayley fibrations
 Anna Fino (Florida International University), Strong G2structures with torsion
 Dominik Gutwein (Humboldt Universität zu Berlin), G2instantons over generalised Kummer constructions via finite group actions
 Saman Habib Esfahani (Duke), On the DonaldsonScaduto conjecture
 Thorsten Hertl (Freiberg), On the Moduli Space of G2 Manifolds
 Yang Li (MIT), Special Lagrangian pairs of pants
 Zhenhua Liu (Stony Brook), General behavior of areaminimizing submanifolds
 Jason Lotay (Oxford), G2instantons, the heterotic G2 system and generalized geometry
 Victor Majewski (Humboldt Universität zu Berlin), Resolutions of Spin(7)Orbifolds
 Greg Parker (Stanford), A Fredholm framework for singular deformation problems
 Daniel Platt (Imperial College London), New Spin(7)instantons on compact manifolds
 Eirik Svanes (University of Stavanger), Heterotic distance conjectures and symplectic cohomology
 Yuanqi Wang (University of Kansas), Spectral theory of singular G2instantons
 Emily Windes (University of Oregon), The Moduli Space of Graphical Associative Submanifolds
 Albert Wood (King’s College London), InfiniteTime Singularities of Lagrangian Mean Curvature Flow
Albert Wood: Lectures
May 15, 2024
TITLE: InfiniteTime Singularities of Lagrangian Mean Curvature Flow
ABSTRACT: Lagrangian mean curvature flow is the name given to the phenomenon that, in a CalabiYau 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 WeiBo Su and ChungJun 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 infinitetime singularity of Lagrangian mean curvature flow. Our construction is a parabolic analogue of work of Dominic Joyce and YngIng 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.
Thorsten Hertl: Lectures
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 G2holonomy metrics, but also proved that the moduli space of all G2metrics on a closed manifold is itself a finitedimensional 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 nontrivial 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 nontrivial example might also descend to the (full) moduli space.
This talk is based on ongoing joint work with Sebastian Goette.
Dominik Gutwein: Lectures
May 14, 2024
TITLE: G2instantons over generalised Kummer constructions via finite group actions
ABSTRACT: This talk explanes a method for producing new examples of G2instantons over generalised Kummer constructions. This method is based on an extension of Walpuski’s original gluing theorem for G2instantons 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 (preglued) almostinstanton to be invariant under such a group action. In order for the linear equation inside the fixpoint 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.
Emily Windes: Lectures
May 17, 2024
TITLE: The Moduli Space of Graphical Associative Submanifolds
ABSTRACT: In this talk, I discuss an infinitedimensional Lagrangemultipliers 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.
Shubham Dwivedi: Lectures
May 17, 2024
TITLE: Geometric flows of G_2 and Spin(7)structures
ABSTRACT: We will discuss a family of flows of G_2structures on seven dimensional Riemannian manifolds. These flows are negative gradient flows of natural energy functionals involving various torsion components of G_2structures. We will prove shorttime 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.
Gilles Englebert: Lectures
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.
Viktor Majewski: Lectures
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 Z2singular 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.
Yuanqi Wang: Lectures
May 14, 2024
TITLE: Spectral theory of singular G2instantons
ABSTRACT: G2instantons on 7dimensional manifolds generalize
both flat connections in dimension 3, and anti selfdual connections in dimension 4. DonaldsonSegal program expects a certain count of G2instantons 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 5dimensional 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 G2instantons with 1dimensional singularities. This is the preliminary of a joint project with Thomas Walpuski and Henrique Sá Earp.
Evyatar Sabag: Lectures
September 10, 2023
TITLE: G2 Manifolds from 4d N=1 Theories
ABSTRACT: We propose new G2holonomy manifolds, which geometrize the GaiottoKim 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 Mtheory on a noncompact singular CalabiYau threefold and its Kahler cone. We construct the 7manifold that realizes the domain wall in Mtheory by fibering the CalabiYau threefold over a real line, whilst varying its K¨ahler parameters as prescribed by the domain wall construction. In particular this requires the CalabiYau 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 7manifold to have holonomy G2. Indeed, for simple domain wall theories, this construction results in 7manifolds, which are known to admit torsionfree G2holonomy metrics. We develop several generalizations to new 7manifolds, which realize domain walls in 5d SQCD theories and walls between 5d theories which are UVdual.