Over the last few decades, a very strong interaction between mathematics and computer science has developed. Thus, classical topics such as geometry, topology, algebra, category theory or probability allow the description and understanding of discrete structures arising from computer science problems.

Conversely, tools from computer science allow us to better understand certain mathematical properties. The main objectives can be summarized as follows: giving discrete representations of continuous objects, developing mathematical tools to study discrete structures, exploring certain mathematical properties by means of computation, optimizing and specifying the limits of the complexity of the algorithms used, developing a theory of computer-assisted proof.

The Institut de Mathématiques de Toulouse has developed topics at the interface between discrete mathematics and theoretical computer science. Here is a non-exhaustive list of some of the axes developed:

• study of discrete structures (graphs, rational languages) using topological tools ;

• enumerative combinatorics ;

• combinatorial structures in algebra;

• discrete random objects and their continuous limits;

• combinatorial optimization and operations research;

• algorithmics in number theory, information theory, coding and cryptography;

• quantum information theory;

• probabilistic studies of algorithms and algorithms using randomness;

• symbolic dynamics, automata and formal languages, cellular automata, tilings defined by local rules;

• algorithmic complexity of dynamic properties;

• formal calculus;

• language and proof theory.