2023 Simons Collaboration on Special Holonomy in Geometry, Analysis and Physics Annual Meeting

Bobby Acharya
M-theory, particle physics and special singularities of special holonomy spaces: Past, Present and Future

Ricci flat spaces of special holonomy with special kinds of singularities provide models for the extra dimensions of physical space predicted by superstring/M-theory. After reviewing the intimate relationship between the singularities of these spaces and particle physics in four dimensions I will review some of the physics progress made by this Simons collaboration. Then I will review recent work (together with Daniel Baldwin) interpreting (a generalized version of) the Joyce-Karigiannis constructions of G2-manifolds in terms of Higgs and Coulomb phases of four dimensional gauge theories.

I will conclude with some perspectives on the future and some of the important open problems.
 

Camillo De Lellis
Area-minimizing integral currents: singularities and structure

Let T be an area-minimizing integral current of dimension m in a smooth closed Riemannian manifold of dimension m + n. It is known since the work of De Giorgi, Fleming, Almgren, Simons, and Federer in the sixties and seventies that, when n = 1, the (interior) singular set of T has dimension at most m − 7. In higher codimension Almgren’s big regularity paper proved in 1980 that the singular set has dimension at most m − 2, laying the grounds for a theory which has been simplified and extended in the last 15 years. Both theorems are optimal, but at the qualitative level there is a quite important mismatch between the singular sets of the known examples and a general closed set of the same dimension. In a celebrated work in the nineties Simon proved, for n = 1, that the singular set is m − 7- rectifiable and that the tangent cone is unique Hm−7-a.e.. The counterpart of Simon’s theorem in higher codimension has been reached very recently by Paul Minter, Anna Skorobogatova and myself and, independently, by Krummel and Wickramasekera. Even though it would be natural to expect much stronger structural results, our theorem is indeed close to optimal, as a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension α ≤ m − 2.
 

Simon Donaldson
Curves in K3 surfaces and fibred calibrated submanifolds

We will discuss descriptions of calibrated submanifolds near the “adiabatic limit”. The situation we consider is where most of the ambient manifold is fibred by hyperkahler 4-manifolds and the calibrated submanifolds in question are modelled on fibrations with holomorphic curves as fibers. We will review first the case when the holomorphic curves are 2-spheres, which has been the scene of work by a number of authors (and the subject of previous talks in this Collaboration). We will then go on to consider the case of higher genus, which involves the geometry of moduli spaces of curves in K3 surfaces.
 

Sebastian Goette
Homotopy Associative Submanifolds in G₂-manifolds

Associative submanifolds are certain calibrated submanifolds in G₂-manifolds. There is the hope that counting them will reveal subtle information about the underlying G₂-structure. On the other hand, certain singular associatives can be resolved in exactly three different ways, so a naive count will be meaningless. In this talk, we will define homotopy associatives as cobordism classes of three-dimensional submanifolds that are adapted to the G₂-structure in a rather weak sense. We will see that a given cobordism class can be interpreted as a homotopy associative in exactly three different ways. This might help us to define a consistent counting scheme even when the naive number of associatives in a given cobordism class changes due to singularities.
 

Mark Haskins
Recent developments in Laplacian flow

Bryant’s Laplacian flow is a geometric flow of closed positive 3-forms on a 7-manifold that aims to give a parabolic PDE approach to constructing Riemannian manifolds with holonomy $G_2$. My talk will discuss some recent developments and open questions in this area. Along the way I will try to highlight ways in which Laplacian flow has both some similar and some distinctive features compared to more well-known and better-understood geometric flows, like Ricci flow, mean curvature flow and Lagrangian mean curvature flow.
 

Yang Li
Calabi-Yau metrics in the intermediate complex structure limit

Calabi-Yau metrics can degenerate in a 1-parameter family by varying the complex structure, and a basic invariant is the dimension of the essential skeleton, which is an integer between 0 and n. The case of zero is the context of noncollapsed degeneration of Donaldson-Sun theory, while the case of n is the context of the SYZ conjecture. We will discuss how to describe the Kahler potential at the C^0 level in the intermediate case for a large class of complete intersection examples.
 

Jason Lotay
Joyce conjectures for the Lagrangian mean curvature flow of surfaces

Building on the seminal work of Thomas-Yau, Joyce formulated an inspirational collection of conjectures concerning the Lagrangian mean curvature flow in Calabi-Yau manifolds which, in particular, relate the flow to notions of stability and the Fukaya category. I will present an overview of recent progress towards these conjectures in the case of Lagrangian surfaces. This is based on joint works with F. Schulze and G. Szekeleyhidi, and also with G. Oliveira.
 

