This talk summarises a variety of results concerning evolution equations for convex hypersurfaces: Simple proofs are sketched for some of the main results; a few open problems are described, along with some of the clues which suggest what the answers might be; and several applications to geometric problems are given.

We derive the existence of local minimizers of the functional $F_\Omega(E)$ describing the energy of a liquid drop $E \subset \mathbb{R}^3, trapped between two parallel hyperplanes and rotating with constant angular velocity $\sqrt{2\Omega}$, for small $\Omega \gt 0$.

In this paper we seek conditions under which the indefinite integrals of a function $\varphi$ from $L^p(\mathbb{R})$ belong to $L^p(\mathbb{R}) + \mathbb{C}$. We prove that if the spectrum sp($\varphi$) of $\varphi$ is isolated from zero, then it is improperly integrable for $(1 \leq p \lt \infty)$ and its indefinite integrals belong to $L^p(\mathbb{R}) + \mathbb{C}$. Also, we give applications to the differential equation $u^1(x) + \lambda u(x) = \varphi(x)$.

This talk is an exposition of background material for some recent developments in invariant theory carried out in joint work with Toby Bailey and Robin Graham [1]. These developments are based on work of Rod Gover [6]. The programme was initiated by Charles Fefferman (4].

The Hopf Bifurcation Theorem is the simplest result which guarantees the bifurcation of a family of time periodic solutions of an evolution equation from a family of equilibrium solutions. In this paper, we apply the theorem to a class of partial differential equations (PDEs). The usual assumption of differentiability of nonlinear terms is dropped by the employment of topological considerations. The method of parameter functionalization plays a very important role.

We announce the following result: the Jacobian of a locally defined holomorphic mapping between two real-analytic $CR$-manifolds $M_1$ and $M_2$ in $\mathbb{C}^N$ where $M_1$ has a nondegenerate Levi form and satisfies a certain rigidity c~ndition, is identically zero, if it is zero at one point on $M_1$. We also give a description of the image of the mapping in terms of the geometry of $M_2$.

A simple model for drying consisting of an infiltration type of partial differential equation is considered. Whereas the infiltration problem admits plane wave solutions with uniquely determined wave speed, the drying problem exhibits a much more complicated behaviour involving breaking up observed in the pendular state of drying.

Since Coifman, Lions, Meyer and Semmes discovered the link between Hardy space theory and compensated compactness theory, there has been much progresses in developing further connections between them. In this survey we show how the boundedness and compactness of commutators can be used to get the weak convergence and higher integrability of certain bilinear forms which are related to compensated compactness.

We survey some recent results in the theory of quasiregular mappings in higher dimensions, particularly those related to the regularity theory of the nonlinear Cauchy-Riemann and Beltrami systems.

Penalty and path-following methods have been used for solving finitedimensional quadratic programmes. The intention here is to apply such techniques to an infinite-dimensional problem, namely a one-sided obstacle problem, and to develop a method for solving the problem in an infinite-dimensional setting. The numerical methods developed in the infinite-dimensional context, so the convergence rate of the discretisations are (in some sense) independent to the size of the finite-dimensional approximation. These methods are shown to be convergent in appropriate Banach spaces by means of a monotonicity result for the iterates of the associated Newton method. This montonicity carries over to finite-dimensional discretisations for a large class of methods. The overall numerical method developed is based on an exterior penalty fund;ion, and some num."erical results have been obtained.

Our aim here is to describe some recent results concerning nonlinear oblique boundary value problems for two closely related classes of nonlinear elliptic equations-Hessian and curvature equations.

The problem of characterising completely integrable nonlinear partial differential equation (p.d.e) among all p.d.e. is a major current problem in nonlinear dynamics. It is widely believed that the Painlevé property·is a key to the ultimate resolution of this problem. This paper suggests an alternative approach to the characterisation problem which relies on a study of the geometry of differential operator spaces.

Applications of ideas from Kurzweil-Henstock integration to path independence of line integrals is discussed.