Ana Caraiani is a Royal Society University Research Fellow and reader at Imperial College London, working on the Langlands programme. In 2018 she was one of the winners of the Whitehead Prize of the London Mathematical Society. In 2020 she was elected as a Fellow of the American Mathematical Society. She is one of the 2020 winners of the EMS Prize. In this interview we talked with Ana about her work on the Langlands programme, and about her experience being a mathematician.

We would like to start with your work on the Langlands programme. How would you explain your research to a non-specialist?

The Langlands programme is a series of conjectures connecting different areas of mathematics, for example, number theory, representation theory, and harmonic analysis. It’s mostly considered a subfield of number theory, but it really has connections to a lot of different other areas of maths, and it uses techniques from different areas such as arithmetic and algebraic geometry. In a sense, it can be considered a far-reaching generalisation of the law of quadratic reciprocity that was discovered by Gauss.

What drew you to this particular field?

I was always interested in number theory and if you’re interested in number theory at some point you encounter quadratic reciprocity, which seems like a completely magical thing. That’s one reason that drew me to the Langlands problem. Another reason is that another famous instance of this Langlands correspondence is the modularity of elliptic curves, which played a key role in the proof of Fermat’s last theorem. I wanted to do number theory and was interested in these connections between different areas, so I started reading about this. In Princeton, I did my senior thesis with Andrew Wiles and I guess that was the point at which I really knew that I wanted to continue working in this area.

Wiles gave me a very hard problem. It was clear I wasn’t really going to solve it in my thesis, but I think it was a good preparation for a research mathematician: facing a hard problem, thinking about it for a year and realising you have made almost no progress…

You won several medals in the International Mathematical Olympiads (IMO). Has this played any role in your later career as a mathematician?

I grew up in Romania and, if you’re interested in maths, participating in the Olympiads was the obvious thing to do, there weren’t many other things at the time and they’re quite popular.

It was a great experience because I travelled all over the world and met people my age from other countries who were interested in math. The experience and my results in the IMO helped me immensely. It would have been very hard to be accepted into a top university like Princeton without high-profile international experience.

Was your family supportive in your decision to move to the other side of the ocean? Was it difficult?

They were supportive, even if it was difficult because I am their only child. But in Romania, it is very common for young people to go study abroad and they accepted the fact that going abroad was an opportunity to have a career. I think that they were quite encouraging.

How would you describe your experience of being a mathematician?

I like the intellectual freedom that comes with being a mathematician, the freedom to play around with ideas, the creativity, the possibility of using all kind of different techniques. I feel that the things that appeal to me in life I can also find within maths.

For instance, I study the geometry of certain spaces that have a lot of symmetries and I find it aesthetically very pleasing: I like to be able to visualise things. I visualise spaces and try to see how they interact. That’s a part of doing maths that I really like.

I also collaborate with other people a lot and I like this social aspect. I like meeting people from all over the world, with different experiences, and getting to know them through talking about maths and working together. And I also like the flexibility, you can be anywhere if you’re doing maths, you don’t have to be in a particular place. This is a double-edged sword and sometimes the experience is very intense: if you’re stuck on a problem you can be anywhere, doing anything, but you’re just obsessing over that problem.

Is there a particular problem that you dream of solving?

I don’t know if there is a particular problem that I dream of solving. There are things that I would like to do in the next, say, five/ten years, but I prefer not to focus on just one thing or a particular problem to solve.

I am more interested in understanding the connections between things that have a different nature. The project that I’m most excited about now is establishing a connection between objects that are constructed differently. I don’t know if that will solve a big problem, but it’s something that’s very satisfying for me.

Speaking of drawing connections between different types of objects, can you tell us more about the ten author paper about potential automorphy?

Just to give a brief summary, Calegari and Geraghty had a great insight in 2011 for how to extend the famous Taylor-Wiles method for modularity lifting to a very general setting. This depended on three very difficult conjectures about torsion in the cohomology of locally symmetric spaces. In 2013, Scholze proved the first of those conjectures, by constructing Galois representations associated to those torsion classes. By early 2016, Scholze and I had made definitive progress towards the second conjecture.