Sakura Schafer-Nameki
Categorical Symmetries and Geometric Engineering

I will provide an overview of recent works on realizing generalized global symmetries (or categorical symmetries) in quantum field theories, that are constructed in string theory by geometric engineering.

Participation in the meeting falls into the following four categories. An individual’s participation category is communicated via their letter of invitation.

Group A – PIs and Speakers
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses.

Group B – Out-of-town Participants
The foundation will arrange and pay for all air and train travel to the conference as well as hotel accommodations and reimbursement of local expenses.

Group C – Local Participants
Individuals in Group C will not receive financial support, but are encouraged to enjoy all conference-hosted meals.

Group D – Remote Participants
Individuals in Group D will participate in the meeting remotely. Please register at the link above and a remote participation link will be sent to you approximately two weeks prior to the meeting.

Air and Train
For individuals in Groups A and B the foundation will arrange and pay for round-trip travel from their home city to the conference.

All travel and hotel arrangements must be booked through the Simons Foundation’s preferred travel agency. Travel arrangements not booked through the preferred agency must be pre-approved by the Simons Foundation and a reimbursement quote must be obtained through the foundation’s travel agency. Travel specifications can be provided by clicking the registration link above.

Personal Car
Personal car trips over 250 miles each way require prior approval from the Simons Foundation via email.

The James NoMad Hotel offers valet parking. Please note there are no in-and-out privileges when using the hotel’s garage, therefore it is encouraged that participants walk or take public transportation to the Simons Foundation.

Hotel
Participants in Groups A & B who require accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel. Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link above.

The James NoMad Hotel
22 E 29th St
New York, NY 10016
(between 28th and 29th Streets)
https://www.jameshotels.com/new-york-nomad/

For driving directions to The James NoMad, please click here.

Attendance
In-person participants and speakers are expected to attend all meeting days. Partial participation is permitted so long as the individual fully attends the first day, which is typically Thursday for two-day meetings. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.

COVID-19 Vaccination
Individuals accessing Simons Foundation and Flatiron Institute buildings must be fully vaccinated against COVID-19.

Entry & Building Access
Upon arrival, guests will be required to show their photo ID to enter the Simons Foundation and Flatiron Institute buildings. After checking-in at the meeting reception desk, guests will be able to show their meeting name badge to re-enter the building. If you forget your name badge, you will need to provide your photo ID.

The Simons Foundation and Flatiron Institute buildings are not considered “open campuses” and meeting participants will only have access to the spaces in which the meeting will take place. All other areas are off limits without prior approval.

If you require a private space to conduct a phone call or remote meeting, please contact your meeting manager at least 48-hours ahead of time so that they may book a space for you within the foundation’s room reservation system.

Guests
Meeting participants are required to give 24 hour advance notice of any guests meeting them at the Simons Foundation either before or after the meeting. Outside guests are discouraged from joining meeting activities, including meals.

Ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are discouraged.

Children under the age of 18 are not permitted to attend meetings at the Simons Foundation. Furthermore, the Simons Foundation does not provide childcare facilities or support of any kind. Special accommodations will be made for nursing parents.

Reimbursement
Individuals in Groups A & B will be reimbursed for meals and local expenses including ground transportation. Expenses should be submitted through the foundation’s online expense reimbursement platform after the meeting’s conclusion.

Expenses accrued as a result of meetings not directly related to the Simons Foundation-hosted meeting (a satellite collaboration meeting held at another institution, for example) will not be reimbursed by the Simons Foundation and should be paid by other sources.

Below are key reimbursement takeaways; a full policy will be provided with the final logistics email circulated approximately 2 weeks prior to the meeting’s start.

Meals
The daily meal limit is $125 and itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses.

• Meals taken on travel days are reimbursable.
• Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner) are not reimbursable.
• If a meal was not provided on a meeting day, dinner for example, that expense is reimbursable.
• Meals taken on days not associated with Simons Foundation-coordinated events are not reimbursable.
• Minibar expenses are not reimbursable
• Meal expenses for a non-foundation guest are not reimbursable.
• Group meals consisting of fellow meeting participants paid by a single person will be reimbursed up to $65 per person per meal and the amount will count towards each individual’s $125 daily meal limit.

Ground Transportation
Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport) as well as local transportation. While in NYC, individuals are encouraged to use public transportation and not use taxi, Uber or Lyft services.

Registration and Travel Assistance
Ovation Travel Group
sfnevents@ovationtravel.com
(917) 408-8384 (24-Hours)
www.ovationtravel.com

Meeting Questions and Assistance
Emily Klein
Assistant Manager, Events, MPS, Simons Foundation
eklein@simonsfoundation.org
(646) 751-1262