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Session I Boundary value problems, such as Dirichlet or Neumann problems for the Laplacian or other elliptic operators, belong to the traditional knowledge in partial differential equations. The ideas on solvability are in fluenced by numerous elds of Math ematics, such as complex analysis, potential theory, functional analysis, special functions, distribution theory, index theory, and pseudo-differential operators. It is a fascinating new point of the development of the past few decades that boundary value problems have opened the door to the analysis on manifolds with geometric singularities and to other elds of Mathematics, such as Geometry and Topology, and also to new applications in natural sciences. In this talk we start with basic observations on differential operators in an open set of Euclidean space, and we look at ellipticity with respect to the principal symbolic structure, composition and inversion rules with respect to the Leibniz multiplication between symbols and at parametrices of operators based on the Fourier transform. Then we pass to operators in a domain with smooth boundary and show how a differential operator acquires from the boundary a second symbolic structure, called the boundary symbol, which is operator-valued and acts in Sobolev spaces on the inner normal to the boundary. We show that the boundary symbol is responsible for elliptic boundary conditions and the construction of solutions via an inversion process of symbols. We finally develop a few elements of the 2x2 block matrix algebra of pseudo-differential boundary value problems in the frame of Boutet de Monvel's calculus, explain the role of Green, trace, and potential operators, we illustrate the principle of reducing boundary conditions to the boundary which leads to the famous Agranovich-Dynin formula for the Fredholm index. Moreover, we look at the Dirichlet-to-Neumann operator on the boundary which is a classical elliptic first order pseudo-differential operator that has not the transmission property at any interface on the boundary.
Session II We show how a new look at boundary value problems opens new research elds of geometric analysis on manifolds with singularities. First we develop the concept of edge Sobolev spaces, and derive basic properties by new methods in terms of operatorvalued symbols with twisted symbolic estimates. Then we inspect ideas from the boundary symbolic structures to interpret the half axis as a manifold with conical singularities and a manifold with boundary as a manifold with edge. Here we rephrase Fourier-based operators as operators based on the Mellin transform with holomorphic symbols. We show how the classical smoothness of solutions up to the boundary is to be replaced by an asymptotic property in the distance variable to the boundary, where the origin on the inner normal is regarded as the tip of the cone. The asymptotic data such as weights and complex exponents in the distance variable to the boundary, including logarithmic powers, are coordinated with poles and multiplicities of inverses of operator-valued leading conormal symbols that appear in symbol inversion and parametrices of Mellin pseudo-dierential operators. We finally give an idea, how cone and edge pseudo-differential algebras look like, including their principal symbolic hierarchies which are coordinated with an iterative construction of higher corner manifolds, appearing in geometric analysis.