Once Taylor found out about this, he suggested to me that we organise a week-long working group to see if we can implement the Calegari-Geraghty method unconditionally, even in some very restricted settings. The working group was held late 2016 at the Institute for Advanced Study (IAS) in Princeton. It was successful beyond our expectations, and we were able to push the argument through in significant generality. By Thursday afternoon, we were done doing mathematics and went on a long hike in the IAS woods. We wrote up the results over the next two years in the so-called “ten author paper”.

One of the things that I am happiest about regarding the ten author paper is that I think it was successful as a collaboration, mixing younger and older, more established mathematicians and the younger people especially contributed a lot. I also think that this paper is just the beginning, people are already pushing the methods we developed further, and I look forward to seeing where this will all go.

You collected a number of prizes: what do awards and prizes mean to you?

I think that’s something incredibly important for a young mathematician to feel you’re recognised by the community and that your work is appreciated. It’s encouraging. Sometimes you’re stuck on a problem for a long time and maybe you’re having a bad day, but then you can look back and say: Ok, but I was able to solve that problem, I was given that prize. And you get the confidence to continue.

In your CV you mention that you turned down a fellowship because of maternity leave, and we think it’s important you wrote it, because people should be aware of these issues.

I got pregnant right at the time when we moved to London and it was a complicated period. Something I want to stress about this experience is that my husband, who is also a mathematician, takes on more of the childcare than I do. And this makes a huge difference to my ability to continue my work as a mathematician.

He really does more than half the work and I cannot understate how important this is for me. You can never really do half and half, but it’s so common that women take on more and you’d really want there to be more couples where the men share more.

You started working at Imperial College around the same time you had a child. How was your experience moving there?

I have a very close mentor and collaborator working at Imperial, Toby Gee, who recruited me to come here. My experience is great, and it is great also because he worked very hard to make sure that I feel welcome here.

Speaking of mentorship, you mentioned earlier how Wiles giving you a very difficult problem while you were an undergraduate had an impact on your way of being a mathematician. Are there other experiences that played a big role in your career?

I guess the hardest point in my career was when I finished my PhD. At that point, I had no idea what to work on next. I really had no idea: I finished this one problem and I did not know what I would be able to work on next. And I also felt that I didn’t have anyone older who could give me advice. So that was a really, really hard point in my career.

But, I ended up with a great post-doctoral mentor who put me on a slightly different path from the one I had followed during my PhD and so I could go on. And that was a great experience because I learned something new: my thesis was on Shimura varieties, but afterwards I wrote a paper on p-adic Langlands and that taught me some new mathematics and it also taught me that I am good at learning new mathematics quickly.

It was a very hard experience, but once I was able to overcome it, new horizons really opened up: I felt I did not have to work on just one particular subfield and I could pretty much choose to work on anything I wanted. And I think I probably wouldn’t have been as successful as a mathematician today if I had just continued working in only one subfield.

So, in some sense, you’re saying that it’s important to meet the right people to mentor you.

Yes, it’s absolutely important. And I don’t want to say it was just luck because it wasn’t just luck. There was a mathematician at the University of Chicago, Matthew Emerton, and I knew that he was extremely friendly and extremely generous with his ideas and I decided I wanted to do a post-doc with him. I went to Chicago for one year as a post-doc while my husband stayed back in Boston for his post-doc. I really felt that it was important to go where someone would expose me to new ideas. After my year in Chicago, I moved to Princeton and Sophie Morel became my unofficial mentor.

Are you also mentoring someone? How is your experience being on the other side?

I have three students and one postdoc and I’m getting a few more next year: I really try to do for them what my mentors have done for me, I try to give back. Probably it will take some time until I become a really good mentor but I’m excited about this. I like talking to younger people and I think they’re stronger than I was at their age.

Something I also really enjoyed was the European girls maths Olympiad in Romania in 2016, where I was invited to give a guest lecture. Talking to hundreds of girls who are really interested in maths about what kind of mathematics there is beyond the Olympiads, it’s one of the most rewarding things I have done.

I think that it is important to get young women interested in math, but it is also important to support them at the PhD level and at the post-doc level because that’s where I felt it was hardest for me. Now that my daughter is a little bit older and I have more time, one thing that I want to do is try and establish a mentoring program for PhD students in London, because we actually have a good percentage of women PhD students in number theory and in geometry.