From EWM Newsletter 21 (2012/2)

I was born in December 1979 in France. I received my PhD in December 2005 under the direction of Gerard Laumon at the Universite de Paris Sud in Orsay. The title of my thesis was “Complexes d’intersection des compactifications de Baily-Borel – le cas des groupes unitaires sur Q”. I was a member of the Institute for Advanced Study for the academic years 2006-2007, 2007-2008, 2008-2009 and 2010-2011. In August 2009, I came to Harvard as a visitor, and I became a professor there in December 2009. In September 2012 I moved to Princeton University. I have also been a Clay Research Fellow from October 2006 to October 2011.

EWM: Can you describe your research area and what in particular you find most interesting to study?

SM: I work in what is called the Langlands program. The Langlands program is a vast network of conjectures (some of them theorems now) that involves objects from algebraic number theory, arithmetic geometry and representation theory (both for real or complex Lie groups and for locally compact disconnected groups). There are many different flavours. For example, the local Langlands correspondence relates irreducible representations of the group of invertible n by n matrices with coefficients in Q_p (the field of p-adic numbers) and n-dimensional representations of the absolute Galois group of Q_p (that is, the group of automorphisms of an algebraic closure of Q_p that fix Q_p). As we have different kinds of information about the two sides, this allows us to better understand them both. There is also a (partly conjectural) global Langlands correspondence that involves representations of the absolute Galois group of the field of rational numbers Q and certain representations of the group of invertible n by n matrices with adelic coefficients (called automorphic representations). The proofs of many of these statements involve the study of the cohomology of algebraic varieties, especially so-called Shimura varieties, the simplest example of which are modular curves. Here the cohomology can be Betti cohomology, etale cohomology, L^2 cohomology or the cohomology of well-chosen vector bundles. Another big player is the Arthur-Selberg trace formula, that is a statement (or rather a long list of very hard theorems) in representation theory and analysis. And then there are versions of the Langlands conjectures over function fields, from functions fields over finite fields to functions fields over the complex numbers, where we get the geometric Langlands program, that contains a lot more algebraic geometry. So it is a very technical field because there is so much you must know, but it also contains problems for many different kind of tastes. It is also about finding all these surprising connections between objects that seemed very different, and that is deeply satisfying. I am probably most interested in etale cohomology of Shimura varieties at this time, but I have also done work with the trace formula to apply some of my results.

EWM: You were recently awarded a prize at the recent ECM, can you tell us something about the work that won this prize? How do you think that this prize will influence your career?

SM: I assume that the work that won this prize is what I did in my PhD dissertation, since that has been my most original work so far. It was about calculating the intersection cohomology of singular compactifications of smooth varieties (mostly Shimura varieties). Intersection cohomology (first introduced by Goresky and MacPherson) is a cohomology theory that is the same as Betti or etale cohomology for smooth varieties, but behaves much better for singular varieties; the only problem is that it can be very difficult to calculate in general. I was lucky enough to stumble on a very simple trick that, for the compactifications of Shimura varieties, allowed to easily reduce the calculation to known results about cohomology with compact support of Shimura varieties.

As for the influence on my career, I am not sure what to tell you. I have permanent positions in good places for almost three years now. I think that if you want a prize to help people’s careers, then you have to give it to younger people who are still struggling to get a good job.

EWM: You have studied and worked in France and USA. What can you tell us about your living/working experience in these environments?

SM: To be precise, I have studied in Paris and worked in Princeton and Harvard. I doubt that I have a good idea of the American system in general. I liked being a student in France, but then I went to the Ecole Normale Superieure, and that means that I did not have to attend a class in a university until I was almost a grad student. I found being a teaching assistant in Paris quite depressing, while teaching at Harvard was very enjoyable, even the lower-level classes for non-mathematicians. Positions in France are (almost) all permanent, so I was afraid of the American tenure-track system and I am far from sure that I would have stayed here if I had had to go through it. My salary here is much higher than what it would have been in France, which I must admit is nice, though not the reason I went into mathematics research, and also not the reason I left France.

EWM: Can you share with us about being a (young) professor at Harvard? What about being the only female senior research professor at the Math Department?

SM: Each I only was a Harvard professor for two and a half years, and I often felt that, as a French person, I did not have the same reverence for Harvard as Americans seem to have. In any case, I never really felt that I was imposing enough for the job; it probably does not help that I look much younger than I am (or so I am told by the people who routinely ask where I go to college). When people outside of work ask me what I do, I usually answer something like “I teach maths” and try to leave it at that, unless provoked. I always felt better at ease with the junior faculty – postdocs, assistant professors and preceptors – and the graduate students than with the other professors; if they were annoyed by my presence, they were gracious enough not to let it show. Maybe that would have changed as I got older, but then I left.

As for being the only female research professor, I might not even have noticed if it had not been pointed it out to me so many times. I just prefer not to talk about it; being hired by Harvard was nice, but being the first woman was just a matter of chance and pretty meaningless as far as I am concerned.

EWM: What do you like to do in your free time?

SM: My main activity is probably reading. I like reading novels and learning foreign languages, then reading the same novel in different languages. A few years ago I took up running as a way to keep in shape, then got hooked and started running half marathons and marathons. In the summer, I prefer to hike in the mountains in France (but not in the USA, as I am very afraid of bears). I do other things too, but those three take up the most time.