Methods of computer algebra provide new ways to study classical problem in geometry.
This includes discrete analogues of geometric objects in combinatorics.
Classical problems in geometry can be revisited over finite fields.
Infinite families of objects can be found by generalizing examples over finite fields.
These infinite families may lift to examples over infinite fields, for which classification up to isomorphism
is impossible. This way, combinatorial and algorithmic techniques contribute to a classical field of study.
Algorithms for isomorphism testing can be established for small finite fields. The isomorphisms
can be lifted to isomorphisms of the generic objects from the infinite families.
The work over finite fields may give the cue to the general proof, but the final proof is
valid for the infinite family of objects based on symbolic computations.
The symbolic computation in linear groups is inspired by the computations over finite fields.
The finite field algorithms are based on poset classification, which allows to
classify the orbits of finite groups acting on finite sets. Such techniques have led to new results in
the theory of cubic surfaces as well as the theory of objects in projective planes.
Applications of this and related work lie in the fields of cryptography and coding theory.
Discrete geometric objects play an important role in post quantum cryptography, where
hard mathematical problems are used to design quantum resistant algorithms for cryptography.
Many of these algorithms rely on the hardness of certain problems in finite and discrete geometry.
Examples can be found in lattice based and code based cryptosystems.
The security of these systems is not dependent on the hardness of integer factoring,
which means that they are quantum resistant.