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Organizing Committee
Northwestern University
Case Western Reserve University
Wayne State University
University of Washington
UCLA
Abstract
Spheres are the basic building blocks of geometry. More complicated geometric objects can be built by attaching spheres to each other along continuous maps. For many purposes, such constructions depend only on the homotopy classes of these continuous maps. A fundamental problem in algebraic topology is to compute these homotopy classes, i.e., to compute the homotopy groups of spheres. Machines can be used to great effect in the exhaustive computation of these fundamental invariants of homotopy theory.
The workshop will study several software packages that are specifically designed for this purpose. Participants will have the opportunity to interact directly with codebases in work groups led by experienced programmers. The workshop will also introduce a variety of projects in homotopy theory that rely on computers.
This program will foster the community of researchers who are working with machines to solve long-standing problems in homotopy theory. It will also connect that community with researchers who are using machines in related fields, such as applied topology, category theory, commutative algebra, and number theory.
Application Deadline March 13, 2026
The workshop will study several software packages that are specifically designed for this purpose. Participants will have the opportunity to interact directly with codebases in work groups led by experienced programmers. The workshop will also introduce a variety of projects in homotopy theory that rely on computers.
This program will foster the community of researchers who are working with machines to solve long-standing problems in homotopy theory. It will also connect that community with researchers who are using machines in related fields, such as applied topology, category theory, commutative algebra, and number theory.
Application Deadline March 13, 2026
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Application Information
The deadline to submit an application to this program has passed.
Your Visit to ICERM
- ICERM Facilities
- ICERM is located on the 10th & 11th floors of 121 South Main Street in Providence, Rhode Island. ICERM's business hours are 8:30am - 5:00pm during this event. See our facilities page for more info.
- Traveling to ICERM
- ICERM is located at Brown University in Providence, Rhode Island. Providence's T.F. Green Airport (15 minutes south) and Boston's Logan Airport (1 hour north) are the closest airports. Providence is also on Amtrak's Northeast Corridor. See our travel page for full directions.
- Lodging
- Confirmed participants will receive an email to book a room at our preferred hotel, the Hampton Inn & Suites Providence Downtown. Contact programstaff@icerm.brown.edu with questions.Only use hotel links from ICERM’s official emails or this page. ICERM never uses third-party booking vendors.
- Childcare/Schools
- Those traveling with family can contact housing@icerm.brown.edu for childcare and school information.
- Technology Resources
- Wireless internet access and wireless printing are available for all ICERM visitors. Eduroam is supported. Thin clients provide access to web browsers, SSH terminals, and printing. See our Technology Resources page for setup info.
- Accessibility
- For special services, accommodations, or assistance, please contact accessibility@icerm.brown.edu in advance. Thank you.
- Discrimination and Harassment Policy
- ICERM is committed to a safe, professional, and welcoming environment. Brown University's Code of Conduct and related policies apply to all ICERM participants. Contact the ICERM Director or Associate Director Jenna Sousa for concerns.
- Fundamental Research
- ICERM programs promote Fundamental Research and mathematical education. Participants involved in sensitive or proprietary work should be aware of U.S. export control restrictions.
- Exploring Providence
- Providence offers world-class dining and attractions. Visit our Explore Providence page to see what’s near ICERM.
Visa Information
Contact visa@icerm.brown.edu for assistance.
- Eligible to be reimbursed
- B-1 or Visa Waiver Business (WB)
- Ineligible to be reimbursed
- B-2 or Visa Waiver Tourist (WT)
- Already in the US?
F-1 and J-1 not sponsored by ICERM: need to obtain a letter approving reimbursement from the International Office of your home institution prior to travel.
H-1B holders do not need a letter of approval.
All other visas: alert ICERM staff immediately about your situation.
ICERM does not reimburse visa fees. This chart is to inform visitors whether the visa they enter the US on allows them to receive reimbursement for the items outlined in their invitation letter.
