2025 Simons Collaboration on New Structures in Low-Dimensional Topology Annual Meeting

Mike Miller Eismeier
University of Vermont

ASD Connections & Cosmetic Surgery

The cosmetic surgery conjecture predicts that, given a knot K in a 3-manifold, the oriented diffeomorphism type of surgery on K determines the surgery slope (up to oriented diffeomorphism). For knots in the 3-sphere, a sequence of restrictions coming from Heegaard Floer homology implies that if the cosmetic surgery is false for K, then r-surgery on K is not oriented diffeomorphic to (-r)-surgery, for some r in {2, 1, 1/2, 1/3, …}, but Heegaard Floer homology techniques reach a limit here.

Mike Miller Eismeier will discuss how a quantitative enhancement of instanton homology rules out the cases r = 1/n, leaving only the possibility of 2-surgery. Miller Eismeier will discuss limitations of this approach as well as possible future developments.
 

Sergei Gukov
Caltech

AI and AC

An alternative title for this talk could be “Learning Hardness.” While it comes as no surprise that solving challenging research-level math problems drives progress in mathematics, it is far less obvious that solving such long-standing open problems also plays a crucial role in shaping the next generation of AI systems. We live in an exciting time where mathematics and AI can greatly benefit each other, and the goal of the talk is to explain how and why, drawing on specific examples from knot theory and combinatorial group theory. The notion of “hardness” — as in hard ribbon knots or hard AC presentations — plays a central role.

This talk is largely based on recent work with A.Shehper, A.Medina-Mardones, L.Fagan, B.Lewandowski, A.Gruen, Y.Qiu, P.Kucharski, and Z.Wang.
 

Gary Matthew Guth
Stanford University

Real Heegaard Floer Homology

There has been a burst of interest in gauge theoretic invariants of 3- and 4-manifolds equipped with an involution, developed in various contexts by Tian-Wang, Nakamura, Konno-Miyazawa-Taniguchi, and Li. Notably, Miyazawa proved the existence of an infinite family of exotic RP^2-knots using real Seiberg-Witten theory. In joint work with Ciprian Manolescu, we construct an invariant of based 3-manifolds with an involution, called real Heegaard Floer homology. This is the analogue of Li’s real monopole Floer homology. Our construction is a particular case of a real version of Lagrangian Floer homology, which may be of independent interest to symplectic geometers. In the case of Heegaard Floer homology, the construction starts from a Heegaard diagram where the involution swaps the alpha and beta curves. We prove that real Heegaard Floer homology is indeed a topological invariant of the underlying pointed real 3-manifold. Further, we study the Euler characteristic of our theory, which is the Heegaard Floer analogue of Miyazawa’s invariant for twist-spun 2-knots. This quantity is algorithmically computable and, indeed, appears to agree with Miyazawa’s invariant.
 

Peter Kronheimer
Harvard University

SO(3) Versus SU(3) in the Instanton Homology for Webs and Foams

In joint work with Tom Mrowka, an instanton homology for webs and foams was constructed previously using SO(3) gauge theory. Among other motivations, there were close connections with the foam evaluations in subsequent work of Khovanov-Robert for example. It turns out that the gauge group SO(3) can be replaced by SU(3) with surprisingly little change. A non-vanishing theorem for the SO(3) theory no longer holds, but other results carry over, and the structure simplifies. This talk will describe the relationship.
 

Lisa Piccirillo
Univeraity of Texas at Austin

Exotic Phenomena in Dimension 4

In favorable circumstances, topological 4-manifolds can be classified. In contrast, smooth 4-manifolds remain poorly understood. Several hard questions in 4-manifold topology boil down to asking whether there can be distinct smooth (sub)manifolds with the same topological type; such manifolds are called exotic. In this talk, Lisa Piccirillo will discuss a few methods to demonstrate exotic phenomena and comment on how these methods might be adapted towards addressing some of these questions.
 

Lev Rozansky
University of North Carolina Chapel Hill

Quiver-Like Varieties for the Geometric Categorification of Super-Algebras Gl(M|N) and the Associated Link Invariants

We show how a string-theory inspired reinterpretation of the Hamiltonian reduction involved in the definition of the Nakajima quiver varieties leads to a wider class of varieties, some of which should be associated with the super-algebras gl(m|n).

This is a joint project in progress with A. Oblomkov and Li Han.
 

Andras Stipsicz
Alfréd Rényi Institute of Mathematics

Smooth Structures on Closed Four-Manifolds with Non-Trivial Fundamental Group

It is a key question of four-dimensional topology to determine which closed topological four-manifolds admit more than one (or even, infinitely many) smooth structures.

