L3

There is a well-known relationship between the theory of formal group schemes and stable homotopy theory, with Ravenel's chromatic filtration and the nilpotence theorem of Hopkins, Devinatz and Smith playing a central role. It is also familiar that one can sometimes get a more geometric understanding of homotopical phenomena by examining how they interact with group actions. In this talk we will explore this interaction from the chromatic point of view.

Let $(M,\omega)$ be a symplectic manifold, and $g$ a Riemannian metric on $M$ compatible with $\omega$. If $L$ is a compact Lagrangian submanifold of $(M,\omega)$, we can compute the volume Vol$(L)$ of $L$ using $g$. A Lagrangian $L$ is called {\it Hamiltonian stationary} if it is a stationary point of the volume functional amongst Lagrangians Hamiltonian isotopic to $L$.

Suppose $L'$ is a compact Lagrangian in ${\mathbb C}^n$ which is Hamiltonian stationary and {\it rigid}, that is, all infinitesimal Hamiltonian deformations of $L$ as a Hamiltonian stationary Lagrangian come from rigid motions of ${\mathbb C}^n$. An example of such $L'$ is the $n$-torus $ \bigl\{(z_1,\ldots,z_n)\in{\mathbb C}^n:\vert z_1\vert=a_1, \ldots,\vert z_n\vert=a_n\bigr\}$, for small $a_1,\ldots,a_n>0$.

I will explain a construction of Hamiltonian stationary Lagrangians in any compact symplectic manifold $(M,\omega)$, which works by `gluing in' $tL'$ near a point $p$ in $M$ for small $t>0$.