Financial Support
This section is for general purposes only and does not indicate that all attendees receive funding. Please refer to your personalized invitation to review your offer.
- ORCID iD
- As this program is funded by the National Science Foundation (NSF), ICERM is required to collect your ORCID iD if you are receiving funding to attend this program. Be sure to add your ORCID iD to your Cube profile as soon as possible to avoid delaying your reimbursement.
- Acceptable Costs
- 1 roundtrip between your home institute and ICERM
- Flights on U.S. or E.U. airlines – economy class to either Providence airport (PVD) or Boston airport (BOS)
- Ground Transportation to and from airports and ICERM.
- Unacceptable Costs
- Flights on non-U.S. or non-E.U. airlines
- Flights on U.K. airlines
- Seats in economy plus, business class, or first class
- Change ticket fees of any kind
- Multi-use bus passes
- Meals or incidentals
- Advance Approval Required
- Personal car travel to ICERM from outside New England
- Multiple-destination plane ticket; does not include layovers to reach ICERM
- Arriving or departing from ICERM more than a day before or day after the program
- Multiple trips to ICERM
- Rental car to/from ICERM
- Flights on a Swiss, Japanese, or Australian airline
- Arriving or departing from airport other than PVD/BOS or home institution’s local airport
- 2 one-way plane tickets to create a roundtrip (often purchased from Expedia, Orbitz, etc.)
- Travel Maximum Contributions
- New England: $350
- Other contiguous US: $850
- Asia & Oceania: $2,000
- All other locations: $1,500
- Note these rates were updated in Spring 2023 and supersede any prior invitation rates.
- Reimbursement Requests
Request Reimbursement with Cube
Refer to the back of your ID badge for more information. Checklists are available at the front desk and in the Reimbursement section of Cube.
- Reimbursement Tips
- Scanned original receipts are required for all expenses
- Airfare receipt must show full itinerary and payment
- ICERM does not offer per diem or meal reimbursement
- Allowable mileage is reimbursed at prevailing IRS Business Rate and trip documented via Google Maps PDF
- Keep all documentation until you receive your reimbursement!
- Reimbursement Timing
6–8 weeks after all documentation is sent to ICERM. All reimbursements are reviewed by multiple Brown University offices.
- Reimbursement Deadline
Submissions must be received within 30 days of ICERM departure to avoid applicable taxes. No submissions are accepted after six months.
Projects
Spectral sequences of algebras in SageMath
John Palmieri (University of Washington)
SageMath includes an implementation of commutative differential graded algebras over a field. The proposed project will build on this to set up a framework for computations of spectral sequences of algebras. This will start with brainstorming --- what do we want such a framework to entail, what can we hope to compute, what should the interface be, what sort of input and outputs might we want? Prerequisites: for the brainstorming, a little familiarity with SageMath wouldn't hurt but is not necessary. For implementing the ideas, you need to know Python.
Equivariant algebra and Cp Mackey Functors
Ben Spitz (Indiana University Bloomington)
The CpMackeyFunctors package for Macaulay2 was recently developed by the two project leaders, together with Chan, Mudrak, Vogeli, Wang, Zeng, and Zotine. It provides computational support for working with the abelian category of Mackey functors over a cyclic group of prime order, including random constructors and the ability to compute resolutions, ext and tor. In this project, we will expand the functionality of this package to include Green functors and modules over them, equivariant ideals, and other constructions over Cp. This project is suitable for coders of all levels and anyone interested in working with computational equivariant algebra.
Mackey functors in OSCAR
Thomas Brazelton (Harvard University)
Working closely with the Cp Mackey functors project, we will port and expand existing computational methods for Mackey functors into the OSCAR framework in Julia. Using the GAP interface, we will define data types which allow us to work with G-Mackey functors for arbitrary finite groups G. This project is suitable for more advanced coders, or those who have experience in Julia.
