Past Topology Seminar

The filtration on the infinite symmetric product of spheres by number of

factors provides a sequence of spectra between the sphere spectrum and

the integral Eilenberg-Mac Lane spectrum. This filtration has received a

lot of attention and the subquotients are interesting stable homotopy

types.

In this talk I will discuss the equivariant stable homotopy types, for

finite groups, obtained from this filtration for the infinite symmetric

product of representation spheres. The filtration is more complicated

than in the non-equivariant case, and already on the zeroth homotopy

groups an interesting filtration of the augmentation ideal of the Burnside

rings arises. Our method is by `global' homotopy theory, i.e., we study

the simultaneous behaviour for all finite groups at once. In this context,

the equivariant subquotients are no longer rationally trivial, nor even

concentrated in dimension 0.

Knot Floer homology (introduced by Ozsvath-Szabo and independently by

Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In

particular, it gives rise to a numerical invariant, which provides a

nontrivial lower bound on the 4-dimensional genus of the knot. By deforming

the definition of knot Floer homology by a real number t from [0,2], we define

a family of homologies, and derive a family of numerical invariants with

similar properties. The resulting invariants provide a family of

homomorphisms on the concordance group. One of these homomorphisms can be

used to estimate the unoriented 4-dimensional genus of the knot. We will

review the basic constructions for knot Floer homology and the deformed

theories and discuss some of the applications. This is joint work with

P. Ozsvath and Z. Szabo.

An invariant random subgroup in a (finitely generated) group is a

probability measure on the space of subgroups of the group invariant under

the inner automorphisms of the group. It is a natural generalization of the

the notion of normal subgroup. I’ll give an introduction into this actively

developing subject and then discuss in more detail examples of invariant

random subgrous in groups of intermediate growth. The last part of the talk

will be based on a recent joint work with Mustafa Benli and Rostislav

Grigorchuk.

In order to study the group of (outer) automorphisms of

any group G by geometric methods one needs a well-behaved "outer

space" with an interesting action of Out(G). If G is free abelian, the

classic symmetric space SL(n,R)/SO(n) serves this role, and if G is

free non-abelian an appropriate outer space was introduced in the

1980's. I will recall these constructions and then introduce joint

work with Ruth Charney on constructing an outer space for any

right-angled Artin group.

Open 2d TCFTs correspond to cyclic A-infinity algebras, and Costello showed

that any open theory has a universal extension to an open-closed theory in

which the closed state space (the value of the functor on a circle) is the

Hochschild homology of the open algebra.  We will give a G-equivariant

generalization of this theorem, meaning that the surfaces are now equipped

with principal G-bundles.  Equivariant Hochschild homology and a new ribbon

graph decomposition of the moduli space of surfaces with G-bundles are the

principal ingredients.  This is joint work with Ramses Fernandez-Valencia.

The A-theory characteristic of a fibration is a

map to Waldhausen's algebraic K-theory of spaces which

can be regarded as a parametrized Euler characteristic of

the fibers. Regarding the classifying space of the cobordism

category as a moduli space of smooth manifolds, stable under

extensions by cobordisms, it is natural to ask whether the

A-theory characteristic can be extended to the cobordism

category. A candidate such extension was proposed by Bökstedt

and Madsen who defined an infinite loop map from the d-dimensional

cobordism category to the algebraic K-theory of BO(d). I will

discuss the connections between this map, the A-theory

characteristic and the smooth Riemann-Roch theorem of Dwyer,

Weiss and Williams.

Trees are not just combinatorial structures: they are also

biological structures, both in the obvious way but also in the

study of evolution. Starting from DNA samples from living

species, biologists use increasingly sophisticated mathematical

techniques to reconstruct the most likely “phylogenetic tree”

describing how these species evolved from earlier ones. In their

work on this subject, they have encountered an interesting

example of an operad, which is obtained by applying a variant of

the Boardmann–Vogt “W construction” to the operad for

commutative monoids. The operations in this operad are labelled

trees of a certain sort, and it plays a universal role in the

study of stochastic processes that involve branching. It also

shows up in tropical algebra. This talk is based on work in

progress with Nina Otter [www.fair-fish.ch].

We will present a somewhat different proof of Agol's theorem that

3-manifolds 

with RFRS fundamental group admit a finite cover which fibers over S^1.

This is joint work with Takahiro Kitayama.

We will consider infinite translation surfaces which are abelian covers of

compact surfaces with a (singular) flat metric and focus on the dynamical

properties of their flat geodesics. A motivation come from mathematical

physics, since flat geodesics on these surfaces can be obtained by unfolding

certain mathematical billiards. A notable example of such billiards is  the

Ehrenfest model, which consists of a particle bouncing off the walls of a

periodic planar array of rectangular scatterers.

