# Using analysis to find projective modules

Gabriel Dorfsman-Hopkins
Thursday, February 22, 2018 - 2:30pm to 3:20pm
PDL C-401
We will give an analytic proof of a new algebraic result, which says that vector bundles on a type of space arising in number theory (called the perfectoid unit polydisk) all have a very simple structure (in particular, they are all isomorphic to the trivial vector bundle).  The vector bundles on a space control it's geometry, and one should expect that simple spaces should have simple vector bundles, so this result says that the perfectoid polydisk has a straightforward analytic structure.

Although the result comes from algebraic geometry and number theory, the proof makes an interesting use of analytic methods, so we will try to keep the algebraic preliminaries to a minimum.  In particular, we will introduce the notion of a nonarchimedean Banach algebra (which is a ring and a metric space).  In algebraic geometry there is a well known correspondence between vector bundles and projective modules (which we can defined as images of idempotent matrices), and this correspondence will allow us to reduce the problem to studying the analytic properties of certain classes of Banach algebras, using an application of Newton's method for finding zeros of a function.

The intersection of analysis and algebraic geometry can lead to very powerful tools, which I hope will be the main takeaway of this talk.
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