Symplectic Geometry Seminar

Dylan Cant (Université de Montréal) Title: On the spectral diameter of the Grassmannians Abstract: For a compact symplectic manifold 𝑀 there is a spectral pseudometric on the universal cover of 𝐻𝑎𝑚(𝑀), built from Floer theory. A folklore conjecture says its diameter should be infinite whenever the symplectic form vanishes on 𝑝𝑖2(𝑀), on the other hand the presence of symplectic spheres can force the diameter to be finite, as happens for 𝐶𝑃𝑛. I will discuss what happens for the complex Grassmannians 𝐺𝑟(𝑘, 𝑛). We show that the spectral diam- eter of 𝐺𝑟(2, 𝑛) is finite when 𝑛 is prime, and that the spectral diameter of 𝐺𝑟(2𝑘, 2𝑛) is infinite when 𝑘 < 𝑛 (both over a field of characteristic zero; in nonzero characteristic there is a more refined statement for 𝐺𝑟(2, 𝑛)). The finiteness comes from the quantum cohomology of the Grassmannian, and the infiniteness comes from Lagrangian submanifolds. This is joint work with Habib Alizadeh, Marcelo Atallah and Jianqiao Shang.