Recent advances and challenges
in partial differential equations

The conference

The conference will be held at the Institute of Mathematics of Polish Academy of Sciences in Warsaw on 2-5 September 2025.

The event will be a special occasion to celebrate Professor Andrzej Szulkin's 80th birthday, in recognition of his significant contributions to nonlinear analysis and critical point theory.

Topics

The conference will serve as a chance to unite some renowned experts in the field of partial differential equations from different countries. There will also be the opportunity for young researchers to present their results. We believe this is an important chance to consolidate already existing collaborations and create new ones, as well as promote the circulation of innovative ideas. There will also be room for informal discussions. The main topics of the conference will be the following.

• Variational and topological methods in partial differential equations
  Many problems with a variational formulation have origins in mathematical physics. Attention is drawn, e.g., to the search for standing-wave solutions, travelling-wave solutions, and normalized solutions, which are of great interest in applications. There will be a particular focus on nonlinear Schrödinger equations, Born–Infeld equations, and curl-curl-type problems. Nonlocal counterparts and extensions, such as fractional Schrödinger equations and Choquard-type equations, will also be considered.
• Spectral properties of differential operators
  It is well known that there is a relation between the differential operator of a certain equation and its domain. Eigenvalue problems will be discussed and, in particular, properties of theirs that depend on the shape of the domain. The focus will be on the properties of the Laplacian with different boundary conditions (Dirichlet, Neumann, Robin), but other operators such as the Schrödinger operator or nonlocal ones (fractional Laplacian, fractional Hartree operator...) will not be excluded.
• Fixed points and topological invariants
  As it is well known, many differential equations can be reformulated as fixed point problems, that is, a solution is a fixed point of a map that acts on a Banach space. Then, fixed-point theorems can be used to solve partial or ordinary differential equations via Brouwer or Leray–Schauder theories. Likewise, some variational problems can be studied through the Conley index theory. The conference, thus, will be an opportunity to develop new approaches at the board of fixed point theory and variational methods.