Institut Camille Jordan

• Analysis of kinetic and fluid mechanics equations

Applications of functional spaces to PDEs. Qualitative properties of solutions (stability, asymptotic behavior, etc.). Boltzmann, Navier-Stokes and Euler equations. Non-Newtonian fluids, fluids in rotation, capillarity problems. Kinetic theories with and without collisions. Applications of harmonic analysis to PDEs. Approximation of incompressible flows with weakly compressible flows. Shallow water wave equations and formation of singularities.

• Elliptic equations, Analysis on manifolds

Qualitative properties of solutions of nonlinear elliptic PDEs. Stability and instability. Study of the 3D Ginzburg-Landau equation : structure of the natural boundary data, asymptotic behavior of the energy. Questions in functional analysis (Sobolev spaces between manifolds, general theory of degree). Minimal surfaces and stationary varifolds. Classification of singularities, global regularity, calibrations.

• Riemannian geometry. Functional inequalities and convergence to equilibrium. Optimal transport

Studying geometric and functional inequalities (such as Poincaré and logarithmic Sobolev inequalities), we analyze the asymptotic behavior of diffusion equations, such as Fokker-Planck or Ornstein-Uhlenbeck equations. Optimal transport and Ricci curvature. Regularity questions in optimal transport.

• Functional and Harmonic Analysis, Operator theory and algebras

We combine methods from complex analysis, functional analysis, harmonic analysis and of Banach spaces geometry, to study questions from interpolation theory, operator theory, reproducing kernels, approximation, etc.

• Quantum Probability and Mechanics

Quantum noise, mechanics, statistical mechanics, stochastic calculus and random walks.

• Optimization. Optimal control and regularity in variational calculus, topological methods

The main directions are : geometric measure theory ; necessary optimality conditions in optimizations ; Hamilton-Jacobi equations in control theory ; construction of state feedbacks in optimal and stabilizing control ; Hilbert-Haar theory in calculus of variations. Variational inequalities and applications to the stability of stationary solutions. Coercivity properties of non smooth functionals. Differential inclusions. Use of the theory of asymptotic functions for optimization and for variational inequalities (new penalty and barrier methods, new techniques for the convergence analysis) ; new optimization algorithms, based on non-euclidian projections which take constraints into account. Hyperbolic problems with low regularity coefficients : well-posedness and control.

• PDE-Analysis Seminar : on Tuesdays.
• Working seminar "Operator theory and complex analysis" : on Wednesdays.
• Analysis Meetings : ICJ (Lyon 1), LATP (Marseille), IRMP (Louvain-la-Neuve), IF (Grenoble), Laboratoire de Mathématiques (Clermont-Ferrand).
• EDP-Rhone-Alpes-Auvergne Days : A regional conference on PDEs.