Spotlight

A Fascination for Prime Numbers: Interview with Prof. Peter Schneider

Prof. Peter Schneider from the University of Muenster in Germany is one of the leading mathematicians working on p-adic methods in arithmetic and geometry. In recognition of his contributions to the field, he has been awarded the prestigious Leibniz Prize and the French-German Humboldt Prize. He was also the Clay Research Scholar in 1995.

During his stay at IISER-Pune as a JC Bose Chair, he is teaching an advanced course on “Local Fields” to undergraduate students in addition to giving two lectures that are open to all. In this conversation with Guhan and Shanti, Prof. Schneider talks about his academic career and work that is close to his heart. He says most people have a misconception that mathematicians use a lot of numbers, while many such as him define and investigate notions which are more conceptual and philosophical in nature. Here, Prof. Schneider walks us through his career as a mathematician even as he tries to reach out to as many people as possible by avoiding technical jargon.

When did your interest in Mathematics take shape?

This is easy to answer! This happened when I was a young high-school boy; I was fascinated by prime numbers. These are irregular, you never know when the next prime number comes, this was very mysterious and this fascinated me. From that time on, I think I knew this was something I would like to learn more about.

Any teachers who influenced your interest?

Not in high school, but later at the graduate level, I had teachers who were very important in shaping my interests. My interest in Mathematics was natural—although at that time I was naïve and did not know much. I would get into arguments with teachers since I couldn’t believe in something that they did.

My high school was in my home town, it was a small town called Weissenburg in the southern state of Bavaria in Germany. At that time, although the admission into Mathematics at a university was not restricted in any way, there was a central system for distributing students into different universities; one could not freely choose the university. We usually went to the nearest one, as it was in my case. I went to Erlangen, which was about 70 km north of Weissenburg and is very close to the city of Nuremberg. I did my Diploma at Erlangen and in the meantime, I had also spent a year at a different university studying number theory with a famous old number theorist.

You then went on to do your Ph.D.?

At that time in Germany (although it’s better now), the problem was that there were no particular fellowships for Ph.D. students.  You needed either a stipend from some organization or you could work as an assistant to a professor. You need to earn, need some money to live! So I approached

the professor I had done Diploma with for advice, he suggested Prof. Neukirch who had an assistant position in his group. This was in Regensburg, this is where I did my Ph.D.

How would you describe your Ph.D. thesis work?

My Ph.D. thesis was about Galois Cohomology. Like many other theories in mathematics, this was named after the famous mathematician Galois. This is about certain groups dealing with number theory and associating some new datum with these groups.

Prof. Neukirch got me in contact with Prof. Coates, now in Cambridge, who was then in Paris. He accepted, I got a stipend from DFG – the German equivalent to the NSF, and I spent a year there.

These two mathematicians, Prof Neukirch and Prof. Coates, had an important influence on my work. The work I did with Prof. Coates is what formed my habilitation thesis which I wrote up after returning to Regensburg.

I spent another year or so at Regensburg and went to Harvard for a year on a stipend. I returned to take up my first professor position in Heidelberg; one year later I moved to Cologne on a Chair position.

What did you begin to work on? In what way was it related to your Ph.D. or habilitation thesis?

My area of work had already diverged somewhat from number theory towards geometric objects from my Ph.D. thesis to my habilitation thesis which was about the Iwasawa Theory of Abelian Varities.

I was always motivated by number theory, but slowly and gradually in these last 25 years or so I moved away into more algebraic connections. This is motivated by a big program in number theory named after the famous mathematician who proposed it: the Langlands Program. It’s a vast vision of what could be true in a certain branch of Mathematics. Many dozens of mathematicians have been working on this for many years. The power of this vision is that it connects together many different branches of theoretical mathematics and I got very much fascinated by this program.

Langlands Program was surely a vision, but it turned out that there was something missing from the vision. There was one aspect, the p-adic aspect, which was completely neglected by Langlands. Over the last 15 years or so, I have worked to fill this gap. We are still far from having filled it, but we were able to construct the language or the foundation needed to extend the vision.

Can you describe what was missing from Langlands Program and what your contribution was to fill this void?

Sure, but I can’t do this without getting a little technical!! I should first say what the basic idea of Langlands Program is. It’s about studying objects in number theory, namely the Galois groups—certain groups number theorists want to investigate. One method to investigate a group is through representation theory: that means, naively speaking, you have a group which you do not understand, you compare it to other groups which you understand very well. This process of comparison is called representation of a group. Langlands proposed to relate the representations of Galois groups to a completely different kind of groups in another field of mathematics. So that’s the basic structure of Langlands Program.

Now, when I said known groups, you may have many lists of known groups and you select some of them. Langlands selected specific lists of known groups and completely omitted few others. These other groups, so called p-adic ones, they have in the last 10 years or so, turned out to be very important for other reasons. So it’s clear that one should extend the Langlands Program.

But you see the other part of the Langlands Program—comparing those representations with different groups—the objects did not exist in this new direction. I collaborated with Teitelbaum, a mathematician at Chicago and we constructed these missing objects from the other side, we developed a theory for these missing objects.

People talk about certain areas of work being “in” and being “fashionable”. Do you believe this or have you seen this occur?

This question is good, because this actually happened!! I had this dream of extending the Langlands Program in a certain direction. I think for many years, may be 7-8 years, Teitelbaum and I had worked on this, laying the foundations for this. The mathematical community found it interesting but did not think it was fashionable. Then suddenly, about 10 years ago, it became fashionable and many young people got into this field and started working on this. Something that the two of us started working more or less alone… we were convinced that it was important to study this, but it was not until many years later that it became important for many others as well. I witnessed this change in perception during my time!!

It must be rather gratifying to see your work being a part of the change.

Yes, but competition increases too (laughs) !!

Your work is largely in the realm of Pure Mathematics. Do you think students often tend to prefer Applied Mathematics over Pure Mathematics?

It is not considered politically correct to say Pure mathematics anymore due to the connotation that the word “pure” has. The preferred description is theoretical mathematics. I think students’ choices have varied over the years, and I don’t think I have noticed any pattern. Like it was for me, I think the choice to study mathematics and a specific topic in it are probably at an individual level.

How has your work influenced your teaching over the years?

I have mostly taught general courses at undergraduate level. As far as the advanced courses are concerned, I often use the opportunity to teach to learn a new topic myself. Since you have to explain it to your students, you have to be much more systematic, you have to go through it much more carefully-so you understand it better. So teaching has been a good way to learn for me.

-As told in a conversation with Guhan Venkat and Shanti Kalipatnapu

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