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Organizing Committee
Bucknell University
UFRO Universidad de La Frontera
University of California San Diego
Università degli Studi di Napoli 'Federico II'
University of California San Diego
Carleton University
Rijksuniversiteit Groningen
Université de Poitiers
Rice University
Abstract
The rapid development of computer algebra in recent years has contributed to great progress in algebraic and arithmetic geometry, both practical and theoretical. For every curve over a number field, there is a finite description of the set of rational points. Recent algorithmic advances have made it possible to give such a description explicitly in many cases that were previously out of reach.
The case of surfaces over number fields is more difficult, and the arithmetic of most surfaces (those of general type) is beyond the reach of current mathematics. K3 surfaces occupy an intermediate position where the geometric structure is sufficiently accessible to allow progress to be made, but a full description of the set of rational points is impossible. They are also important examples in complex algebraic geometry and play an important role in mathematical physics, particularly in string theory and conformal field theory. Results on K3 surfaces also inspire work on Calabi-Yau and other varieties of higher dimension, whose geometry is much less well understood. Thus, this program will focus on the arithmetic of K3 surfaces, though not to the exclusion of other related surfaces and varieties of higher dimension.
The main areas of interest of the program are the Brauer-Manin obstruction; analogies with higher-dimensional varieties, in particular Calabi-Yau varieties and cubic fourfolds; mathematical problems arising from string theory; computation and study of moduli spaces of families of K3 surfaces and their compactifications; the Batyrev-Manin conjecture and other approaches to estimating the growth of rational points; and holomorphic symplectic manifolds. The determination of the elliptic fibrations on a K3 surface underlies most of these. Although much is known about this problem, there is still much to be done.
The program will unite researchers from algebraic and complex geometry, arithmetic geometry, Hodge theory, and mathematical physics. It will bring theoretically and computationally-oriented researchers together, expecting that computations will illuminate conjectures made by the theorists and that theory will enlarge the range of what can be computed. We also intend to develop databases of certain types of K3 surfaces for the L-Functions and Modular Forms Database and to promote the development of software for computations on K3 surfaces in Magma, SageMath, or other systems for public release.
The case of surfaces over number fields is more difficult, and the arithmetic of most surfaces (those of general type) is beyond the reach of current mathematics. K3 surfaces occupy an intermediate position where the geometric structure is sufficiently accessible to allow progress to be made, but a full description of the set of rational points is impossible. They are also important examples in complex algebraic geometry and play an important role in mathematical physics, particularly in string theory and conformal field theory. Results on K3 surfaces also inspire work on Calabi-Yau and other varieties of higher dimension, whose geometry is much less well understood. Thus, this program will focus on the arithmetic of K3 surfaces, though not to the exclusion of other related surfaces and varieties of higher dimension.
The main areas of interest of the program are the Brauer-Manin obstruction; analogies with higher-dimensional varieties, in particular Calabi-Yau varieties and cubic fourfolds; mathematical problems arising from string theory; computation and study of moduli spaces of families of K3 surfaces and their compactifications; the Batyrev-Manin conjecture and other approaches to estimating the growth of rational points; and holomorphic symplectic manifolds. The determination of the elliptic fibrations on a K3 surface underlies most of these. Although much is known about this problem, there is still much to be done.
The program will unite researchers from algebraic and complex geometry, arithmetic geometry, Hodge theory, and mathematical physics. It will bring theoretically and computationally-oriented researchers together, expecting that computations will illuminate conjectures made by the theorists and that theory will enlarge the range of what can be computed. We also intend to develop databases of certain types of K3 surfaces for the L-Functions and Modular Forms Database and to promote the development of software for computations on K3 surfaces in Magma, SageMath, or other systems for public release.
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Application Information
ICERM welcomes applications from faculty, postdocs, graduate students, industry scientists, and other researchers who wish to participate. Some funding may be available for travel and lodging. Graduate students who apply must have their advisor submit a statement of support in order to be considered. In addition, graduate students will only be considered if they can remain in-residence for a minimum of six weeks.
ICERM Funding for Fellowships
ICERM has funding available for fellowships to support faculty who attend one of our semester programs. These are often used to cover teaching buyouts so that participants can be in residence throughout a semester program. We welcome requests from those not based at R1 research universities and those based at minority-serving institutions, including HBCUs, HSIs, TCUs, and AAPIs. The standard amount for a full semester is $30,000. Email ICERM’s Director for more information.
ICERM has funding available for fellowships to support faculty who attend one of our semester programs. These are often used to cover teaching buyouts so that participants can be in residence throughout a semester program. We welcome requests from those not based at R1 research universities and those based at minority-serving institutions, including HBCUs, HSIs, TCUs, and AAPIs. The standard amount for a full semester is $30,000. Email ICERM’s Director for more information.
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/Housing
- Visiting ICERM for longer than a week? ICERM staff helps participants locate suitable accommodations. Contact housing@icerm.brown.edu for details.
- 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.
- Need a US Visa?
- J-1 visa requested via ICERM staff
- Eligible to be reimbursed
- B-1 or Visa Waiver Business (WB) – if you already have either visa, contact ICERM staff for a visa-specific invitation letter.
- Ineligible to be reimbursed
- B-2 or Visa Waiver Tourist (WT)
- Already in the US?
F-1 and J-1 not sponsored by ICERM: 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.
