Bernhelm Booß-Bavnbek

Elliptic boundary problems for Dirac operators

The major goal of this book is to make the theory of elliptic boundary problems accessible to mathematicians and physicists working in global analysis and operator algebras. The book is about operators of Dirac type on manifolds with boundary....whats?

The Calderón projection: New definition and applications

Abstract We consider an arbitrary linear elliptic first-order differential operator A with smooth coefficients acting between sections of complex vector bundles E , F over a compact smooth manifold M with smooth boundary Σ . We describe the analytic and topological properties of A in a collar neighborhood U of Σ and analyze various ways of writing A ↾ U in product form. We discuss the sectorial projections of the corresponding tangential operator, construct various invertible doubles of A by suitable local boundary conditions, obtain Poisson type operators with different mapping properties, and provide a canonical construction of the Calderon projection. We apply our construction to generalize the Cobordism Theorem and to determine sufficient conditions for continuous variation of the Calderon projection and of well-posed self-adjoint Fredholm extensions under continuous variation of the data.

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying symplectic Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.

Unique Continuation Property for Dirac Operators

We give a direct proof of the unique continuation property of a Dirac operator by exploiting its simple product decomposition.

Perturbation of sectorial projections of elliptic pseudo-differential operators

Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderón projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley’s original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.

Geometric and Electromagnetic Aspects of Fusion Pore Making

For regulated exocytosis, we model the morphology and dynamics of the making of the fusion pore or porosome as a cup-shaped lipoprotein structure (a dimple or pit) on the cytosol side of the plasma membrane. We describe the formation of the dimple by a free boundary problem. We discuss the various forces acting and analyse the magnetic character of the wandering electromagnetic field wave produced by intracellular spatially distributed pulsating (and well-observed) release and binding of Ca2+ ions anteceding the bilayer membrane vesicle fusion of exocytosis. Our approach explains the energy efficiency of the dimple formation prior to hemifusion and fusion pore and the observed flickering in secretion. It provides a frame to relate characteristic time length of exocytosis to the frequency, amplitude and direction of propagation of the underlying electromagnetic field wave. We sketch a comprehensive experimental programme to verify – or falsify – our mathematical and physical assumptions and conclusions where conclusive evidence still is missing for pancreatic β-cells.

Clifford Algebras and Clifford Modules

We study bundles over a point, recalling the definition of the Clifford algebra Cl(V, q) of a real vector space V of dimension m equipped with a positive definite inner product q; the ℤ2-grading of Clifford algebras is shown, followed by an introduction of complex representations of Clifford algebras and the concept of complex Cl(V, q)-modules and of Clifford multiplication; the isomorphism classes of irreducible Cl(V, q)-modules are studied.

Dirac Operators on the Two-Sphere

We show that the index of any elliptic operator on any closed Riemann surface or even-dimensional sphere is completely determined by the pasting of trivial pieces of the operator. Our example gives a full analysis of the index on S 2 and determines the index of the generalized Riemann-Hilbert problem.

Clifford Bundles and Compatible Connections

We show how the natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics. For a Riemannian manifold X (with or without boundary), we obtain the Clifford bundle Cl(X) ≔ Cl(TX, g). We show that there exists a connection D for any bundle S of complex left modules over Cl(X) which is compatible with Clifford multiplication and extends the Riemannian connection on X to S.

Bojarski’s Theorem. General Linear Conjugation Problems

We recover the index of a Dirac operator A over a closed partitioned manifold M =X + ∪ X − with ∂X + = ∂X − = X + ∩ X − = Y from the Fredholm pair of Cauchy data spaces along Y. Similarly, the index of the linear conjugation (or transmission) problem As ± = 0 in X±\Y and s −|y = Φ (s + |y) is given by twisting the Cauchy data spaces with Φ. Related local elliptic boundary conditions for systems of Dirac operators are considered