In the lecture, we collect several constructions for producing ‘exotic’ smooth structures, which lead to a satisfactory answer in some cases with non-trivial fundamental group. We also examine the potential source of exoticness through the genus function on the second homology group.
 

Paul Wedrich
University of Hamburg

From Link Homology to Topological Quantum Field Theories

Quantum topology first revealed that the Jones polynomial—and many other knot and link invariants—originate from braided monoidal categories of quantum group representations, providing a foundation for associated 3- and 4-dimensional topological quantum field theories (TQFTs).

Khovanov’s categorification of the Jones polynomial suggests an analogous higher categorical structure, hinting at connections to 4- and 5-dimensional TQFTs via braided monoidal 2-categories. In this talk, I will outline four types of TQFTs emerging from link homology—4d and 5d, linear and derived—and survey the current landscape of concrete examples.

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

The Simons Foundation will never ask for credit card information or require payment for registration to our events.

Group A – PIs and Speakers

Economy Class: For flights that are three hours or less to your destination, the maximum allowable class of service is Economy class.
Premium Economy Class: For flights where the total air travel time (excluding connection time) is more than three hours and less than seven hours per segment to your destination, the maximum allowable class of service is premium economy.
Business Class: When traveling internationally (or to Hawaii/Alaska) travelers are permitted to travel in Business Class on those segments that are seven hours or more. If the routing is over budget, a premium economy or mixed-class ticket will be booked.

Group B – Funded Participants

The foundation will arrange and pay for round-trip air or train travel to the conference as well as hotel accommodations and reimbursement of local expenses. Economy-class airfare will be booked for all flights.

Group C – Unfunded 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.

Air and Rail

For funded individuals, 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 Deviations

The following travel specifications are considered deviations and will only be accommodated if the cost is less than or equal to the amount the Simons Foundation would pay for a standard round-trip ticket from your home city to the conference city:

• Preferred airline
• Preferred travel class
• Specific flights/flight times
• Travel dates outside those associated with the conference
• Arriving or departing from an airport other than your home city or conference city airports, i.e. multi-segment or triangle trips.

All deviations must be reviewed and approved by the Simons Foundation and, if the cost is in excess of what would normally be paid, a reimbursement quote must be obtained through the foundation’s travel agency before proceeding to booking and paying for travel out of pocket. All reimbursements for travel booked directly will be paid after the conclusion of the meeting.

Changes After Ticketing

All costs related to changes made to ticketed travel are to be paid for by the participant and are not reimbursable. Please contact the foundation’s travel agency for further assistance.

Personal & Rental Cars

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

Rental cars must be pre-approved by the Simons Foundation.

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

Funded individuals who require hotel accommodations are hosted by the foundation for a maximum of three nights at The James NoMad Hotel, arriving one day before the meeting and departing one day after the meeting.

Any additional nights are at the attendee’s own expense. To arrange accommodations, please register at the link included in your invitation.

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.

Overview

In-person participants 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 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; itemized receipts are required for expenses over $24 USD. The foundation DOES NOT provide a meal per diem and only reimburses actual meal expenses up the following amounts.

• Breakfast $20
• Lunch $30
• Dinner $75

Allowable Meal Expenses

• Meals taken on travel days (when you traveled by air or train).
• Meals not provided on a meeting day, dinner on Friday for example.
• Group dinners consisting of fellow meeting participants paid by a single person will be reimbursed up to $75 per person and the amount will count towards each individual’s $125 daily meal limit.

Unallowable Meal Expenses

• Meals taken outside those provided by the foundation (breakfast, lunch, breaks and/or dinner).
• Meals taken on days not associated with Simons Foundation-coordinated events.
• Minibar expenses.
• Meal expenses for a non-foundation guest.

• Ubers, Lyfts, taxis, etc., taken to and from restaurants in Manhattan.

  • Accommodations will be made for those with mobility restrictions.

Ground Transportation

Expenses for ground transportation will be reimbursed for travel days (i.e. traveling to/from the airport or train station) as well as subway and bus fares while in Manhattan are reimbursable.

Transportation to/from satellite meetings are not reimbursable.

Attendance

In-person participants and speakers are expected to attend all meeting days. Participants receiving hotel and travel support wishing to arrive on meeting days which conclude at 2:00 PM will be asked to attend remotely.

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 & Children

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.

With the exception of Simons Foundation and Flatiron Institute staff, ad hoc meeting participants who did not receive a meeting invitation directly from the Simons Foundation are not permitted.

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.

Meeting & Policy Questions

Meghan Fazzi
Senior Manager, Events & Administration, MPS
[email protected]

Travel & Hotel Support

FCM Travel Meetings & Events
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