Subventions et des contributions :

Subvention ou bourse octroyée s'appliquant à plus d'un exercice financier. (2017-2018 à 2022-2023)

Theoretical computer science has given rise to a number of important problems
that can be attacked with techniques of the subfield of mathematics known as
linear algebra. Our proposal is to use a number of related techniques in
linear algebra applied to "graph theory", to study such problems. The
connection of linear algebra to such problems is well known. Our proposal
seeks to strengthen the known techniques in linear algebra, with a view towards
their applications, to solve problems in theoretical computer science.

Such techniques can be used also to solve problems in a number of fields of
mathematics; indeed, the problems in computer science of interest to us are
related to a number of areas of mathematics and physics. In the other
direction, these related areas have research results which we can sometimes
borrow to enhance our knowledge of the parts of linear algebra and computer
science of interest to us.

Our proposed research is motived by two areas of theoretical computer science,
one known as "expander graphs", another known as "complexity theory". The
areas of linear algebra that we work with are the "eigenvalues" and "spectral
theory" of certain matrices arising in to "graph theory", and the related
notion of "sheaf theory". While sheaf theory is often viewed as a field of
algebraic topology, our interest in sheaf theory is in "sheaves of vector
spaces," which is a type of linear algebra that is enhanced by the structure of
a graph.

Our research in "expander graphs" focuses on "relative expansion," which is a
way of building larger networks from smaller ones, in a way where one can
understand expansion of the large network in terms of the smaller network and
the way the large network lies "over" the smaller one. Recently there has been
a lot of results and constructions of new networks from smaller ones. The
foundations of relative expansion began in a paper of ours from 2003, motivated
by work of Alexander Grothendieck in topology and algebraic geometry.

Our research in sheaf theory allows us to compare structures in linear algebra
that lie over the same graph; this type of sheaf theory is in its infancy. We
began to study this theory motivated by complexity theory, but have used it to
solve the Hanna Neumann Conjecture of the 1950's regarding group theory. This
sheaf theory was inspired by work of Grothendieck and his colleagues.

Hence our project exploits the fruitful connection between areas of computer
science and mathematics.