L3

In this talk I describe some results obtained in collaboration with

J.F. Lafont and A. Sisto, which concern rigidity theorems for a class of

manifolds which are ``mostly'' non-positively curved, but may not support

any actual non-positively curved metric.

More precisely, we define a class of manifolds which contains

non-positively curved examples.

Building on techniques coming from geometric group theory, we show

that smooth rigidity holds within our class of manifolds

(in fact, they are also topologically rigid - i.e. they satisfy the Borel

conjecture - but this fact won't be discussed in my talk).

We also discuss some results concerning the quasi-isometry type of the

fundamental groups

of mostly non-positively curved manifolds.

I will explain how to embed arbitrary RAAGs (Right Angled

Artin Groups) in Ham (the group of hamiltonian symplectomorphisms of

the 2-sphere). The proof is combination of topology, geometry and

analysis: We will start with embeddings of RAAGs in the mapping class

groups of hyperbolic surfaces (topology), then will promote these

embeddings to faithful hamiltonian actions on the 2-sphere (hyperbolic

geometry and analysis).

Surfaces of large genus are intriguing objects. Their geometry

has been studied by finding geometric properties that hold for all

surfaces of the same genus, and by finding families of surfaces with

unexpected or extreme geometric behavior. A classical example of this is

the size of systoles where on the one hand Gromov showed that there exists

a universal constant $C$ such that any (orientable) surface of genus $g$

with area normalized to $g$ has a homotopically non-trivial loop (a

systole) of length less than $C log(g)$. On the other hand, Buser and

Sarnak constructed a family of hyperbolic surfaces where the systole

roughly grows like $log(g)$. Another important example, in particular for

the study of hyperbolic surfaces and the related study of Teichmüller

spaces, is the study of short pants decompositions, first studied by Bers.

The talk will discuss two ideas on how to further the understanding of

surfaces of large genus. The first part will be about joint results with

F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of

pants decompositions and related questions. In particular we investigate

how to find short pants decompositions on punctured spheres, and how to

find families of homologically independent short curves. The second part,

joint with L. Guth and R. Young, will be about how to construct surfaces

with large pants decompositions using random constructions.

All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.

What is a random group? What does it look like? In Gromov's few relator
and density models (with density < 1/2) a random group is a hyperbolic
group whose boundary at infinity is homeomorphic to a Menger curve.
Pansu's conformal dimension is an invariant of the boundary of a
hyperbolic group which can capture more information than just the
topology. I will discuss some new bounds on the conformal dimension of the
boundary of a small cancellation group, and apply them in the context of
random few relator groups, and random groups at densities less than 1/24.

We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.