Mathematical Interests
1. Geometrical Analysis of Optimal Shapes and PDEs. Geometric variational problems arise in various applications e.g. in physics, engineering and in biology. Classical examples are the shape of solid bodies minimizing air resistance, area minimizing surfaces with prescribed boundary properties or the shape of membranes giving rise to the lowest fundamental mode. In current mathematical language, these and other geometric variational problems can be formulated as optimization problems for domain dependent functionals on given classes of subdomains of Rn or of Riemannian, Hyperbolic manifolds with or without boundary. They are usually called shape optimization problems and most generally we are interested by geometrical analysis of optimal shapes problems that involve other types of questionings such as symmetry, convexity, over determined problems, isoperimetric inequalities etc.
The main questions in the study of shape optimization problems are the existence of optimizers, their classifications and qualitative properties of the solutions provided they exist. While in some cases, existence results are guaranteed by the direct method of the calculus of variations, in some others several, interesting questions remain open. However, since, in general, optimal sets belong in the class of stationary sets of the energy (domain dependent functional: perimeter, eigenvalues, etc.) under consideration, it is natural to investigate, say to characterize, all stationary sets. This in general leads to nonlinear PDEs (prescribed constant mean curvature problem) or overdetermined PDEs (for the Dirichlet, Neumann eigenvalues, etc.). Our attention is drawn by these main questions driven by local or non local operators.
Let's point out the Cheeger constant, having link with overdetermined problems, that is a geometric quantity introduced by Cheeger in 1960 to get lower bound of eigenvalues on Riemannian manifolds. It has several applications for instance in image processing and land slides. We will describe free boundary geometric problems involving Cheeger constant motivated by models from plate failure. The study of open questions related to non local operators and isoperimetric problems draw our attention. Inverse problems are involved in lot of important questionings. And there are a lot of ones that can be formulated as overdetermined problems.
The Senegalese coastal is among the most vulnerable sectors to climate change. Interesting works concerning costal erosion or encroachment have been done. We can quote, on the one hand, those for the non local models due to A.C. Fowler P. Azérad and al, and on the other hand, works on local models proposed by I. Faye, E. Frénod and D. Seck. One of the goals (Tasks 1, 2, 3, 4) of the NLAGA will be to go on developing mathematical analysis to understand complex systems and to consequently build forecasts of it, through evolution Partial Differential Equations about the involved processes that are the main causes of coastal erosion.
The advance of the deserts is another important and interesting problematic on which we already work. This question, also, directly implies negative consequences on life and the development of the human corporations (rarity of water, impossibility to take agricultural activities, exodus of the populations ...) specially in Western Africa where one notices a notorious progression of the Sahara desert.
2. Optimization ad Geometric Analysis. These last years we have investigated questionings related to optimization and optimal mass transportation. We plan to go on these efforts by adding other questions that can be linked to branched transport and shape optimization. Branched transport is a domain of optimal transport problems, say a variant of the classical Monge Problem, where the cost for shipping a mass m over a length l is proportional to l but is not linear in m. Its name branched stands for the typical shape of the optimal solutions, where masses are shipped together as far as possible and only branch at the end of the transportation to reach their respective destinations. This model fits particularly well the costs of realizing a road network, but is also used by geophysicists as a reference for river basins and models biological systems (such as blood vessels) as well.
A part of the NLAGA project is devoted to optimal mass transportation, optimization and numerical aspects related. At first, we focus on the numerical aspects of urban traffic problems. In fact, our effort will be concentrated on the numerical questions to locate the optimal network in a given domain and to optimize amenities location. Another problem that we will be interested is to give contribution on the numerical approach of the Monge and Kantorovitch problems that are not completely investigated. Secondly, for the cashew sector in developing country such as Senegal, we aim to propose a quantitative approach for quickly making operational decisions; and illustrate how can be developed medium and short term plans while focusing on the three facets economic, societal and logistics.
3. Geometry and Applications. One topic of our research concerns the resolution of Differential Equations's in Lie groups, and remains a logical continuation of the previous work on automorphisms of cotangent bundles of Lie groups. We search triplets J1; J2; J3 of left invariant tensors on a Lie algebra G that satisfy the conditions of quaternions. One says that they form a hyper complex structure if in addition, they are all integrable. Additionally, we must assume the existence of a metric g compatible with all the Ji at the same time, then we say that we have a hyper Kähler structure.
We aim also to analyse the local and global properties of almost-Riemannian structures. A n-dimensional almost-Riemannian structure is a generalized Riemannian structure defined locally by n vector fields that play the role of an orthonormal frame, but could become collinear on some set Z called the singular set. Hopf's conjecture on induced metrics by the graph of k smooth functions on the manifold S*S is a question we would like to investigate. Another problem on which we intend to work revolves around the conjecture of Carathéodory. This conjecture states that any convex surface M in R^3has at least two ombilic points. Concerning symbolic dynamics and dynamics negatively curved geometry, a celebrated result of G. Margulis said that the number N(T) of geodesic primitive of length T>0 is finite and is equivalent to exp(dT)/T, when T goes to infinity.
It is also well known that the geodesic flow on the tangent bundle TM is encoded by a symbolic dynamic system, and specifically, a flow of suspension above a offset on a finite alphabet. S. Lalley shows that a typical periodic geodesic on a compact surface with negative curvature, self-intersects proportionally to the square of its length, where the proportionality constant C0 is the number of self- intersection of the Liouville measure. The recent article of the same author titled : Statistical regularities of self-intersection counts for geodesics on negatively curved surfaces, studied the fluctuations of typical self-intersections around the average. We focused to understand the self-intersections via coding. Considering the geometric structure of complex manifolds, we are interested by the study of PDE in Cn. Mainly, for a domain in a complex analytic manifold X of dimension n, the study of the solvability of the d-bar operator for differentials forms of class C1 with boundary value in the sense of currents will draw our attention.
We are also interested by the algorithmic aspect of solving systems of polynomial equations in several unknowns. Polynomial models are ubiquitous and widely applied across the sciences. They arise in optimization, shape optimization, mathematical biology, robotics, and numerous other areas. The set of solutions to a system of polynomial equations is an algebraic variety, the basic object of algebraic geometry. Exciting recent developments in symbolic algebra and numerical software for geometric calculations have revolutionized the field, making formerly inaccessible problems tractable, and providing fertile ground for experimentation and conjecture. The case of a polynomial with two unknows with real coefficients was intensively studied in the 2000s, leading to efficient symbolic-numerical algorithms to compute the topology of its real locus. On the other hand, fewer improvements have been made for systems with more than two unknows. We focused on the topology computation of algebraic surfaces.