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.