##### Department of Mathematics,

University of California San Diego

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### Final Defense

## Shaunak Das

#### UCSD

## Vector Bundles on Perfectoid Spaces

##### Abstract:

Perfectoid spaces were introduced to provide a geometric framework to the field of norms isomorphism from $p$-adic Hodge Theory, however, have proven their value well beyond this old result. For this reason, the geometry of perfectoid spaces is worth studying, for its own sake. In this talk, we explicitly compute the Picard group for the projectivoid line. With the desire to generalize this result to other perfectoid spaces, as well as to classify higher-rank vector bundles on these $p$-adic analytic spaces, we ask whether an appropriate GAGA Theorem holds for perfections of proper schemes over a base nonarchimedean field of characteristic $p >0$.

### June 1, 2016

### 3:00 PM

### AP&M 7321

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