**Title of the course:** Arf Rings

**Instructor:** Prof. Ali Sinan Sertöz

**Institution:** Bilkent Ü.

**Dates:** 26 August – 1 September 2019

**Prerequisites:** Arf uses only elementary techniques but with an admirable degree of mathematical maturity. Therefore a must prerequisite is willingness to learn something new. For undergrads, if you know what a ring is then you are set to go. If not, I will define that too!

**Level:** Graduate, advanced undergraduate

**Abstract:** The aim of the course is to explain the original definition and the use of Arf rings as Arf himself did. We will start by defining the basics of curve singularities, their resolutions and the multiplicities involved. The motivating problem is to find these multiplicities that occur during the resolution process. I will explain Du Val’s geometric solution. At this point Du Val poses the challenge of finding these multiplicities once the local parametrization of the singularity is given. Arf solves this algebraic problem by defining some special rings which he called canonical rings and which are today known as Arf rings. I will go step by step through Arf’s original article and explain his solution and calculations. At the end, if time permits I may talk about Lipman’s generalization of Arf rings in the language of commutative algebra.

**Language: **EN, TR