Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Fourth Annual Meeting

The Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics met remotely to provide research highlights from the past year. Director Robert Bryant provided a collaboration overview, while Bobby Acharya gave a talk on the collaboration’s progress in connections of special holonomy with physics and Simon Donaldson gave a talk on differential geometric developments in special holonomy involving gluing, singularities, collapsing, and flows. The meeting was complimented by a Simons Foundation Lecture given by S.-T. Yau entitled “Existence of Canonical Metrics in Non-Kähler Geometry”.

An online workshop was held September 14, 16 and 18. The workshop’s full agenda, abstracts, and talk videos may be viewed at the collaboration’s website, here.
 

1. Introduction. This is a brief update on the activities of our Simons Collaboration that have taken place since our fourth Annual Report was filed on June 30, 2020. There have been two events: a brief (1.25 hours) oral presentation with slides via Zoom to the Simons Foundation MPS on September 9 and a weeklong virtual workshop by Zoom and YouTube held September 14–18.

In the brief report, the director, Robert Bryant, gave a 30-minute presentation on the activities of the Collaboration, particularly highlighting how the pandemic has caused us to reconfigure our meeting schedule, to virtualize our June meetings and the plans to reschedule and virtualize the joint workshop with the American Institute of Mathematics, originally scheduled for the end of March and the beginning of April. This was followed by two 15-minute presentations, the first by Simon Donaldson on the mathematical progress and challenges being addressed by the Collaboration and the second by Bobby Acharya on the implications for theoretical physics and the Collaboration activities centered on the expected interactions between mathematics and physics.

The format of the weeklong workshop was our now-traditional “Progress and Open Problems.” See below for an expanded description.

2. Briefing replacing the Fourth Annual Meeting. September 9, 2020, 3:00–4:15 p.m. ET, via Zoom (Simons Foundation, NY, NY)

• Themes: Background and overview of special holonomy progress, applications in physics, topology and invariants of G₂-manifolds, new constructions methods, coassociative fibrations and adiabatic limits, twistor theory and special holonomy, gauge theory and calibrated geometry.
• Speakers (the presentation slides have been submitted to the foundation):
  • Robert Bryant (director & PI), 30 minutes on Collaboration operations, new PIs and plans for future meetings.
  • Simon Donaldson (PI), 15 minutes on the mathematical progress in the past year and challenges for the coming year
  • Bobby Acharya (PI), 15 minutes on envisioned physics applications and interactions with mathematical developments
• Participants: All of the Collaboration PIs were in attendance via Zoom for the briefing.

3. Initial Year-Five Workshop. September 14, 16 & 18, 2020 (virtual via Zoom through University of Pennsylvania account and YouTube Live Streaming)

This three-day workshop was organized as a collection of 12 presentations on recent progress and challenges that it would be desirable to address interspersed with eight curated discussion sessions. All the talks were recorded and posted on the Collaboration website. The discussion sessions, though planned to be focused on specific topics, were more informal and, hence, were not recorded.

• Title: Progress and Open Problems
• Organizers: Bobby Acharya, Jason Lotay and Sakura Schafer-Nameki
• Themes: Enumerative G₂-geometry and invariants, almost complex curves on the 6-sphere, solitons for the G₂-flow, the weak SYZ conjecture, G₂-instantons, Higgs bundles, and singular G₂-metrics and their field theory interpretations.
• Agenda: The schedule, along with speakers, titles and abstracts, is available on the Collaboration’s website. See Meeting Page for September 2020. The videos of the lectures are posted there.