Galois Representations and Arithmetic Geometry

This award supports research in arithmetic geometry, a field that studies number systems using geometric and algebraic methods. These tools help understand the structure of equations and their solutions, with applications in areas such as cryptography, secure communication, and digital verification. The project investigates questions involving Galois representations and algebraic cycles -- central objects in the modern mathematical understanding of arithmetic -- and also supports the training of graduate students in advanced research settings. The research will examine problems related to Galois representations and algebraic cycles. Specific directions include structural analysis of mod p reductions of crystalline representations, the study of algebraic cycles through prismatic cohomology, and exploration of de Rham analogues of the Mumford–Tate conjecture. The project also addresses questions in the arithmetic of Shimura varieties and the independence of l in Weil–Deligne representations. These efforts are grounded in current methods in p-adic Hodge theory and align with current directions in number theory and arithmetic geometry. 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. NSF Award ID: 2502326 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Mark Kisin | Institution: Harvard University, CAMBRIDGE, MA | Award Amount: $255,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2502326 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2502326.html