Local problems in low-dimensional topology from the perspective of multicurves

• Applicant: Professor Dr. Claudius Zibrowius
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 505125645

My research pertains to low-dimensional topology, an area of pure mathematics which encompasses the study of geometric and topological properties of 3- and 4-manifolds as well as knots, links, and surfaces embedded therein. Over the past decades, research in this area has been shaped by the arrival of modern homological invariants, which have contributed new perspectives on old problems through ties with algebraic geometry, representation theory, and physics. Arguably the most influential of these new homology theories—and the most relevant for my own work—are Khovanov homology and Heegaard Floer homology.In recent years, certain multicurve invariants have emerged that offer geometric interpretations of local versions of Khovanov and Heegaard Floer theory in terms of Fukaya categories of simple surfaces. I was directly involved in the definition of three of these invariants and in their application to the resolution of several open conjectures in the field.The purpose of my research programme is to advance our understanding of this new technology in order to investigate other open problems in low-dimensional topology. The main focus will lie on the following three conjectures: the Mutation Conjecture for Khovanov homology, the Baker-Moore Conjecture on L-space knots, and the Cosmetic Crossing Conjecture and its generalizations.Towards these goals, I will pursue the following three lines of basic research: Firstly, I will study the properties of the new invariants and their relation to more classical invariants such as the Seifert genus and the fundamental group. Secondly, I will explore the existence of local versions of spectral sequences that are known to relate the global invariants. Finally, the techniques that underlie the definition of the multicurve invariants apply in broad generality. I am interested in using them to study related theories and exploiting the intuition built by multicurves to study higher-dimensional settings.

• DFG Programme: Research Grants