Past Topology Seminar

The goal of this talk is to construct new examples of hyperbolic

aspherical complexes. More precisely, given an aspherical simplicial

complex P and a subcomplex Q of P, we are looking for conditions under

which the complex obtained by attaching a cone of base Q on P remains

aspherical. If Q is a set of loops of a 2-dimensional complex, J.H.C.

Whitehead proved that this new complex is aspherical if and only if the

elements of the fundamental group of P represented by Q do not satisfy

any identity. To deal with higher dimensional subcomplexes we use small

cancellation theory and extend the geometric point of view developed by

T. Delzant and M. Gromov to rotation families of groups. In particular

we obtain hyperbolic aspherical complexes obtained by attaching a cone

over the "real part" of a hyperbolic complex manifold.

We give an introduction to the Kakimizu complex of a link,

covering a number of recent results. In particular we will see that the

Kakimizu complex of a knot may be locally infinite, that the Alexander

polynomial of an alternating link carries information about its Seifert

surfaces, and that the Kakimizu complex of a special alternating link is

understood.

I'll describe an approach to perturbative quantum field theory
which is philosophically similar to the deformation quantization approach
to quantum mechanics. The algebraic objects which appear in our approach --
factorization algebras -- also play an important role in some recent work
in topology (by Francis, Lurie and others).  This is joint work with Owen
Gwilliam.

Coarse geometry provides a very useful organising point of view on the study
of geometry and analysis of discrete metric spaces, and has been very
successful in the context of geometric group theory and its applications. On
the other hand, the work of Carlsson, Ghrist and others on persistent
homology has paved the way for applications of topological methods to the
study of broadly understood data sets. This talk will provide an
introduction to this fascinating topic and will give an overview of possible
interactions between the two.

Topological spaces and manifolds are commonly used to model configuration
spaces of systems of various nature. However, many practical tasks, such as
those dealing with the modelling, control and design of large systems, lead
to topological problems having mixed topological-probabilistic character
when spaces and manifolds depend on many random parameters.
In my talk I will describe several models of stochastic algebraic topology.
I will also mention some open problems and results known so far.

Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
double helix
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.

We thus give a taxonomy of local DNA axis configurations. More precisely, we
characterise
all rational tangles obtained from a given rational tangle via a rational
subtangle
replacement (RSR). This builds on work of Berge and Gabai. 
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners.

Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
categorisation.

Joint work with Ken Baker

The curve complex on an orientable surface, introduced by William Harvey about 30 years ago, is the abstract simplicial complex whose vertices are isotopy classes of simple close curves. A set of vertices forms a simplex if they can be represented by pairwise disjoint elements. The mapping class group of S acts on this complex in a natural way, inducing a homomorphism from the mapping class group to the group of automorphisms of the curve complex. A remarkable theorem of Nikolai V. Ivanov says that this natural homomorphism is an isomorphism. From this fact, some algebraic properties of the mapping class group has been proved. In the last twenty years, this result has been extended in various directions. In the joint work with Ferihe Atalan, we have proved the corresponding theorem for non-orientable surfaces: the natural map from the mapping class group of a nonorientable surface to the automorphism group of the curve compex is an isomorphism. I will discuss the proof of this theorem and possible applications to the structure of the mapping class groups.

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 prove that no Riemannian manifold quasiisometric to

complex hyperbolic plane can have a better curvature pinching. The proof

uses cup-products in $L^p$-cohomology.

Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$

of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove

that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct

product of $k$ copies of the fundamental group of $G$. This is joint work with

Jose Manuel Gomez and Juan Souto.

There is a close but underexploited analogy between the Euler characteristic

of a topological space and the cardinality of a set. I will give a quite

general definition of the "magnitude" of a mathematical structure, framed

categorically. From this single definition can be derived many

cardinality-like invariants (some old, some new): the Euler characteristic

of a manifold or orbifold, the Euler characteristic of a category, the

magnitude of a metric space, the Euler characteristic of a Koszul algebra,

and others. A conjecture states that this purely categorical definition

also produces the classical invariants of integral geometry: volume, surface

area, perimeter, .... No specialist knowledge will be assumed.

The is the second part of the Analysis and Geometry Seminar today.

Abstract: We will explain a certain natural way to project elements of

the mapping class to simple closed curves on subsurfaces. Generalizing

a coordinate system on hyperbolic space, we will use these projections

to describe a way to characterize elements of the mapping class group

in terms of these projections. This point of view is useful in several

applications; time permitting we shall discuss how we have used this

to prove the Rapid Decay property for the mapping class group. This

talk will include joint work with Kleiner, Minksy, and Mosher.