Dr. Nicolas Mascot

  Algebra   Algebraic geometry   ALGORITHMS   Arithmetic geometry   Computational mathematics, discrete mathematics   Computer Science/Engineering   Galois representations   Mathematics of computing   Number Theory   PARI/GP   Pure mathematics

I am a computational number theorist. In a nutshell, I invent and implement algorithms to compute with mathematical objects which occur in algebra, number theory, and arithmetic geometry. I then publish these articles in peer-reviewed journals, and also make these algorithms accessible to the mathematics community by integrating them in number theory computer packages - mostly PARI/GP, a major open-source computer algebra system which received the 2021 Richard D. Jenks Memorial Prize for Excellence in Computer Engineering Applied to Computer Algebra, and to which I have contributed ca. 20,000 lines of code so far. My research is driven by the fact that central objects in modern number theory, such as Galois representations and etale cohomology, remain mysterious in spite of their major theoretical role, because they are never explicitly visualised, and are thus shrouded in a reputation of abstractness and difficulty. Yet I firmly believe that profound understanding of mathematics requires a crystal clear representation of the objects at play, which is best achieved by the study of explicit examples. My research therefore focuses on computing explicitly such objects, so as to lift the veil of abstract mystery off them, to get better acquainted with them, thus helping the community to progress by clarifying the big theoretical picture (Langlands programme) in which they play an essential role. This is essential, as the field is largely driven by phenomenology, so that computational tools are critical. More specifically, my work has mostly focused so far on explicit methods to handle Jacobian varieties of curves. These Jacobians are the instance of etale cohomology corresponding to the specific case of curves. Although they are directly attached to curves, they are immensely less accessible and more difficult to handle algorithmically than the curves themselves. My research has succeeded in developing efficient algorithms to overcome this. Recently, I have even managed to generalise my algorithms to the etale cohomology of higher-dimensional objects (such as surfaces instead of curves).