GeoSpec

Scientific description

Mathematical complexity theory involves classifying objects or problems based on the difficulty of understanding or solving them. It is a well-known area of research in applied mathematics, particularly in numerical analysis and combinatorics, where estimating the number of steps required to compute a quantity is crucial for computer implementation. 

In more fundamental areas of mathematics, such as geometry or dynamical systems, the exact geometry of an object is rarely known with sufficient precision: it may, for example, change over time, as is the case with many mechanical parts, or simply be unknown a priori, such as the structure of the Universe. For these reasons, the characteristics of an object are often evaluated in terms of associated algebraic or analytical quantities (for example, the rate of decay of the solution to a partial differential equation, etc.), which pave the way for a measure of its complexity.

The goal of this project is to foster collaboration between researchers in pure and applied mathematics in order to study important problems related to the complexity of two-dimensional objects (such as surfaces) and to initiate research in higher dimensions, where the situation remains largely unknown for most problems. In this project, we aim to study the geometric complexity of a mathematical object (for example, a manifold) from three perspectives: its metric (i.e., the way distances are measured on the object), its dynamics (i.e., the trajectories followed by particles moving on it), and its spectrum (i.e., its resonance frequencies). More specifically, we plan to study extremal manifolds for the invariants describing this complexity in each of the above aspects.

Focus on a result

The study of isometric embeddings is closely linked to the regularity of the objects in question. As suggested by Nash and Kuiper’s theorem, any admissible embedding is expected to exhibit irregular behavior, such as that seen in Cantor sets. Our first contribution concerns the parameterization, using as few parameters as possible, of such irregular surfaces. Below, we illustrate a recursive process capable of generating irregular objects with only a few thousand parameters. The bottleneck of this approach lay in the combination of computational geometry tools and sparse representation techniques. This breakthrough is promising and crucial for the analysis of the spectral properties of integrated surfaces.

Scientific activities

Baptist Trey began his doctoral thesis under the supervision of E. Russ (IF) and B. Velichkov (LJK) in September 2016.

We have already invited two candidates to apply for our funded postdoctoral fellowship for 2017–2018. 

We are organizing a CIRM meeting in Luminy in February 2017 and a summer school in Grenoble in 2018. 

Our team is already holding a weekly meeting as part of the working group on “Isoperimetric inequalities in metric spaces” organized by H. Pajot and G. Besson.

Coordinators

Gérard Besson (Fourier Institute)

Edouard Oudet (LJK)

Members

Dorin Bucur (LAMA)

Charles Dapogny (LJK)

Pierre Dehornoy (Fourier Institute)

Erwan Lanneau (Fourier Institute)

Emmanuel Russ (Fourier Institute)

Boris Thibert (LJK)

Bozhidar Velichkov (LJK)

Notable publications

M. Bonafini, G. Orlandi, and E. Oudet, “Variational approximation of functionals defined on 1-dimensional connected sets: the planar case,” submitted.

F. Hamel, E. Russ, "Comparison results for semilinear elliptic equations using a new symmetrization method," to be published in Math. Ann.

E. Lanneau, D.-M. Nguyen, and A. Wright, “Finiteness of Teichmüller curves in non-arithmetic rank-1 orbit closures,” to appear in the American Journal of Mathematics (2016).

E. Lanneau, F. Valdez, “Computing the Teichmüller Polynomial,” to appear in J. Eur. Math. Soc. (2016).