# Geometry in number theory

​Broadly speaking, Manh Hung Tran works with solutions of algebraic equations. In his PhD thesis, he has studied the equations which define curves of genus one – the natural generalizations of elliptic curves.

Pure mathematics is abstract and hard to feel or “touch” even with classical subject like equations. To get a better intuition of solutions of equations, they can be drawn as geometric objects. Arithmetic and algebraic problems will then be transformed through a “mirror” to the field of geometry. Among many geometric objects, elliptic curves are interesting ones which appear in many areas of mathematics such as number theory, algebraic geometry and complex analysis. Moreover, they have many applications outside of mathematics such as in computer science and cryptography.

## The natural generalisations of elliptic curves

Genus one curves are natural general forms of elliptic curves. In his thesis, Hung studies two different but connected important problems regarding equations that define these curves. The first problem concerns the density of points over the rational numbers on genus one curves. The goal is to be able to uniformly estimate the number of points for a large class of curves which does not individually depend on the given equations. This study might be used in cryptography, where it is important to know the structure of solutions of equations in order to create a strong security code. On this subject, Hung held a popular science talk in the Science Festival last year. But he himself works entirely on the theoretical side.

The second problem is about the structure of invariants of genus one models. These invariants are algebraic objects which is stable under natural transformations. Hung has tried to get a correct formula for the two generators of these invariants.

## A good working environment

Hung has always liked mathematics, he says he has “followed the flow”. Mathematics is not an easy subject. You must bear in mind that it is fine to be stressful or to get stuck sometimes. On the way, you need to learn also what is not so interesting, or far away from your expertise, in order to get further in mathematics. After finishing his master in Italy, Hung applied for several PhD positions. One of the offers he got was from Chalmers. He did not know much about Sweden but decided to give it a chance. He has not regretted this, as he has found Sweden to be a really good mathematical country, though it is maybe not commonly known. There is also a great working environment, with a much better salary and amenities than many other European countries.

Now, Hung is looking for a postdoctoral position. He searches all over the world but would really like to stay in Sweden, or at least in one of the Nordic countries. However, in pure mathematics at least, the postdoctors are strongly encouraged to go abroad. This can be difficult when the postdoctor, as Hung, has a family, and he believes that is one reason why many mathematicians rather go to companies than stay in academia.

Manh Hung Tran will defend his PhD thesis “The density of rational points and invariants of genus one curves” on January 29 at 13.15 in the room Pascal, Hörsalsvägen 1. Supervisor is Dennis Eriksson.

Text and photo: Setta Aspström
Picture: Manh Hung Tran. Geometry of elliptic curves

Page manager Published: Mon 27 Jan 2020.