The PI will investigate representations of finite dimensional algebras by a variety of geometric, homological, and combinatorial methods, the latter including adaptations of Groebner techniques. Her first project targets representations with fixed dimension and fixed top. For that purpose, the PI is supplementing the classical geometric methods in the field -- predominantly affine -- with a projective setting inside Grassmannian varieties. Combining the different viewpoints, she will explore moduli problems for representations with fixed top or fixed radical layering and work on a classification of those degenerations of a given representation which leave the top or, more strongly, the radical layering unchanged. The second project is homological and focuses on a class of tame algebras (string algebras), which first surfaced in work of Gelfand-Ponomarev on representations of the Lorentz group. Her goal is to extend the (at this point well-understood) homological behavior of finite dimensional representations to infinite dimensional ones. In particular, she will address the structure of modules of finite projective dimension and the placement -- in terms of in- and outgoing maps -- of the full subcategory of such modules within the entire module category. The third project is aimed at derived categories. In this direction, the PI will continue prior research on the geometry of chain complexes. She expects applications to a longstanding homological problem known as the `No Loop Conjecture'.
Under a wider angle, the project can be located as follows: The majority of our physical models for the universe live in finite dimensional vector spaces. The most interesting of these spaces carry additional structure, such as a Lie or associative algebra multiplication, for example. Associative finite dimensional algebras are the objects of the investigator's research. One approach towards understanding them is to explore their 'representations', which can be thought of as linearized snapshots of the algebra which are realized within algebras of matrices -- both coarse-grained and fine-grained snapshots are of interest to highlight different features of the algebra.
