The French mathematical community represented at the 9th European Congress of Mathematics (ECM 2024)

● Hello, can you introduce yourself?

I've been a Professor at Université Paris Cité since 2010, carrying out my research within the "Groups, representations and geometry" team at the Institut de Mathématiques de Jussieu-Paris Rive Gauche. Before that, I was a CNRS research fellow from 2005 to 2010.

● What is your area of research?

My research area is the theory of representations, in particular infinite-dimensional Lie algebras and quantum groups, in connection with problems of a geometric, combinatorial or mathematical-physics nature.

● What drew you to mathematics? Were there any decisive encounters in your career, or results that profoundly marked your relationship with mathematics?

I've always loved science, and mathematics in particular. But it was only at the Ecole Normale Supérieure that I discovered that it was possible to become a professional mathematician. I hesitated for a long time over physics, which I studied right up to my master's degree. So I was delighted to discover a field of fundamental mathematics, algebraic and geometric in nature, in touch with questions of theoretical physics. In addition to Marc Rosso, who directed my thesis, other leading experts in the field played a decisive role in my career. Some went on to become collaborators. I'll mention just a few: Vyjayanthi Chari, Corrado De Concini, Edward Frenkel, Michio Jimbo, Bernard Leclerc, Hiraku Nakajima, Nicolai Reshetikhin and Olivier Schiffmann. More recently, collaborations with Hironori Oya and Ryo Fujita, whom I was fortunate enough to have on my team as post-docs, have enabled me to resolve long-standing issues.

Some of their articles have also been remarkable. For example, in a 1998 publication, Edward Frenkel and Nicolai Reshetikhin highlighted the deep links between certain categories of quantum group representations and a wide range of fields (integrable systems, Langlands programs, W-algebras, etc.). I've read and studied this article many times, yet I continue, year after year, to discover its subtleties. One of my main results, obtained in 2015, is in fact the proof of one of the conjectures stated therein.

● What do you love about being a mathematician? How would you describe it to someone outside research?

I like many aspects of being a mathematician, first and foremost the fact of being in contact with the raw material, mathematics itself, and being able to explore new paths that no one else has taken. I also like to see general results gradually emerge, and to put into perspective examples manipulated by hand. Contact with students, whether thesis students or younger, is also one of the pleasures of this job, as is working in collaboration with colleagues in France and abroad, which brings the great joy of finding things together. Understanding how mathematics is thought of from one country to another is also one of the very enriching aspects of these exchanges.

● Do you already know what you'll be talking about at ECM in July? What does this congress mean to you?

I haven't yet chosen the topic of my talk, but I'll try to highlight the variety of themes with which the categories I'm interested in interact. The ECM is a prestigious congress that provides a great opportunity to bring together mathematicians working on a variety of themes; it's also a chance to affirm the importance of the European scale for the mathematics community.

● Is there a message you'd like to pass on?

Don't be unnecessarily impressed or intimidated - if you love mathematics, go for it!

● Hello, can you introduce yourself?

After a master's degree in Timisoara, Romania, I did a DEA in Nice, followed by a thesis at the Ecole des Mines de Paris. In 1995 I was recruited as a research fellow at the CNRS in Besançon, and since 2002 I've been a professor, first in Metz, and now at the Université Savoie Mont Blanc. In 2014 I became a senior member of the Institut Universitaire de France for 5 years.

● What is your area of research?

I work on mathematical questions at the interface between theory and applications, which engineers might call "shape optimization". In more precise language, these are analytical questions around problems with free boundaries or discontinuities. This is part of the broad field of applications of the calculus of variations in the study of questions (often) arising from physics, whose modeling mixes geometry with partial differential equations. I pay particular attention to problems related to spectral geometry.

● What drew you to mathematics? Were there any decisive encounters in your career, or results that profoundly marked your relationship with mathematics?

By chance. Following a family tradition, I was destined to go into medicine, but a family friend, a math teacher, urged me to take part in the Maths Olympiads. And I liked it...

● What do you like about being a mathematician? How would you describe it to someone outside research?

Research work in mathematics, even if we don't say so, is often a competition. It's just about the only discipline where there's no ambiguity about results, and subjectivity in their interpretation is virtually non-existent. If you demonstrate a result, whether you're at Harvard or Chambéry, it's worth the same. When it comes to teaching, I like to think of each lesson as a performance in front of an audience who, in the end, must be happy. It's hard, and at the same time, when it works, it's a joy.

● Do you already know what you'll be talking about at the ECM in July? What does this congress mean to you?

