| Abstract: |
| A fundamental problem in applied algebraic topology is the efficient computation of minimal presentations and minimal free resolutions of homology modules arising from multigraded torsion-free complexes. While substantial progress has been made in the bigraded setting, the general multigraded case remains largely open.
After a brief introduction to the algebraic framework of applied algebraic topology, I will present joint work with Fritz Grimpen on the construction and minimization of free--injective presentations of multigraded modules. I will then discuss ongoing work with Matthias Orth and Fritz Grimpen on developing relative Groebner basis techniques for submodules of quotients of free modules, and in particular for submodules of injective modules. These methods provide new tools for computing minimal resolutions of such modules, including, as special cases, modules given either as homology of chain complexes or via free--injective presentations. |
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