NSF Award
2505220
Symplectic topology is a field of mathematics that originated from classical physics. In the 1980s a well-known conjecture in this field, known as the nearby Lagrangian conjecture, was formulated. To this day the conjecture remains open as one of the fundamental problems that play a central role in modern mathematics. The overarching research goal of this project is to obtain new results in the direction of this conjecture by exploiting an interaction between two different areas of mathematics. This process will naturally lead to new problems and related applications opening future directions of research. The project aims to make the subject more accessible to early career researchers through inclusive learning seminars delivered in a hybrid mode accompanied by expository writings, fostering collaboration and community along the way. The PI is committed to undergraduate mentoring and to promoting diversity. The project includes problems suitable for undergraduate research. <br/><br/>This project aims to study the nearby Lagrangian conjecture, using tools and ideas from algebraic K-theory and probe various K-theoretic aspects of the conjecture. One aspect concerns the normal invariant of nearby Lagrangians, which encodes the tangential obstruction for the projection of the nearby Lagrangian to the base, which is a simple homotopy equivalence, to be homotopic to a diffeomorphism. Using tools from parametrized Morse theory, algebraic K-theory, and stable homotopy theory it will be shown that the normal invariant of a nearby Lagrangian factors through an eta map, which in particular implies it is 2-torsion. This will constitute the first known restriction on the smooth structure of nearby Lagrangians (beyond the simple homotopy equivalence condition) when the base is not a homotopy sphere. Another aspect concerns the higher Whitehead torsion of nearby Lagrangians. This invariant is defined more generally for Legendrians in 1-jet spaces which admit generating functions on tube bundles (or even twisted tube bundles), and can be shown to be nontrivial for this class of Legendrians. A natural conjecture is that the higher Whitehead torsion of nearby Lagrangians vanishes, and to confirm this conjecture constitutes another goal of this project. An application would be a restriction on the stable isomorphism classes of tube bundles which may be used to generate nearby Lagrangians. A number of related questions will be studied along the way, such as the homotopical calculation needed to apply the h-principle for the simplification of caustics in this context (this is needed to show that we may take our generating families to be generalized Morse) as well as the interaction with the analogous Floer theoretic invariant. There are other K-theoretic aspects of the nearby Lagrangian conjecture which may be considered in this project once the above aspects are understood, such as the possibility of developing a 1-parametric Cerf-Hatcher-Wagoner theory for the difference function of nearby Lagrangians.<br/><br/>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.
