Bernhelm Booss-Bavnbek

Unbounded Fredholm Operators and Spectral Flow

We study the gap (= projection norm = graph distance) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.

WEAK UCP AND PERTURBED MONOPOLE EQUATIONS

We give a simple proof of weak Unique Continuation Property for perturbed Dirac operators, using the Carleman inequality. We apply the result to a class of perturbations of the Seiberg–Witten monopole equations that arise in Floer theory.

The ζ-Determinant and C-Determinant on the Grassmannian in Dimension One

We discuss the relation of the ζ-determinant of the Dirac operator on the interval to the canonical determinant, which appears naturally in this situation. This Letter is a pilot for papers in preparation on the ζ determinant defined on the infinite-dimensional Grassmannian of elliptic boundary problems for a Dirac operator in dimensions greater than one.

General spectral flow formula for fixed maximal domain

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by the Cauchy data spaces. We provide rigorous definitions of the underlying concepts of spectral theory and symplectic analysis and give a full (and surprisingly short) proof of our General Spectral Flow Formula for the case of fixed maximal domain. As a side result, we establish local stability of weak inner unique continuation property (UCP) and explain its role for parameter dependent spectral theory.

The Invertible Double of Elliptic Operators

First, we review the Dirac operator folklore about basic analytic and geometrical properties of operators of Dirac type on compact manifolds with smooth boundary and on closed partitioned manifolds and show how these properties depend on the construction of a canonical invertible double and are related to the concept of the Calderón projection. Then we summarize a recent construction of a canonical invertible double for general first order elliptic differential operators over smooth compact manifolds with boundary. We derive a natural formula for the Calderón projection which yields a generalization of the famous Cobordism Theorem. We provide a list of assumptions to obtain a continuous variation of the Calderón projection under smooth variation of the coefficients. That yields various new spectral flow theorems. Finally, we sketch a research program for confining, respectively closing, the last remaining gaps between the geometric Dirac operator type situation and the general linear elliptic case.

Spectral flow of paths of self-adjoint Fredholm operators

Abstract First we discuss some difficulties with the currently available definitions of spectral flow (SF). Then we use the Cayley transform to study the topology of the space CF sa (H) of (generally unbounded) self-adjoint Fredholm operators in a fixed complex separable Hilbert space H and give two different (but equivalent) rigorous definitions of SF as a homotopy invariant for continuous paths in CF sa (H). Our study is based on the gap (= projection or graph norm) topology. As examples, we consider families of operators of Dirac type on a compact manifold M with boundary, acting on sections of a fixed Hermitian vector bundle E with domains defined by varying global well-posed boundary conditions. Such families are continuous families in CF sa (L2(M;E)) if the coefficients of the Dirac operators and the boundary conditions vary continuously. No additional assumptions are required.

Quantum gravity: Unification of principles and interactions, and promises of spectral geometry

Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e., quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme.

The Geometry of Cauchy Data Spaces

First we summarize two different concepts of Cauchy data (‘Hardy’) spaces of elliptic differential operators of first order on smooth compact manifolds with boundary: the L2-definition by the range of the pseudo-differential Calderon-Szego projection and the ‘natural’ definition by projecting the kernel into the (distributional) quotient of the maximal and the minimal domain. We explain the interrelation between the two definitions. Second we give various applications for the study of topological, differential, and spectral invariants of Dirac operators and families of Dirac operators on partitioned manifolds.

Dirac operators and manifolds with boundary

We deal with various elliptic analogies to the classical Dirac equation; we explain our main analytical tools: invertible extension, Calderon projector, and twisted orthogonality of Cauchy data spaces; we investigate natural spaces of global elliptic boundary value problems for Dirac operators; and we develop an index theory for transmission problems and give additivity and non-additivity theorems for the index and the η-invariant under cutting and pasting of Dirac operators over partitioned manifolds. The explicit formulas rely on Clifford multiplication with vectors normal to the cutting submanifold.

The mathematization of the individual sciences — Revisited

We recall major findings of a systematic investigation of the mathematization of the individual sciences, conducted by the author in Bielefeld some 35 years ago under the direction of Klaus Krickeberg, and confront them with recent developments in physics, medicine, economics, and spectral geometry.