ACAT 2005

General mathematical software systems like Mathematica, Maple, Derive, etc.
and special mathematical systems like FORM, Singular, Macaulay etc. are
having a major impact on the way how science, in particular physics, can be
done today. The tremendous progress made in the sophistication and
efficiency of these systems is based, first, on dramatic advances in
mathematical research over the past four decades that lead to new and
powerful mathematical algorithms  and, also, on advances in software
technology that made these systems easy to use. The mathematical field on
which the algorithms in mathematical software sytems are based is called
"Symbolic Computation".

In this talk, we will first mention a few high-lights of current symbolic
computation. We will go into some more detail about the speaker's own
Groebner bases theory for algorithmic problems related to systems of
multivariate polynomials. Then we will draw a few lines into what we think
will be the future of symbolic computation: 

- algorithmization of abstract fields of mathematics, e.g. Hilbert space theory
- symbiosis of algebraic and numeric algorithms
- interaction between computer algebra and computational logic
- mathematical knowledge management and computer-supported mathematical
  theory exploration
- computer-supported mathematical invention

The talk will expose the present status of research in quantum
computing, concerning both experimental implementations and algorithms (with
an emphasis on physics applications).

Pulsed neural networks encode information as firing patterns.
In order to realize functions of information processing the firing
pattern generation process has to be controlled by dedicated network
dynamics. We use adaptive synapses and simple integrate-and-fire neurons
as dynamical elements. Networks of these are shown to perform as universal
feature detectors as well as robust memories. A head recognizer will be
demonstrated.

I will discuss how FORM can currently be used and what lies in its
future.