New tools from combinatorial topology, sheaf theory and homological algebra for the study of hyperplane arrangements and oriented matroids

• Applicant: Dr. Paul Mücksch
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539874788

Hyperplane arrangements and (oriented) matroids - their abstract combinatorial counterparts - are classical objects in discrete geometry and combinatorics. Their study reveals surprising and deep connections between algebra, combinatorics, algebraic geometry and topology. During the years, remarkable results were obtained regarding the interplay of geometric, topological, algebraic and combinatorial invariants of arrangements and (oriented) matroids, and only recently seminal results, using a modern synthesis of methods from algebraic geometry, topology and combinatorics resolved long-standing conjectures. This proposal follows this philosophy of combining methods form diverse fields and aims to yield new results towards main open problems regarding the interplay of algebra, topology and combinatorics of arrangements, (oriented) matroids, related varieties and topological spaces, such as the K(pi,1)-problem for the complement manifold of a complex hyperplane arrangement, the relation of topological invariants of Milnor fibers to the combinatorics of (oriented) matroids, and Terao's freeness conjecture on the combinatorial nature of the commutative-algebraic properties of logarithmic derivations. We will develop new tools towards a deeper understanding of these intriguing open problems, adopting methods from homological algebra, sheaf theory, abstract homotopy theory and combinatorial topology, in particular emphasizing a relative point of view. At the center of our proposal are new approaches via combinatorial models of topological fibrations and sheaves on partially ordered sets. An important part will be the implementation of our new structures lying at the intersection of algebra, topology and combinatorics in established computer algebra systems such as SageMath, GAP and OSCAR. This will enable us to produce databases of these structures and their invariants which, in turn, will be used to discover new connections. In particular, one of our central theoretical results we aim for will yield a concrete complex cosntructed from oriented matroid data to compute the (co)homology groups of Milnor fibers of real arrangements. This will lead to a new computational approach towards the long-standing open problem regarding the combinatorial nature of the Betti numbers of Milnor fibers.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies
• International Connection: Japan, Switzerland
• Cooperation Partners: Professor Dr. Takuro Abe; Professor Dr. Emanuele Delucchi; Professor Dr. Masahiko Yoshinaga
• Co-Investigator: Professor Dr. Lukas Kühne