Floer homology theories have had an enormous impact on low-dimensional topology over the last 2-3 decades.  The goal of this talk is to introduce two Floer homology theories -- Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings) -- and to sketch a proof of the equivalence of the two.  This is joint work with Vincent Colin and Paolo Ghiggini.

The talk will be accessible to beginning graduate students.

We first survey some development regarding the study of holomorphic functions on manifolds. We insist mostly on Liouville theorems or, more generally, dimension estimates for the space of polynomially growing holomorphic functions. Then we present some recent joint work with Jiaping Wang on this topic. Our work is motivated by the study of Ricci solitons in the theory of Ricci flow. However, the most general results we have do not require any knowledge of curvature.

Each locally compact group, commutative or not,
(this includes finite groups and Lie groups) has a dual object which
completely determines it. This object is a commutative
semigroup which is partially ordered and convex. This duality
theory generalizes the Pontryagin-Van Kampen duality for abelian,
locally compact groups in a natural way. We will give a short history,
some examples and indications of proofs.

A tree-strip is the product of a finite set (graph) with an infinite
tree. For a tree-strip of finite cone type, the tree is of finite cone
type and constructed starting from a root with certain substitution rules.
For a vertex of such a tree one can consider the cone of descendants and
the term 'finite cone type' refers to the fact that there are only
finitely many different
non-isomorphic cones of descendants.
On a certain class of such trees we obtain absolutely continuous
spectrum for the Anderson model for low disorder. The proof is based in
an Implicit Function Theorem in a very abstract Banach space.

The most recent result considers the Fibonacci tree-strip which is quite
special.
For the original set up, an essential assumption needed is the fact that
each vertex has at least 2 children, i.e. the tree can not have short
line segments. This is the key assumption that excludes quasi-one
dimensional Anderson models on strips for which Anderson localization is
known.
The Fibonacci tree, whose number of vertices in the n-th generation
corresponds to the n-th Fibonacci number, violates this assumption. But
with certain modifications this special case can also be treated.
The Fibonacci tree-strip is the first tree-strip where the tree has
short line segments and absolutely continuous spectrum for random
operators could be established.