Analysis, Geometry and Applications

Analysis, geometry and applications

The “Analysis, geometry and applications” team works on two main lines of research.

  1. Analysis of PDE and optimization
  2. Algebraic geometry of projective spaces and low-dimensional topology

Analysis of PDE and optimization

  • Mathematical analysis of nonlinear degenerate elliptic or parabolic equation problems, hyperbolic conservation law problems with possible parabolic/hyperbolic coupled systems along an interface, deterministic versions of pseudo-parabolic equation problems. Study of singular solutions and qualitative behavior of the solutions to quasi-linear elliptic problems.
  • Mathematical analysis of certain equations resulting from fluid mechanics (Navier Stokes, Stokes, Oseen).
  • Problems related to dynamic systems in biomathematics, applied to medicine in particular: study of new mathematical models of leukemia.
  • Optimization - Calculus of variations: nonsmooth analysis, calculus of variations. Problems related to nonsmooth analysis (optimality conditions, sub-gradients, generalized Hessian). Existence of and at extremal properties in the case of singularity functions. A second theme concerns problems posed by generalized calculus of variations (stochastic, fractional, non-differentiable, singular).

 

Algebraic geometry of projective spaces and low-dimensional topology

  • Arrangement of hyperplanes.
  • Complex plane algebraic curves, topological invariants of their embedding potential (fundamental group, characteristic varieties, Alexander modules).
  • Logarithmic bundles and derivation modules associated with an arrangement.
  • Derived categories and their uses in moduli spaces of bundles.
  • Knot theory, braid groups and their representations.
  • Determinantal varieties, Cohen-Macaulay and Ulrich modules.