I haven't prepared my presentation yet, but it will probably revolve around my work on optimal shapes in spectral geometry. I'll be taking particular care to ensure that my talk is not too specialized and that it's accessible to a wider audience.

I am, of course, very honored to have been invited. Yet I have an unsentimental and rather technical approach to my participation.

● Is there a message you'd like to get across?

Whenever the opportunity arises, typically during "Mathematics Week", I'm delighted to give talks in high schools to promote mathematics. In today's world, mathematics is taking on a central role. At the same time, young people's lack of interest in mathematics is becoming increasingly marked. So I'd say to my fellow mathematicians: don't hesitate to give up your time, go into high schools and colleges and promote our beautiful discipline and its applications in everyday life.

● Hello, what's your area of research?

My main area of research is topological data analysis, commonly known as TDA forTopological Data Analysis. It's a field straddling mathematics and computer science whose aim is to provide methods based on topological invariants for data mining and machine learning. What distinguishes this approach to data analysis is that it is based on theoretical guarantees of invariance, stability and (to a certain extent) interpretability, and can be integrated with existing learning models, whether statistical or neural network-based, to enrich them. To achieve this, TDA uses and adapts a wide range of mathematical tools from fields as diverse as algebra and statistics, topology and geometry, non-smooth analysis, optimization and optimal transport.

● What drew you to mathematics? Were there any decisive encounters in your career, or results that profoundly marked your relationship with mathematics?

Let's just say that I came to the mathematics I practice today via the schoolboy route, and that my career path is intimately linked to the evolution of my field.

I trained as a computer scientist, with a thesis in algorithmic geometry. It was during my postdoc, in the mid-2000s, that I met the small group of enthusiasts interested in the then nascent subject of ADT, led by Gunnar Carlsson (Stanford) and Herbert Edelsbrunner (Duke). Enthusiasm being contagious, I joined them, along with many other researchers from very different backgrounds, and together over the last twenty years we have developed the mathematical and algorithmic foundations of ADT.

We can't do justice to all those who have contributed to this collective effort, but in my direct circle I can at least mention Frédéric Chazal, with whom I had a very fruitful collaboration for several years on some of the key concepts and results of the field. I'd also like to underline the essential contribution of the younger generation to the most recent developments in ADT, particularly on algebraic aspects.

In addition, a series of meetings between our community and experts from other fields have strongly influenced the subject and opened it up to the outside world, including algebra representation theory with William Crawley-Boevey (Bielefeld) and Steffen Oppermann (NTNU), beam theory with Pierre Schapira (IMJ) and François Petit (then at the University of Luxembourg), optimal transport with Marco Cuturi (then at ENSAE), statistics and geometric inference with Pascal Massart (LMO) and Larry Wasserman (Carnegie Mellon). At the same time, TDA has been able to produce a number of notions and results of its own, such as the concept of interleaving, which makes it possible to define metrics between objects of an algebraic nature, notions that have spread to other communities, bringing a fresh perspective to some of the questions that arise there.

● What do you like about being a mathematician? How would you describe it to someone outside research?

What I like most is being able to link together concepts or results from different fields, and TDA is very conducive to this. For example, the objects it mainly looks at at the algebraic level, called persistence modules, can be seen in various ways: as modules on algebras, as functors, as representations of quivers, or as bundles. Each point of view is linked to a particular domain and gives access to a specific toolbox. For example, seeing persistence modules as bundles enables us, under certain conditions, to use Grothendieck's six operations formalism to manipulate them. On the other hand, when possible, immersing beams in persistence modules enables us to benefit from optimized computational tools for efficient decomposition and resolution. In this way, fruitful interactions are established between fields that are a priori different, and which benefit everyone.

● Do you already know what you'll be talking about at ECM in July? What does this congress mean to you?

The fact that the scientific committee has chosen to include a presentation on ADD at ECM is, I think, a great recognition for the field. As a cross-disciplinary subject, TDA has taken a long time to find its place in the mathematical landscape, but today we have a pretty clear idea of what it borrows from other subjects and what it can bring to them in return. A remarkable example of this is a recent result by Lev Buhovski and colleagues (https://arxiv.org/abs/2206.06347), which combines the theory of persistence modules with analytical tools to generalize Courant's theorem on the counting of nodal domains of eigenfunctions of the Laplacian operator. It's a result that is at once surprising, non-trivial, and whose proof has led as well to interesting developments for TDA itself. I mention it because Lev Buhovski is also a guest speaker at this year's ECM and I'm confident he'll talk about it there.

