**Office:** 999 Evans

**Office Hours:** TBA

**Prerequisites:** Math 256

**Required Text:** none

**Recommended Reading:** Vladimir Berkovich, Spectral Theory and Analytic Geometry over Non-Archimedean Fields

**Syllabus:** Non-Archimedean geometry is motivated by the desire to
develop a theory of analytic spaces over non-Archimedean (for example
p-adic) fields analogous to the classical theory of complex manifolds
(or, more generally, complex analytic spaces). There are several
different approaches to the subject, including Tate's theory of rigid
analytic spaces, Raynaud's theory of admissible formal schemes,
Berkovich's theory of non-Archimedean analytic spaces, and Huber's
theory of adic spaces. We will touch on all of these, focusing mainly
on Berkovich's theory. We will also discuss (in varying levels of
detail) applications of non-Archimedean geometry to areas such as Galois
representations, tropical geometry, and dynamical systems. Students
will be expected to be familiar with the basics of scheme theory (as
covered in Chapter II of Hartshorne) and we will present many of the
foundational results concerning affinoid algebras without proof, but
otherwise we will try to keep the material relatively self-contained.

**Grading:** Students will be expected to help write lecture notes
which will be made available to the public at the end of the course.
Grades will be based on class participation, assistance in writing up
notes based on the lectures, and suggested improvements to the lecture
notes.

**Homework:** see above