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# Pre-Seminar for the Ruth Moufang Lectures

## July 9, 2021 at 16:00 – 17:45 CEST

In the very first instalment of the Ruth Moufang Lectures, Jennifer Balakrishnan will talk about rational points on curves. In the tradition of the “Gingko-Seminar”, we offer a “pre-seminar” as preparation consisting of two 45 minute talks. The pre-seminar is aimed at PhD students and PostDocs, as well as interested Bachelor and Master students.

**1. What is… a p-adic number? (Theresa Kumpitsch)**

The p-adic numbers were invented (or discovered depending on your philosophy) at the beginning of the twentieth century by Kurt Hensel following the observation that that numbers behave similar to functions in many ways. In this short introduction to p-adic numbers we want to explore this analogy, learn about different ways of expressing p-adic numbers, look at lots of examples and get a sense of why they play a role in the theory of Diophantine equations.

**2. What is… an algebraic curve? (Martin Lüdtke)**

Roughly, an algebraic curve is a 1-dimensional shape defined by polynomial equations. Examples are parabolas, hyperbolas, elliptic curves, or the Fermat curves defined by x^n + y^n = z^n. We want to explore first the geometry of curves and discover the genus as a discrete invariant. We then turn to the problem of finding rational solutions to equations in two variables. We discuss several examples and see how the set of rational solutions is governed by the geometry of the associated algebraic curve.

Zoom coordinates:

https://uni-frankfurt.zoom.us/j/92910007294?pwd=MTFwU2VKR3MvU1JWanZIUDlmYms4UT09

Meeting ID: 929 1000 7294

Passcode: 931095