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For a sequence of Hilbert spaces and continuous linear operators the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Of particular interest are those sequences for which the curvature is "small" at each step, e.g., belongs to a fixed operator ideal. In this context we elaborate the theory of Fredholm sequences and show how to introduce the Lefschetz number.
In order to characterise the C*-algebra generated by the singular Bochner-Martinelli integral over a smooth closed hypersurfaces in Cn, we compute its principal symbol. We show then that the Szegö projection belongs to the strong closure of the algebra generated by the singular Bochner-Martinelli integral.
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernel of this domain. The Lefschetz number is proved to be the sum of the usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points
Anisotropic edge problems
(2002)
We define the Dirichlet to Neumann operator for an elliptic complex of first order differential operators on a compact Riemannian manifold with boundary. Under reasonable conditions the Betti numbers of the complex prove to be completely determined by the Dirichlet to Neumann operator on the boundary.
The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments.
We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
We consider a boundary value problem for an elliptic differential operator of order 2m in a domain D ⊂ n. The boundary of D is smooth outside a finite number of conical points, and the Lopatinskii condition is fulfilled on the smooth part of δD. The corresponding spaces are weighted Sobolev spaces H(up s,Υ)(D), and this allows one to define ellipticity of weight Υ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(Υ + 2m)/n, then the root functions of the problem are complete in L²(D). In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary.
We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernal of this domain. The Lefschetz number is proved to be the sum of usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.