Aniruddh Venkatesan L&S Math & Physical Sciences
The Geometry of the Fargues-Fontaine Curve
Many of the most interesting and difficult problems in number theory and arithmetic geometry arise in a setting called mixed characteristic. This refers to situations where different parts of a geometric object behave according to different number systems—leading to phenomena that are often surprising and poorly understood.
To address these challenges, mathematicians in the late 20th century developed what is now known as p-adic Hodge theory, a powerful framework for studying such problems. However, many of the constructions in this theory initially appeared ad hoc or lacked clear geometric motivation.
This changed with the groundbreaking work of Fargues and Fontaine, who discovered a new geometric object—the Fargues-Fontaine curve—that provides a natural and unifying setting for p-adic Hodge theory. Their insight has reshaped our understanding of the field.
The goal of my project is to explore the geometry of the Fargues-Fontaine curve and related spaces. In particular, I aim to investigate whether classical results from algebraic geometry can be extended to this new and intriguing context.
Message To Sponsor
Dear Sponsor, Thank you so much for funding my project! During my years I have become increasingly certain in my desire to go to graduate school to pursue a PhD in math, and working on this project this summer has helped me not only develop my research skills, but also understand which areas of math I am interested in. I hope to use some of the funding I have recieved to attend conferences and present my findings to the global community of mathematicians.