Aniruddh Venkatesan L&S Math & Physical Sciences
The Geometry of the Fargues-Fontaine Curve
Often in arithmetic algebraic geometry, there are two prime numbers of interest – one prime “controls the geometry”, while the other “controls the algebra”. When these two primes differ, often called the “ l =/= p” case”, the theory is much more classical and well-behaved, and was mostly worked out in the 60’s and 70’s.
However, when the two primes are the same, called the “l = p” case, most of the well-behaved structure of the theory goes away. A major goal of late 20th century arithmetic geometry was to find a way to resolve this problem and understand which results from the “l =/= p” theory had appropriate analogues in the “l = p” theory. The first key breakthrough in this effort was the formalism of p-adic Hodge theory initiated by Jean-Marc Fontaine in the 70’s and 80’s.
One of the downsides of p-adic Hodge theory is its unmotivatedness – many constructions are made just because “they work”. This issue was resolved in 2010 by Laurent Fargues – in joint work with Fontaine, the two discovered a curve that is a natural setting for many of the constructions of p-adic Hodge theory. The goal of my research project is to try to understand the geometry of this curve and other related geometric objects.
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.