##### Department of Mathematics,

University of California San Diego

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### Special Colloquium

## Andrew Obus

#### Columbia University

## The Oort Conjecture and the Local Lifting Problem

##### Abstract:

Whenever a mathematical structure is given in characteristic $p$, one can ask whether it is the reduction, in some sense, of an analogous structure in characteristic zero. If so, the structure in characteristic zero is called a ``lift'' of the structure in characteristic $p$. The most famous example is Hensel's Lemma about lifting solutions of polynomials in $\mathbb{Z}/p$ to solutions in the $p$-adic integers $\mathbb{Z}_p$. We will consider a more geometric problem: given a curve $X$ in characteristic $p$ with an action of a finite group $G$, is there a curve in characteristic zero with $G$-action that reduces to $X$? Oort conjectured that this could be done when $G$ is cyclic, and his conjecture was recently proven by the speaker, Stefan Wewers, and Florian Pop. It turns out that this question reduces to a more ``local'' question about automorphisms of power series rings in one variable. This local question will occupy most of the talk. Many examples will be given throughout.

Host: Kiran S. Kedlaya

### January 16, 2013

### 1:00 PM

### AP&M 6402

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