We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R. Bott) and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.

In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.

A locally non-trivial holomorphic family of Riemann surfaces over a hyperbolic Riemann surface B determines a homomorphism (monodromy) of the fundamental group of B to the modular group. We give a necessary condition for a homomorphism to be a mondromy of a holomorphic family of Riemann surfaces. And we estimate the number of holomorphic families in terms of topological information for holomorphic families over Riemann surfaces of genus 0.

We describe a method of constructing explicitly eigenfunctions of the Laplacian with a prescribed boundary behaviour on a class of manifolds of hyperbolic type. These are manifolds of the form M=X×R×+, where X is an n-dimensional Riemannian manifold and the metric of M is a perturbation of the hyperbolic one.

It is shown that the spectrum of a one-dimensional Dirac operator with a potential q tending to infinity at infinity, and such that the positive variation of 1/q is bounded, covers the whole real line and is purely absolutely continuous. An example is given to show that in general, pure absolute continuity is lost if the condition on the positive variation is dropped. The appendix contains a direct proof for the special case of subordinacy theory used.

Let X(t) (t ∈ RN) be a fractional Brownian motion in Rd of index α. Let GrX([0, 1]N) be the graph of X and let

formula here

and

formula here

It is proved that if N<αd, then almost surely

formula here

and if N>αd, then almost surely

formula here

where ϕ-m is the ϕ-Hausdorff measure and K1, K2 are positive finite constants. The exact Hausdorff measure of the image and graph of certain Gaussian random fields with independent fractional Brownian motion components are also obtained.

14 October 1996 In the Cockcroft Lecture Theatre, New Museums Site.

28 October 1996 In the Cockcroft Lecture Theatre, New Museums Site.

ANNUAL GENERAL MEETING 28 October 1996 In the Arts School.

11 November 1996 In the Cockcroft Lecture Theatre, New Museums Site.

25 November 1996 In the Cockcroft Lecture Theatre, New Museums Site.

20 January 1997 In the Cockcroft Lecture Theatre, New Museums Site.

3 February 1997 In the Cockcroft Lecture Theatre, New Museums Site.

17 February 1997 In the Cockcroft Lecture Theatre, New Museums Site.

3 March 1997 In the Cockcroft Lecture Theatre, New Museums Site.

19 May 1997 In the Arts School.

21 July 1997 In the Arts School.