Linear structures are fundamental in all branches of mathematics and many applications. The study of such structures, linear algebra, is one of the corner stones of modern mathematics and the main objects of study are vector spaces and linear maps.
Structures occurring both in the mathematical and in the physical world are usually non-linear. A prominent example is the group structure of the set of symmetries of some physical object. While such a group is not inherently an object of linear algebra, it may be studied by considering an action of the group on a vector space. Equivalently, the abstract group is realized explicitly as a collection of matrices, with matrix multiplication as group operation. This gives a framework for studying non-linear structures using linear methods.
The above generalizes to representation theory of groups. In particular, the representation theory of finite groups is well developed. In the second half of the previous century, ideas from this theory gave impetus to the development of representation theory in a more general context: that of finite dimensional algebras.
This project deals with some of the most prominent open questions in representation theory of finite dimensional algebras. In particular, we attack the homological conjectures of Bass and Nakayama from the 60s. We apply computational methods, and part of the project deals with further development of computer software (QPA = Quivers and path algebras). We also
apply and further develop reduction techniques, motivated by the potential of generalizing ideas of cluster combinatorics to general finite dimensional algebras. Two PhD-students and a postdoctoral fellow are hired in the project.
The main aim is to further develop and apply reduction techniques to prominent open problems in Representation theory.
There are three work packages.
WP1: Apply reduction techniques to study problems related to tau-tilting, motivated by the potiential of generalizing ideas of cluster combinatorics to general finite dimensional algebras.
WP2: Further develop and apply QPA to investigate problems concerning computiations of projective resolutions and in particular in determining the finiteness of projective dimension of modules.
WP3: Use computational methods and ideas from WP2 and reduction techniques studied in WP1, together
with other reduction techniques, to attack one of the long standing conjectures in representation theory: the finitistic dimension conjecture.