Past Topology Seminar

I will describe the history of the homotopy groups of spheres, and some of the many different roles they have come to play in mathematics.

Consider a diagram of the unknot with c crossings. There is a

sequence of Reidemeister

moves taking this to the trivial diagram. But how many moves are required?

In my talk, I will give

an overview of my recent proof that there is there is an upper bound on the

number of moves, which

is a polynomial function of c.

The question of estimating the length of short conjugators in between
elements in a group could be described as an effective version of the
conjugacy problem. Given a finitely generated group $G$ with word metric
$d$, one can ask whether there is a function $f$ such that two elements
$u,v$ in $G$ are conjugate if and only if there exists a conjugator $g$ such
that $d(1,g) \leq f(d(1,u)+d(1,v))$. We investigate this problem in free
solvable groups, showing that f may be cubic. To do this we use the Magnus
embedding, which allows us to see a free solvable group as a subgroup of a
particular wreath product. This makes it helpful to understand conjugacy
length in wreath products as well as metric properties of the Magnus
embedding.

It's well known that the mapping space of two finite dimensional

manifolds can be given the structure of an infinite dimensional manifold

modelled on Frechet spaces (provided the source is compact). However, it is

not that the charts on the original manifolds give the charts on the mapping

space: it is a little bit more complicated than that. These complications

become important when one extends this construction, either to spaces more

general than manifolds or to properties other than being locally linear.

In this talk, I shall show how to describe the type of property needed to

transport local properties of a space to local properties of its mapping

space. As an application, we shall show that applying the mapping

construction to a regular map is again regular.

Topological field theories give a connection between

topology and algebra. This connection can be exploited in both

directions: using algebra to construct topological invariants, or

using topology to prove algebraic theorems. In this talk, I will

explain an interesting example of the latter phenomena. Radford's

theorem, as generalized by Etingof-Nikshych-Ostrik, says that in a

finite tensor category the quadruple dual functor is easy to

understand. It's somewhat mysterious that the double dual is hard to

understand but the quadruple dual is easy. Using topological field

theory, we show that Radford's theorem is exactly the consequence of

the Dirac belt trick in topology. That is, the double dual

corresponds to the generator of $\pi_1(\mathrm{SO}(3))$ and so the

quadruple dual is trivial in an appropriate sense exactly because

$\pi_1(\mathrm{SO}(3)) \cong \mathbb{Z}/2$. This is part of a large

project, joint with Chris Douglas and Chris Schommer-Pries, to

understand local field theories with values in the 3-category of

tensor categories via the cobordism hypothesis.

I will discuss recent joint work with S. Galatius, in which we

generalise the Madsen--Weiss theorem from the case of surfaces to the

case of manifolds of higher even dimension (except 4). In the simplest

case, we study the topological group $\mathcal{D}_g$ of

diffeomorphisms of the manifold $\#^g S^n \times S^n$ which fix a

disc. We have two main results: firstly, a homology stability

theorem---analogous to Harer's stability theorem for the homology of

mapping class groups---which says that the homology groups

$H_i(B\mathcal{D}_g)$ are independent of $g$ for $2i \leq g-4$.

Secondly, an identification of the stable homology

$H_*(B\mathcal{D}_\infty)$ with the homology of a certain explicitly

described infinite loop space---analogous to the Madsen--Weiss

theorem. Together, these give an explicit calculation of the ring

$H^*(B\mathcal{D}_g;\mathbb{Q})$ in the stable range, as a polynomial

algebra on certain explicitly described generators.

I will talk about the fixed-point properties of matrix groups acting CAT(0) paces, spheres and acyclic manifolds. The matrix groups include general linear groups, sympletic groups, orthogonal groups and classical unitary groups over general rings. We will show that for lower dimensional CAT(0) spaces, the group action of a matrix group always has a global fixed point and that for lower dimensional spheres and acyclic manifolds, a group action by homeomorphisms is always trivial. These results give generalizations of results of Farb concerning Chevalley groups over commutative rings and those of Bridson-Vogtmann, Parwani and Zimmermann concerning the special linear groups SL_{n}(Z) and symplectic groups Sp_{2n}(Z).

This is joint work with Dani Wise and builds on his earlier

work. Let M be a compact oriented irreducible 3-manifold which is neither a

graph manifold nor a hyperbolic manifold. We prove that the fundamental

group of M is virtually special. This means that it virtually embeds in a

right angled Artin group, and is in particular linear over Z.

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.

