Constructions in Higher-Dimensional Contact Topology

This research project in pure mathematics focuses on abstract geometric structures in higher dimensions, known as contact and symplectic topology. The field intertwines rigid geometric problems, such as studying periodic orbits or counting curves passing through a given point, with flexible cut-and-paste constructions. Part of the research conducted in this proposal is the development of a combinatorial description of these structures in terms of a diagrammatic calculus, and its applications to these rigid geometric problems. This provides an accessible source of new examples and computations, and establishes the ground for strictly higher-dimensional constructions. Due to its combinatorial nature, this line of research is naturally conducive to participation from undergraduate students and young researchers. Another part of this project uses techniques that combine known elements from algebraic geometry and geometric topology with new ideas from contact and symplectic topology, and the outcome of the proposed research contributes to each of these central fields of mathematical research.

The first part of this project is the development of a Legendrian calculus in the front projection, including higher-dimensional Reidemeister moves, Legendrian handle-slides and criteria for the existence of loose charts. By further establishing a relation with bi-Lefschetz fibrations, this tool will provide effective combinatorial criteria for (sub)flexibility of Weinstein structures and the computation of algebraic invariants such as the Legendrian contact differential graded algebra and the wrapped Fukaya category. This combines ideas from affine algebraic geometry and higher-dimensional contact surgery theory. The second part of this research project focuses on the detection of tightness and over-twistedness of higher-dimensional models, including bordered Legendrian open books and small neighborhoods of over-twisted models. The project will use pseudoholomorphic techniques adapted to each problem in combination with new ideas coming from the study of Legendrian submanifolds, including the study of Legendrian cobordisms and their relation to Weinstein cobordisms. In both these projects the study of compatible open books in special position have a role, and a general existence theorem will be studied with asymptotically holomorphic techniques. In addition, the project will include the study of a flexible class of Engel structures, completely determined by formal homotopy class, and related h-principles.