## Geometry Seminar - Abstracts

### Talk

Tuesday 6 June 2023, 10:30 - 11:30 in HG03.054

**Sid Mathur** (Santiago)

*Degeneration in Algebraic Geometry*

### Abstract

Classical algebraic geometry is the study of sets of solutions of
polynomials in \(\mathbb{C}^n\). These are called complex varieties. In
Grothendieck's modern formulation, we instead consider varieties inside \(R^n\) for
*any* commutative ring \(R\). This perspective not only unites complex geometry and
number theory (for example, when \(R\) is a finite field) but also permits
interaction between the two. The idea underpinning this interplay is
degeneration: the modern formulation allows us to witness the movement of one
variety into another.
In this talk we will survey the technique of degeneration, and focus on two
celebrated examples, one arithmetic and the other geometric in nature. Each
will highlight the rich interplay between complex geometry and number theory.
Time permitting we will explain the technical results that allow us to
transport geometric data between two nearby varieties in a family.

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