In Riemannian geometry there are several notions of rank

defined for non-positively curved manifolds and with natural extensions

for groups acting on non-positively curved spaces.

The talk shall explain how various notions of rank behave for

mapping class groups of surfaces. This is joint work with J. Behrstock.

For a fixed background manifold $M$ and parameter-space $X$, the associated configuration space is the space of $n$-point subsets of $M$ with parameters drawn from $X$ attached to each point of the subset, topologised in a natural way so that points cannot collide. One can either remember or forget the ordering of the n points in the configuration, so there are ordered and unordered versions of each configuration space.

It is a classical result that the sequence of unordered configuration spaces, as $n$ increases, is homologically stable: for each $k$ the degree-$k$ homology is eventually independent of $n$. However, a simple counterexample shows that this result fails for ordered configuration spaces. So one could ask whether it's possible to remember part of the ordering information and still have homological stability.

The goal of this talk is to explain the ideas behind a positive answer to this question, using 'oriented configuration spaces', in which configurations are equipped with an ordering - up to even permutations - of their points. I will also explain how this case differs from the unordered case: for example the 'rate' at which the homology stabilises is strictly slower for oriented configurations.

If time permits, I will also say something about homological stability with twisted coefficients.

Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.

The classical associahedra are cell complexes, in fact polytopes,

introduced by Stasheff to parametrize the multivariate operations

naturally occurring on loop spaces of connected spaces.

They form a topological operad $ Ass_\infty $ (which provides a resolution

of the operad $ Ass $ governing spaces-with-associative-multiplication)

and the complexes of cellular chains on the associahedra form a dg

operad governing $A_\infty$-algebras (that is, a resolution of the

operad governing associative algebras).

In classical applications it was not necessary to consider units for

multiplication, or it was assumed units were strict. The introduction

of non-strict units into the picture was considerably harder:

Fukaya-Ono-Oh-Ohta introduced homotopy units for $A_\infty$-algebras in

their work on Lagrangian intersection Floer theory, and equivalent

descriptions of the dg operad for homotopy unital $A_\infty$-algebras

have now been given, for example, by Lyubashenko and by Milles-Hirsch.

In this talk we present the "missing link": a cellular topological

operad $uAss_\infty$ of "unital associahedra", providing a resolution

for the operad governing topological monoids, such that the cellular

chains on $uAss_\infty$ is precisely the dg operad of

Fukaya-Ono-Oh-Ohta.

(joint work with Fernando Muro, arxiv:1110.1959, to appear Forum Math)

I will discuss some aspects of the simplicial theory of

infinity-categories which originates with Boardman and Vogt, and has

recently been developed by Joyal, Lurie and others. The main purpose of

the talk will be to present an extension of this theory which covers

infinity-operads. It is based on a modification of the notion of

simplicial set, called 'dendroidal set'. One of the main results is that

the category of dendroidal sets carries a monoidal Quillen model

structure, in which the fibrant objects are precisely the infinity

operads,and which contains the Joyal model structure for

infinity-categories as a full subcategory.

(The lecture will be mainly based on joint work with Denis-Charles

Cisinski.)

For a fixed n we will investigate homomorphisms Out(F_n) to

Out(F_m) (i.e. free representations) and Out(F_n) to

GL_m(K) (i.e. K-linear representations). We will

completely classify both kinds of representations (at least for suitable

fields K) for a range of values $m$.

Free groups, free abelian groups and fundamental groups of

closed orientable surfaces are the most basic and well-understood

examples of infinite discrete groups. The automorphism groups of

these groups, in contrast, are some of the most complex and intriguing

groups in all of mathematics. In these lectures I will concentrate

on groups of automorphisms of free groups, while drawing analogies

with the general linear group over the integers and surface mapping

class groups. I will explain modern techniques for studying

automorphism groups of free groups, which include a mixture of

topological, algebraic and geometric methods.

Morgan and Culler proved in the 1980’s that a minimal action of a free group on a tree is

completely determined by its length function. This theorem has been of fundamental importance in the

study of automorphisms of free groups. In particular, it gives rise to a compactification of Culler-Vogtmann's

Outer Space. We prove a 2-dimensional analogue of this theorem for right-angled Artin groups acting on

CAT(0) rectangle complexes. (Joint work with M. Margolis)

The Eilenberg-Ganea conjecture is the statement that every group of cohomological dimension two admits a two-dimensional classifying space.  This problem is unsolved after 50 years.  I shall discuss the background to this question and negative answers to some other related questions.  This includes recent joint work with Martin Fluch.

