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Geometry of generators of homotopy groups of the unstable Lagrangian Grassmannian
Daniel Álvarez-Gavela: email@example.com
The Lagrangian Grassmannian is the space of all linear Lagrangian subspaces in complex Euclidean space C^n. As a homogeneous quotient it can be described as U(n)/O(n) for U(n) the unitary group and O(n) the orthogonal group. Elements of the k-th homotopy group of U(n)/O(n) are families of Lagrangian planes in C^n parametrized by the k-sphere S^k, up to homotopy. These homotopy groups are well-known due to classical computations. For applications to symplectic topology it would be useful to understand some explicit aspects of the geometry of these generators, in particular it would be useful to understand the locus of tangencies of the corresponding family of Lagrangian planes with a fixed Lagrangian plane (for example you can take R^n in C^n as your fixed plane). This locus will depend on the representative you choose for each generator, and the goal is to find the representative which yields the simplest possible locus. So it boils down to writing down some well-chosen formulas and analyzing them by hand. I have figured out most of this but there are a couple of exceptional examples in low dimensions (I think pi_3 and pi_7) which I don't quite understand yet. In this project you would think hard about these generators, starting with well-known formulas in the literature, and ideally get to understand them well enough to describe the simplest locus of tangencies one can achieve. If the outcome is successful we can write a paper together. Even if the outcome is not successful you would learn a little about homotopy groups, the Lagrangian Grassmannian, symplectic geometry and singularity theory.
This is a theoretical project in pure math. Although the work to be done is rather explicit, i.e. writing down some formulas in Euclidean space and performing linear algebra calculations by hand, it would be very helpful to have basic knowledge of point-set topology and multi-variable calculus. Familiarity with smooth manifolds and homotopy groups is a plus but not necessary, we can learn what we need as we go. The goal is to have a publishable result at the end so the project is quite ambitious, but I think not entirely unrealistic.