For my part, I'd like to highlight the transverse side of TDA in my talk. To do this, I plan to talk about recent developments around the definition of a new class of invariants for persistence modules, called homological invariants, which relies on approximations in the sense of relative homological algebra. I hope to show how, with the help of such invariants, we can pose a diffeological space structure on categories of persistence modules for differential calculus and optimization, with interesting applicative results in machine learning.

● Hello, can you introduce yourself?

I am a mathematician at Institut des Hautes Etudes Scientifiques. I grew up in China and came to France for Classe Prépa in Lyon at age 18. After that, I studied at ENS Paris and obtained my PhD from ETH in Zurich in 2019. I then worked as a postdoc at MIT and MSRI in Berkeley before coming back to France. I consider myself a world citizen, a multicultural person, and constantly open to new adventures.

● What is your field of research?

My research lies at the interface of probability, complex analysis, and geometry.

● What drew you to mathematics? Were there any decisive encounters in your career, or results that profoundly marked your relationship with mathematics?

I was curious about many things, but only mathematics has given me a satisfying answer (within the realm of mathematics), both intellectually and aesthetically. Mathematics seeks to understand the fundamentals. It is often a luxury to be allowed to spend as much time as needed to think about the fundamentals and question my understanding again and again. I realized that this is the way of thinking that I like the most. So, I pursued the path toward mathematical research and have felt very little resistance since then. Unfortunately, I don’t have any role model to whom I find myself similar. But I felt since I was young that mathematics was part of my nature. So, after learning that mathematics could be a career, I was happy to try it out.

● What do you love about being a mathematician? How would you describe it to someone outside research?

I really like the freedom to pursue our curiosity with no restriction, the encouragement to challenge our understanding, the euphoria at every moment of getting a bit more clarity, and the community of like-minded people with whom we can share all the pure excitement. I also enjoy being part of the international community, getting to travel around the world, and constantly discovering new ideas and new cultures.

● Do you already know what you'll be talking about at ECM in July? What does this congress mean to you?

I have not completely decided. It is an honor to be invited to ECM. I consider it as an opportunity to report to a broader mathematical audience about a subject I am fascinated about.

● Is there a message you'd like to pass on?

Listen to your heart, and don’t let anyone tell you what you can and cannot do. Don’t judge yourself too soon. The qualities that make an excellent mathematician may be more diverse and complicated than expected. The ones who stayed until the end may not be the quickest or the smartest, but those who truly loved and persisted. 

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● Hello, can you introduce yourself?

I'm a mathematician working at IHES, a research institute in mathematics, theoretical physics and all related sciences, in Bures-sur-Yvette, in the south of Paris.

● What is your area of research?

Broadly speaking, I am interested mathematical physics and algebraic geometry. I often draw my mathematical inspiration from physical theories such as string theory or quantum field theory. Lately, I have been working on aspects of birational geometry, resurgence theory, Floer homology and Chern-Simons theory.

● What drew you to mathematics? Were there any decisive encounters in your career, or results that profoundly marked your relationship with mathematics?

I was drawn to mathematics by my older brother who started to give me mathematical riddles when I was around 10. I also very much enjoyed popular mathematics books such as the ones by Martin Gardner. I went on to attend math classes for gifted students, and participated in the Russian mathematical Olympiads. When I was in my last year of high-school, I was introduced to Israel Gelfand. At University, I started to participate in his seminar which influenced me a lot. Neither program nor end times were set in advance for the lectures, and Gelfand felt free to interrupt the speaker at any moment to ask him to explain certain points in more detail. Scamming the room to check that everyone was keeping up, Gelfand would frequently improvise and launch into unpredictable digressions. After the seminar, older students of Gelfand, such as Alexander Goncharov, often stayed and reexplained the seminar to the younger students. Alexander Goncharov is a friend of mine to this day. He has had a huge influence on me as well.

● What do you like about being a mathematician? How would you describe it to someone outside research?

I very much enjoy the freedom I am granted in my research. This is particularly true at IHES, the institute I am working at since 1995, where I am freed from all administrative tasks and have no teaching obligation, and can devote myself entirely to my research.  I also very much enjoy collaborating with colleagues and students. People often think that mathematics is a lonely activity, but this is far from being true. Most of my papers are written with at least one other colleague of mine.

● Do you already know what you'll be talking about at ECM in July? What does this congress mean to you?

I will talk about the generalized Riemann-Hilbert correspondence, wall-crossing and resurgence. The congress is a good opportunity to learn about work from other colleagues, and to meet and discuss their results with them. 

● Is there any message you'd like to pass on?

My advice to young mathematicians: do not exclusively focus on one problem. If you are stuck, try working on a different problem, and get back to your initial problem later. Working on different problems might help you find analogies or different viewpoints, and find unexpected solutions.

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