Equations and their solutions
One can ask for integral solutions of equations where every component is non-zero. Consider the equation X4 + 2 Y4 = Z4 + 4 W4, for example. By using a computer, one can find a solution, consisting of the numbers 1484801, 1203120, 1169407, and 1157520.
To find these solutions one has to develop a suitable computer program for every equation. We then statistically compare the found solutions with conjectures about numbers and distribution of solutions. This can lead to a corroboration or a change of conjectures.
Surfaces and symmetries
Surfaces of the three-dimensional space like planes, cylinders, and cones are classic objects of study in geometry. They can be described with linear or quadratic equations. These objects are well understood.
Next, one can study surfaces that are given by cubic equations. It becomes apparent that the properties of these surfaces are controlled by a system of 51840 symmetries. If one further increases the degree of the equations, the then occurring surfaces become more and more complex, the number of potential symmetries increases.
The aim of our research is to study these connections. To this end, we develop methods to compute the properties of a given surface and, vice versa, construct surfaces with given properties.