L3

Random partial orders and random linear extensions

Several interesting models of random partial orders can be described via a

process that builds the partial order one step at a time, at each point

adding a new maximal element. This process therefore generates a linear

extension of the partial order in tandem with the partial order itself. A

natural condition to demand of such processes is that, if we condition on

the occurrence of some finite partial order after a given number of steps,

then each linear extension of that partial order is equally likely. This

condition is called "order-invariance".

The class of order-invariant processes includes processes generating a

random infinite partial order, as well as those that amount to taking a

random linear extension of a fixed infinite poset.

Our goal is to study order-invariant processes in general. In this talk, I

shall explain some of the problems that need to be resolved, and discuss

some of the combinatorial problems that arise.

(joint work with Malwina Luczak)

I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations.

This is joint work with Rob Meyerhoff.

The notion of a sutured 3-manifold was introduced by Gabai. It is a powerful tool in 3-dimensional topology. A few years ago, Andras Juhasz defined an invariant of sutured manifolds called sutured Floer homology.

I'll discuss a simpler invariant obtained by taking the Euler characteristic of this theory. This invariant turns out to have many properties in common with the Alexander polynomial. Joint work with Stefan Friedl and Andras Juhasz.

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 $t>0$.