Southwest Conference on Arithmetic Geometry

University Of Arizona, Tucson AZ

Investigators

Abstract

This award supports participation in the Southwest Center for Arithmetic Geometry’s 2026 Arizona Winter School (AWS) and the 2025 Preliminary Arizona Winter School (PAWS). AWS 2026 will be held March 7-11, 2026 at the University of Arizona in Tucson, and PAWS 2025 will be held online September 22-November 21, 2025. AWS 2026 will be organized around the central topic “Computational aspects of arithmetic geometry and cryptography,” and will feature a set of courses and accompanying research projects carefully designed and delivered by leading and emerging experts. The event will be a fusion of traditional mathematics conference and intensive research workshop: the speakers will each design courses of 4-5 lectures and propose research projects for graduate students. Nightly working sessions on these projects, and separate problem sets, will be run by the speakers and postdoctoral fellow assistants, and on the last day, students will present their findings. PAWS 2025, a virtual program aimed at advanced undergraduates and junior graduate students, will feature two six-week-long courses organized around the same topic as the AWS. Participants will engage in weekly problem sessions run by advanced graduate students, and participate in community-building and mentorship activities. Goals of both AWS 2026 and PAWS 2025 are to foster connections among peers, and mentoring relationships between students and senior researchers. Students will likely make concrete strides toward becoming research mathematicians, postdoctoral assistants will gain valuable mentoring experience in their academic careers, and faculty will develop new interests and see new connections. The Southwest Center website shares content from both AWS and PAWS, including lecture notes, project descriptions, and audio and video of lectures. More information about AWS 2026 and PAWS 2025 can be found on the website: http://swc.math.arizona.edu/ Both PAWS 2025 and the AWS 2026 center on the topic “Computational aspects of arithmetic geometry and cryptography.” With ever-more sophisticated computational infrastructure, opportunities arise to develop new mathematics with both intrinsic value and real societal impact. Both PAWS 2025 and AWS 2026 will have a wide scope, covering the fundamentals of analysis of algorithms in number theory and algebraic geometry, while also showcasing recent advances in computational number theory and post-quantum cryptography. Both programs have rosters of speakers who are leading experts: In PAWS 2025, Sabrina Kunzweiler and Juanita Duque-Rosero will give introductions to cryptography and analysis of the algorithms. In AWS 2026, David Harvey and John Voight will take on the algorithmic number theory aspects of the school, while Kirsten Eisentraeger and Chloe Martindale will take on the post-quantum cryptographic aspects of the school. Topics for the lecture series are plentiful, including algorithms for fundamental arithmetic, e.g., Harvey and van der Hoeven's polynomial multiplication over finite fields algorithm in time O(n log n); computing zeta functions of varieties over finite fields; algorithms for quaternion algebras; algorithms for lattice computations, lattice-based cryptography and attacks; quantum algorithms for number theory; the future of isogeny-based cryptography in light of the recent onslaught of successful attacks. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.