Machine computation of the equivariant slice spectral sequence
Danny Shi (University of Washington)
This working group will focus on developing tools and methods for machine-assisted computations for the equivariant slice spectral sequence. The goal is to translate the input data of equivariant stable homotopy theory, including representation gradings, Mackey functors, slice cells, differentials, and extension problems, into forms that can be systematically computed and checked by computer. We will discuss both the mathematical structure underlying these computations and the design of algorithms that can make large-scale slice spectral sequence calculations possible. Examples will be drawn from familiar equivariant spectra such as norms of Real bordism theories and Lubin–Tate theories.
The R-motivic Adams spectral sequence
Eva Belmont (Case Western Reserve University)
The R-motivic Adams spectral sequence converges to the (2-complete) R-motivic homotopy groups of the sphere. The goal is to compute the Adams E2 page in a range using the rho-Bockstein spectral sequence. The E1-page of the rho-Bockstein spectral sequence has been computed in a large range in C-motivic terms. Prior work of Isaksen and myself suggests that the rho-Bockstein spectral sequence is highly constrained: while there isn't exactly an algorithm for determining the differentials, in practice one can determine differentials by using known structure (such as multiplications) along with comparison with the rho-localization, which says that almost all classes must participate in a differential.
The project involves turning the techniques that Isaksen and I used into a constraint satisfaction problem which can then be fed into a SAT solver (an off-the-shelf tool for solving boolean constraint satisfaction problems). A stretch goal or extension project is to do the same for the C2-equivariant rho-Bockstein spectral sequence as described by Guillou and Isaksen, which computes the E2 page of the C2-equivariant Adams spectral sequence.
The code will be written in python. Some familiarity with python will be expected, but a little focused self-study before the workshop should suffice.
The project involves turning the techniques that Isaksen and I used into a constraint satisfaction problem which can then be fed into a SAT solver (an off-the-shelf tool for solving boolean constraint satisfaction problems). A stretch goal or extension project is to do the same for the C2-equivariant rho-Bockstein spectral sequence as described by Guillou and Isaksen, which computes the E2 page of the C2-equivariant Adams spectral sequence.
The code will be written in python. Some familiarity with python will be expected, but a little focused self-study before the workshop should suffice.
Higher (dual) Steenrod algebras
Achim Krause (University of Oslo)
From the adjunction between spectra and the derived category of F_p, one gets a corresponding comonad on D(F_p). Its coalgebras can be regarded as a coherent form of comodules over the dual Steenrod algebra. While we want to eventually make this type of structure fully amenable to computer calculations, in this project we want to focus on developing software to work with its 2-categorical truncation. The resulting theory is equivalent to Baues' secondary Steenrod algebra (and its modules), serving as an important testing ground for generalizing to higher categorical level.
Programming prerequisites: Python
Mathematical prerequisites: some familiarity with higher category theory, stable homotopy theory and Steenrod operations would be ideal
Programming prerequisites: Python
Mathematical prerequisites: some familiarity with higher category theory, stable homotopy theory and Steenrod operations would be ideal
Python bindings for sseq
Hood Chatham (Cloudflare)
sseq is a software library written in Rust to compute minimal resolutions of
chain complexes over the Steenrod Algebra. It is designed to attain a compromise
between ease of use, generality, and performance.
However, Rust is not all that convenient of a language for math experimentation.
Python would be easier for most users. The goal of this project is to create
Python bindings that are sufficiently expressive to translate the Rust code
examples to Python. Alternatively, bring your favorite use case and translate
that to Python.
I am also happy to provide tutorials in the use of the core Ext libraries or to
supervise the addition of a feature.
chain complexes over the Steenrod Algebra. It is designed to attain a compromise
between ease of use, generality, and performance.
However, Rust is not all that convenient of a language for math experimentation.
Python would be easier for most users. The goal of this project is to create
Python bindings that are sufficiently expressive to translate the Rust code
examples to Python. Alternatively, bring your favorite use case and translate
that to Python.
I am also happy to provide tutorials in the use of the core Ext libraries or to
supervise the addition of a feature.