Section: Research Program

Topology in small dimension

Character varieties

The brute force approach to computable objects from topology of small dimension will not allow any significant progress. As explained above, the systems that arise from these problems are simply outside the range of doable computations. We still continue the work in this direction by a four-fold approach, with all three directions deeply inter-related. First, we focus on a couple of especially meaningful (for the applications) cases, in particular the 3-dimensional manifold called Whitehead link complement. At this point, we are able to make steps in the computation and describe part of the solutions [79], [89]; we hope to be able to complete the computation using every piece of information to simplify the system. Second, we continue the theoretical work to understand more properties of these systems [77]. These properties may prove how useful for the mathematical understanding is the resolution of such systems - or at least the extraction of meaningful information. This approach is for example carried on by Falbel and his work on configuration of flags [80], [82]. Third, we position ourselves as experts in the know-how of this kind of computations and natural interlocutors for colleagues coming up with a question on such a computable object (see [87] and [89]). This also allows us to push forward the kind of computation we actually do and make progress in the direction of the second point. We are credible interlocutors because our team has the blend of theoretical knowledge and computational capabilities that grants effective resolutions of the problems we are presented. And last, we use the knowledge already acquired to pursue our theoretical study of the CR-spherical geometry [69], [81], [78].

Another direction of work is the help to the community in experimental mathematics on new objects. It involves downsizing the system we are looking at (for example by going back to systems coming from hyperbolic geometry and not CR-spherical geometry) and get the most out of what we can compute, by studying new objects. An example of this research direction is the work of Guilloux around the volume function on deformation varieties. This is a real-analytic function defined on the varieties we specialized in computing. Being able to do effective computations with this function led first to a conjecture [86]. Then, theoretical discussions around this conjecture led to a paper on a new approach to the Mahler measure of some 2-variables polynomials [88]. In turn, this last paper gave a formula for the Mahler measure in terms of a function akin to the volume function applied at points in an algebraic variety whose moduli of coordinates are 1. The OURAGAN team has the expertise to compute all the objects appearing in this formula, opening the way to another area of application. This area is deeply linked with number theory as well as topology of small dimension. It requires all the tools at disposition within OURAGAN.

Knot theory

We will carry on the exhaustive search for the lexicographic degrees for the rational knots. They correspond to trigonal space curves: computations in the braid group $B_3$, explicit parametrization of trigonal curves corresponding to "dessins d'enfants", etc. The problem seems much more harder when looking for more general knots.

On the other hand, a natural direction would be: given an explicit polynomial space curve, determine the under/over nature of the crossings when projecting, draw it and determine the known knot (for example the first rational knots are listed at https://team.Inria.fr/ouragan/knots) it is isotopic to.

Vizualisation and Computational Geometry

As mentioned above, the drawing of algebraic curves and surfaces is a critical action in OURAGAN since it is a key ingredient in numerous developments. In some cases, one will need a fully certified study of the variety for deciding existence of solutions (for example a region in a robot's parameter's space with solutions to the DKP above or deciding if some variety crosses the unit polydisk for some stability problems in control-theory), in some other cases just a partial but certified approximation of a surface (path planning in robotics, evaluation of non algebraic functions over an algebraic variety for volumes of knot complements in the study of character varieties).

On the one hand, we will contribute to general tools like ISOTOP (https://isotop.gamble.loria.fr) under the supervision of the GAMBLE project-team and, on the other hand, we will propose ad-hoc solutions by gluing some of our basic tools (problems of high degrees in robust control theory). The priority is to provide a first software that implements methods that fit as most as possible the very last complexity results we got on several (theoretical) algorithms for the computation of the topology of plane curves.

A particular effort will be devoted to the resolution of overconstraint bivariate systems which are useful for the studies of singular points and to polynomials systems in 3 variables in the same spirit : avoid the use of Gröbner basis and propose a new algorithm with a state-of-the-art complexity and with a good practical behavior.

In parallel, one will have to carefully study the drawing of graphs of non algebraic functions over algebraic complex surfaces for providing several tools which are useful for mathematicians working on topology in small dimension (a well known example is the drawing of amoebia, a way of representing a complex curve on a sheet of paper).