CAREER: Legendrian and Contact Topology in Higher Dimensions

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The present project studies Legendrian and contact topology in all higher dimensions. Contact structures are the underlying geometric structures reigning the behaviour of light wavefronts in semiclassical ray optics, the evolution of thermodynamical systems and, in general, the study of real-valued functions and their first-order variations. In conjunction with its classical applications to control theory and Hamiltonian dynamics, contact topology has also recently found surprising applications to fluid mechanics and chemistry, including the study of Cholesteric Liquid Crystals. This project aims at providing an understanding of these contact structures through the manipulation of Legendrian wavefronts. These wavefronts are geometric in nature and part of this project aims at systematically describing efficient algorithms to manipulate these fronts. This will significantly enhance our ability to qualitatively study many dynamical systems in terms of the combinatorics of front diagrams. Phrasing problems in terms of front diagrams allows us to bring the strength of the theory of singularities into the study of contact structures. Thus, classical questions such as the stable behaviour of a given Hamiltonian system, the number of periodic orbits and the presence of surfaces of section will potentially be answered through the front-end development perspective offered by the combinatorics of front diagrams.In technical terms, this project shall be studying the classification of codimension-2 contact submanifolds and developing the study of Legendrian Kirby Calculus and Weinstein structures in all higher dimensions. The project addresses existence and non-uniqueness questions in higher-dimensional contact knot theory, the construction and manipulation of Weinstein handlebodies and their higher-dimensional Legendrian wavefronts. The project includes new applications of these goals to complex geometry, through the study of Stein structures, mathematical physics, including applications to Homological Mirror Symmetry, and singularity theory. The techniques proposed in this project will also have immediate applications to the computation of Floer-theoretic invariants in higher-dimensional symplectic topology and the study of Lagrangian fillings and the wrapped Fukaya category. The central tools in the project combine higher-dimensional geometric topology, the theory of h-principles and Legendrian invariants. The project includes a central conjecture on the classification of Lagrangian fillings for algebraic Legendrian links and an existence and uniqueness h-principle for higher-dimensional contact submanifolds. The techniques to be employed in the study of these conjectures include explicit constructions obtained via the manipulation of Legendrian fronts and wrinkled singularities, and the use of pseudo-holomorphic curves in order to compute their contact invariants. In addition, the project will also incorporate the development of invariants for higher-dimensional contact knots and the use of microlocal sheaf theory as a guiding connection to the theory of cluster algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.