## Projects

### Resolutions of length 3

We work on the structure of finite free resolutions of length 3. The approach is based on the paper Generic free resolutions by J. Weyman (2018) which relates the structure of a generic ring for resolutions of length 3 to representations of a Kac- Moody Lie algebra associated to a T-shaped graph $T_{p,q,r}$. In particular we investigate:

- The connection between opposite Schubert varieties and perfect ideals of codimension 3,
- Geometric examples (points in ${\bf P}^3$, curves in ${\bf P}^3$ and ${\bf P}^4$),
- Families of free resolutions we can construct from this approach.

This part of our research is funded by NAWA.

### Gorenstein ideals of codimension 4

We work on the structure of Gorenstein ideals of codimension 4, particularly those with small number of generators. In particular we work on the following questions:

- Analyzing spinor structures on resolutions of Gorenstein ideals of codimension 4.
- Relation of Gorenstein ideals of codimension 4 to certain Schubert varieties in homogeneous spaces.
- The role played by Gorenstein ideals of codimension 4 that arise as ''doublings" of perfect ideals of codimension 3.

This part of our research is funded by the Maestro grant from NCN.

### Koszul modules

We work on the structure of Koszul modules (see e.g. Vanishing resonance and representations of Lie algebras by S. Papadima and A.I. Suciu (2013) and Koszul modules and Greenâ€™s conjecture by M. Aprodu, G. Farkas, S. Papadima, C. Raicu and J. Weyman (2018)). In particular:

- We analyze the equivariant Koszul modules associated with representations with finitely many orbits.
- We analyze Koszul modules related to mapping class group and Johnson filtration of the free group.
- We pursue generalizations of Koszul modules related to higher homology groups of finitely generated groups.

This part of our research is funded by the Maestro grant from NCN.

### D-modules

This part of our research is funded by the Maestro grant from NCN. Joint work with X. Wu (Ghent).