I will discuss quasi-isometries of the free group that preserve an

equivariant pattern of lines.

There is a type of boundary at infinity whose topology determines how

flexible such a line pattern is.

For sufficiently complicated patterns I use this boundary to define a new

metric on the free group with the property that the only pattern preserving

quasi-isometries are actually isometries.

Chataur and Menichi showed that the homology of the free loop space of the classifying space of a compact Lie group admits a rich algebraic structure: It is part of a homological field theory, and so admits operations parametrised by the homology of mapping class groups.  I will present a new construction of this field theory that improves on the original in several ways: It enlarges the family of admissible Lie groups.  It extends the field theory to an open-closed one.  And most importantly, it allows for the construction of co-units in the theory.  This is joint work with Anssi Lahtinen.

I will discuss new rigidity and rationality phenomena

(related to the phenomenon of Arnold tongues) in the theory of

nonabelian group actions on the circle. I will introduce tools that

can translate questions about the existence of actions with prescribed

dynamics, into finite combinatorial questions that can be answered

effectively. There are connections with the theory of Diophantine

approximation, and with the bounded cohomology of free groups. A

special case of this theory gives a very short new proof of Naimi’s

theorem (i.e. the conjecture of Jankins-Neumann) which was the last

step in the classification of taut foliations of Seifert fibered

spaces. This is joint work with Alden Walker.

The concordance group of classical knots C was introduced

over 50 years ago by Fox and Milnor. It is a much-studied and elusive

object which among other things has been a valuable testing ground for

various new topological (and smooth 4-dimensional) invariants. In

this talk I will address the problem of embedding C in a larger group

corresponding to the inclusion of knots in links.

A longstanding question in geometric group theory is the following. Suppose G is a hyperbolic group where all finitely generated subgroups of infinite index are free. Is G the fundamental group of a surface? This question is still open for some otherwise well understood classes of groups. In this talk, I will explain why the answer is affirmative for graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.

Automorphisms of right-angled Artin groups interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. An active area of current research is to extend properties that hold for both the above groups to $Out(A_\Gamma)$ for a general RAAG. After a short survey on the state of the art, we will describe our recent contribution to this program: a study of how higher-rank lattices can act on RAAGs that builds on the work of Margulis in the free abelian case, and of Bridson and the author in the free group case.

We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.

In this talk I'll explain how to build CAT(0) cube complexes and construct Lipschitz maps between them. The existence of suitable Lipschitz maps is used to prove that the asymptotic dimension of a

CAT(0) cube complex is no more than its dimension.

The notion of an E-infinity ring spectrum arose about thirty years ago,

and was studied in depth by Peter May et al, then later reinterpreted

in the framework of EKMM as equivalent to that of a commutative S-algebra.

A great deal of work on the existence of E-infinity structures using

various obstruction theories has led to a considerable enlargement of

the body of known examples. Despite this, there are some gaps in our

knowledge. The question that is a major motivation for this talk is

`Does the Brown-Peterson spectrum BP for a prime p admit an E-infinity

ring structure?'. This has been an important outstanding problem for

almost four decades, despite various attempts to answer it.

I will explain what BP is and give a brief history of the above problem.

Then I will discuss a construction that gives a new E-infinity ring spectrum

which agrees with BP if the latter has an E-infinity structure. However,

I do not know how to prove this without assuming such a structure!

The Farrell-Jones Conjecture predicts a homological formula for K-and L-theory of group rings. Through surgery theory it is important for the classification of manifolds and in particular the Borel conjecture. In this talk I will give an introduction to this conjecture and give an overview about positive results and open questions.

Distortion is an asymptotic invariant of the embeddings

of finitely generated algebras. For group embeddings,

it has been introduced by M.Gromov. The main part of

the talk will be based on a recent work with Yu.Bahturin,

where we consider the behavior of distortion functions

for subalgebras of associative and Lie algebras.

In the 1990's Haagerup discovered a new subfactor, and hence a new topological quantum field theory, that has so far proved inaccessible by the methods of quantum groups and conformal field theory. It was the subfactor of smallest index beyond 4. This led to a classification project-classify all subfactors to as large an index as possible. So far we have gone as far as index 5. It is known that at index 6 wildness phenomena occur which preclude a simple listing of all subfactors of that index. It is possible that wildness occurs at a smaller index value, the main candidate being approximately 5.236.

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.