# ProLOEWE faces

## Professor Dr Jan Hendrik Bruinier The number cruncher

**Professor Dr. Bruinier "Uniformed Structures in Arithmetic and Geometry (USAG)" is the name of your LOEWE cluster which has been funded since 2018. Could you briefly describe what this is about and why it is important to do research on it? **The scientific description of the world we live in is often based on geometric models. For example, in general relativity, space and time are combined in four-dimensional space-time. Gravity is described using the curvature of space-time which leads to complicated geometric spaces.

The idea of uniformization is to replace such complicated spaces with simpler ones without changing the structure in any small way. Thereby important properties of the complicated space can be described by symmetries of the simple space. There are several modern generalizations of this basic idea which we use in the LOEWE cluster to investigate geometrical and arithmetical classification problems. Important practical applications can be found, for example, in encryption methods and digital signatures which are used in Internet banking.

**Can you still remember when and how your passion for mathematics was awakened and what makes it tick? **It was in high school in the seventh or eighth grade. I had chosen number theory as my compulsory elective course. Here, for example, we dealt with prime numbers. I found it fascinating to learn that as early as the third century B.C., Euclid was able to prove that there are an infinite number of prime numbers. Nevertheless, many important questions are still not understood today. It is assumed, for example, that there are also an infinite number of twin prime numbers - these are pairs of prime numbers with a prime gap of 2, such as 5 and 7, or 29 and 31, or 101 and 103. It has still not been possible to prove this, even today. The mathematician, Yitang Zhang, made a major breakthrough in 2013 when he was able to show that there is an infinite number of pairs of prime numbers with a prime gap of less than 70 million. Using similar methods and building on this knowledge, it has been possible to show that there is an infinite number of pairs of prime numbers with a prime gap of less than 246. However, this does not seem to be enough to crack the real twin prime conjecture.

**Wikipedia has the following entry about you: "In 2011 he (Jan Hendrik Bruinier) together with Ken Ono gave a finite algebraic formula for the values of the partition function. Both made a major breakthrough." What does this discovery mean for mathematics and for you personally? **A partition of a natural number n is a representation of n as the sum of natural numbers. The partition function p(n) counts the number of partitions of n. For example

4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1.

So the number 4 has exactly 5 partitions, you also write p(4) = 5. Partitions play an important role in very different places, for example whenever symmetries are involved, in combinatorics or in mathematical physics. The partition function has some surprising properties, for example it grows very fast. You can easily calculate by hand that p(5) = 7 and p(10) = 42, but for the number 100 you can find partitions as early as 190569292 and for 200 almost 4 trillion. A famous formula by Hardy Rademacher and Ramanujan allows to calculate the partition function by an infinite sum. Ken Ono and I have found a new formula that represents the partition function as a finite sum of algebraic numbers. These algebraic numbers are obtained as special values of a certain module function which is characterized by special symmetries.

The formula for the partition function is a nice example of a more general theory that has been developed in recent years. It was a great pleasure for me to contribute to it significantly.

**USAG is now in its third year of LOEWE funding: What do you think is special about the Hessian research funding program? **I find the LOEWE program with its open-topic funding geared towards scientific excellence very attractive. The funding strengthens exciting and promising initiatives, also with regard to other collaborative research programmes. In the case of our USAG focus, LOEWE was particularly well suited. There has already been close cooperation between the working groups in the field of algebra and geometry at the TU Darmstadt and the GU Frankfurt for several years, for example in the context of a joint research seminar and jointly supervised doctorates. LOEWE funding will enable us to further strengthen and deepen this cooperation.

**If you need a break from your everyday university or research life, how do you strike a balance? **I am interested in music and sport. I jog regularly, for example, and this helps clear my mind. We play soccer with the working group every week. We have rented a pitch at the university stadium for this. To increase our motivation, a medal is always awarded for the best goal of the week after the game. (Bruinier laughs).

**Combining a family and career is still a difficult topic for most women today. How do you experience this as a father of a family? **I am very happy to have a family with three children. Of course, children increase the complexity of life as a whole. You have to find compromises again and again, for example, when it comes to mobility and flexibility. It is no longer possible to suddenly accept many invitations to exciting conferences and your working hours have to adapt to family life.

I was lucky to be appointed to a permanent professorship relatively quickly. The planning security that comes with this makes it much easier to organise family life. I believe that the uncertain outlook in the post-PhD phase discourages many young female researchers in particular from pursuing an academic career.

### About the Person

- Spokesman of the LOEWE cluster USAG

### More Information

### Published in ProLOEWE NEWS

## Issue 01.2020 / March

### Topics

The current issue "GLOBAL VIRUS OUTBREAKS, CYBER-ATTACKS, NATURAL DISASTERS - HOW CAN WE PROTECT OURSELVES FROM CRISES IN FUTURE?" focuses on crisis management. Alongside the new LOEWE cluster entitled “Minority studies: language and identity”, we will report on a new professorship for minority languages in the Middle East.

### ProLOEWE faces

You will also find a portrait of Jan Hendrik Bruinier and his work within the LOEWE cluster known as “Uniformized Structures in Arithmetic and Geometry (USAG)”.