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\textbf{Rational points on diagonal cubic surfaces}
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It is known that there are diagonal cubic surfaces defined over
$\bQ$ which do not contain rational points, although for each
prime $p$ they do contain $p$-adic points. The simplest example is
\[ 3X_0^3+4X_1^3+10X_2^3+15X_3^3=0. \]
It is natural to ask what condition additional to solubility in
each $p$-adic field is needed to ensure solubility in $\bQ$; and
there is strong numerical evidence that this condition is that
there is no Brauer-Manin obstruction. However, one does not know
how to exploit such a condition.

Provided one assumes the finiteness of the Tate-\v{S}afarevi\v{c}
groups of elliptic curves, I shall show that there is a
sufficient condition for solubility which is only slightly
stronger than the vanishing of the Brauer-Manin obstruction, and
which can be expressed in very straightforward terms. \vskip 1in


\begin{center}
\textbf{Designing efficient Liapounov functions}
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Let $\dot{x_i}=f_i(x_1,\ldots,x_n)$ for $1\leq i\leq n$ be a
system of first order ordinary differential equations in $\bR$. A
function $V(x_1,\ldots,x_n)$ is called a Liapounov function for
the system if
\[ W=\sum\frac{\partial V}{\partial x_i}f_i \]
satisfies $W<0$ whenever $V\geq0$. Liapounov's theorem in its
simplest form states that if $V$ is a Liapounov function then
(subject to some abuse of language in respect of behaviour at
infinity) every trajectory eventually enters the set $\{V\leq0\}$
and never thereafter leaves it.

The construction of efficient Liapounov functions is currently a
hit-and-miss affair. This seminar is a contribution towards
making the process more systematic.

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