Kiran Kedlaya (MIT)

p-adic differential equations

The p-adic numbers were introduced in the early 20th century to provide a method for transferring techniques from real and complex analysis into number theory. In the early 1960s, Dwork discovered that the solutions of certain differential equations over the p-adic numbers could be used to count solutions of certain polynomial equations over a finite field. This touched off the development of a large but not well-known theory of p-adic differential equations with numerous applications (from complex analytic geometry to cryptography), but also some intriguing basic questions still open.