Here is a short description of my journey as a mathematician. It is mainly aim at students interested in writing a bachelor’s or master’s thesis with me. Please read this, think clearly what your intrests are, and then feel very welcome to get in touch.
As a kid I was trying (unsuccessfully!) to solve Fermat’s Last Theorem. My research interests have changed over the years, but they somehow always come back to number theory and its neighbours. Concretely, I worked on Tate–Shafarevich groups in my master’s thesis (PDF). My Ph.D. was about profinite rigidity of arithmetic lattices (see my first two papers, on profinite completions and Kazhdan’s property (T) and on arithmetic groups with isomorphic finite quotients), and also the beautiful survey by Alan Reid on profinite properties of discrete groups (PDF).
Towards the end of my graduate studies, Uri Shapira introduced me to homogeneous dynamics through the study of continued fractions. See our joint paper “On the evolution of continued fractions in a fixed quadratic field” and its sequel on Continued Fractions of Arithmetic Sequences of Quadratics. Both papers use the connection between geometry, dynamics and number theory that is explained, for example, in Caroline Series’ classic paper The modular surface and continued fractions (JLMS page), in Allen Hatcher’s book Topology of Numbers (book page), and in Chapter 3 and §9.6 of Einsiedler–Ward’s Ergodic Theory with a View Towards Number Theory (Springer).
These themes (group theory, number theory and homogeneous dynamics) come together in my subsequent work with Emmanuel Breuillard, Nicolas de Saxcé and Lior Rosenzweig on Diophantine approximation in Lie groups. See for example our papers Diophantine properties of nilpotent Lie groups (arXiv:1307.1489), On metric Diophantine approximation in matrices and Lie groups (GAFA version, arXiv:1410.3996) and the short companion note in Comptes Rendus (journal page).
Later I became heavily involved in applications of the joinings classification of Einsiedler and Lindenstrauss. My Bourbaki exposé “Joinings classification and applications (after Einsiedler and Lindenstrauss)” surveys this theory and some arithmetic applications. This is still a very active area, and we keep finding new connections. For instance, in joint work with Manuel Luethi, Philippe Michel and Andreas Wieser we studied simultaneous supersingular reductions of CM elliptic curves, while in joint work with Peter Feller, Alison Beth Miller and Andreas Wieser we related binary quadratic forms to knot theory via Seifert surfaces in the four-ball and composition of binary quadratic forms.
Recently I have become very interested in the geometry of unit groups and their “shapes”. One external reference that I particularly like is the paper by Jialun Li, Nihar Gargava and Thi Dang on shapes of unit groups in the cubic setting: “Density of shapes of periodic tori in the cubic case”.
I am also very interested in illustrating mathematics. Students who would like to work on concrete mathematical illustrations (for example of number-theoretic or dynamical phenomena) are very welcome to contact me. A beautiful example in this direction is Katherine Stange’s work on visualizing imaginary quadratic fields (“Visualising the arithmetic of imaginary quadratic fields”). I will participate in the upcoming IHP trimester program Illustration as a Mathematical Research Technique (program page), and I am always happy to supervise projects that combine serious mathematics with serious illustration.