# Computational Mathematics

**Objectives**

### S 5.1: Computer Aspects in Number Theory

Many basic questions in Mathematics go back to computational experiments. They often allow one to see new structures and patterns for which previous theory does not provide much insight. In turn such experiments often lead to new theoretical developments. On the basis of computational experiments lie suitable algorithms that make a certain problem computable in a reasonable amount of time. An old example, predating the time of computers, was the discovery of the distribution of prime number by Gauss done by hand-calculation and proved only much later. A recent example is given by the experiments of Birch and Swinnnerton-Dyer from the 1960s on the still wide open conjecture named after them, where the number of points on elliptic curves over many finite fields was counted. Other recent examples are to modular forms and Galois representations that led to many new exciting developments such as the Serre conjecture (now a theorem) or p-adic families of modular forms.

The area "Computational aspects in Number Theory" seeks Master students who on the one hand wish to acquire deep theoretical knowledge in Number Theory and Arithmetic Geometry and on the other hand want to approach the topic from a computational view point. This requires a strong background in the above themes and at the same time interest in computer algebra, computational issues and computer science. Current themes are questions on modular forms (classical or Drinfeldian), the computation of Heegner points over function fields, and explicit equations of certain deformation rings that occur in number theory.

**S5.2: ****Computational Geometry**

**Computational Geometry**

**Course Offerings / Tentative Study Plan**

**S5.1: ****Computational Aspects in Number Theory**

**Computational Aspects in Number Theory**

**Required background**

Solid knowledge of abstract algebra and Galois theory, some number theory and complex analysis.

**Foundational courses and courses of specialization**

- Algebraic Number Theory I & II
- Algebraic Geometry I & II (ideally one of the two sets was taken in the bachelor program)
- Computer Algebra I (MG19) & II (MG20)
- Modular Forms
- Automorphic Forms.

** Possible courses from Computer Science **

- Databases
- Advanced Parallel Programming

**Seminars & Additional Lectures**

- Seminars and
- topics courses on recent themes in number theory and computational aspects.

**Application field (18 CP according to PO)**

All areas as described in the examination regulations are possible.

**Sample Plan of Study (applicable starting winter semester 2013/14)**

Winter Term 1

- Algebraic Number Theory I
- Course in Computer Science
- Seminar
- Course in Field of Application (SCAP)

Summer Term 1

- Algebraic Number Theory II
- Modular Forms
- Seminar
- Course in Computer Science

Winter Term 2

- Reading Course on Specialization
- Automorphic forms
- Course in Field of Application (SCAP)

Summer Term 2