The dynamics of flat geodesics on compact translation surfaces is now well

understood thanks to the beautiful connection with Teichmueller dynamics. We

will survey some recent advances on the study of infinite translation

surfaces and in particular focus on a joint work with K. Fraczek,  in which

we proved that the Ehrenfest model and more in general geodesic flows on

certain abelain covers have no dense orbits. We will try to convey an

heuristic idea of how Teichmueller dynamics plays a crucial role in the

proofs.

A "bordism representation" (*) is a representation of the abstract

structure formed by manifolds and bordisms between them, and hence of

fundamental interest in topology. I will give an overview of joint work

establishing a simple generators-and-relations presentation of the

3-dimensional oriented bordism bicategory, and also its "signature" central

extension. A representation of this bicategory corresponds in a 2-1 fashion

to a modular category, which must be anomaly-free in the oriented case. J/w

Chris Douglas, Chris Schommer-Pries, Jamie Vicary.

(*) These are also known as "topological quantum field theories".

Computational structures---from simple objects like bits and qubits,

to complex procedures like encryption and quantum teleportation---can

be defined using algebraic structures in a symmetric monoidal

2-category. I will show how this works, and demonstrate how the

representation theory of these structures allows us to recover the

ordinary computational concepts. The structures are topological in

nature, reflecting a close relationship between topology and

computation, and allowing a completely graphical proof style that

makes computations easy to understand. The formalism also gives

insight into contentious issues in the foundations of quantum

computing. No prior knowledge of computer science or category theory

will be required to understand this talk.

Given a triangulated category A, equipped with a differential

Z/2-graded enhancement, and a triangulated oriented marked surface S, we

explain how to define a space X(S,A) which classifies systems of exact

triangles in A parametrized by the triangles of S. The space X(S,A) is

independent, up to essentially unique Morita equivalence, of the choice of

triangulation and is therefore acted upon by the mapping class group of the

surface. We can describe the space X(S,A) as a mapping space Map(F(S),A),

where F(S) is the universal differential Z/2-graded category of exact

triangles parametrized by S. It turns out that F(S) is a purely topological

variant of the Fukaya category of S. Our construction of F(S) can then be

regarded as implementing a 2-dimensional instance of Kontsevich's proposal

on localizing the Fukaya category along a singular Lagrangian spine. As we

will see, these results arise as applications of a general theory of cyclic

2-Segal spaces.

This talk is based on joint work with Mikhail Kapranov.

There are various Floer-theoretical invariants of links and 3-manifolds

which take the form of homology groups which are the E_infinity page of

spectral sequences starting from Khovanov homology. We shall discuss recent

work, joint with Raphael Zentner, and work in progress, joint with John

Baldwin and Matthew Hedden, in investigating and exploiting these spectral

sequences.

The HOMFLY polynomial is an invariant of knots in S^3 which can be

extended to an invariant of tangles in B^3. I'll give a geometrical

description of this invariant for rational tangles, and

explain how this description extends to a more general invariant

(the lambda^k colored HOMFLY polynomial of a rational tangle). I'll then

use this description to sketch a proof of a conjecture of Gukov and Stosic

about the colored HOMFLY homology of rational knots.

Parts of this are joint work with Paul Wedrich and Mihaljo Cevic.

Factorization homology is an invariant of an n-manifold M together with an n-disk algebra A. Should M be

a circle, this recovers the Hochschild complex of A; should A be a commutative algebra, this recovers the

homology of M with coefficients in A. In general, factorization homology retains more information about

a manifold than its underlying homotopy type.

In this talk we will lift Poincare' duality to factorization homology as it intertwines with Koszul

duality for n-disk algebras -- all terms will be explained. We will point out a number of consequences

of this duality, which concern manifold invariants as well as algebra invariants.

This is a report on joint work with John Francis.

The class of sofic groups was introduced by Gromov in 1999. It
includes all residually finite and all amenable groups. In fact, no group has been proved
not to be sofic, so it remains possible that all groups are sofic. Their
defining property is that, roughly speaking, for any finite subset F of
the group G, there is a map from G to a finite symmetric group, which is
approximates to an injective homomorphism on F. The widespread interest in
these group results partly from their connections with other branches of
mathematics, including dynamical systems. In the talk, we will concentrate
on their definition and algebraic properties.

A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the more general notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general commutative ground ring.

Operads provide a very nice way of saying what A-infinity algebras are - they are described by a kind of free resolution of a strictly associative structure. I will explain the analogous result for derived A_infinity algebras - these are obtained in the same manner from a strictly associative structure with an extra differential.

This is joint work with Muriel Livernet and Constanze Roitzheim.

 In this talk I will introduce my joint work with Kreck on a classification of
certain 5-manifolds with fundamental group Z. This result can be interpreted as a
generalization of the classical Browder-Levine's fibering theorem to dimension 5.

A generalized Baumslag-Solitar group is a group G acting co-compactly on a tree X, with all vertex- and edge stabilizers isomorphic to the free abelian group of rank n. We will discuss the $L^p$-metric and $L^p$-equivariant compression of G, and also the quasi-isometric embeddability of G in a finite product of binary trees. Complete results are obtained when either $n=1$, or the quotient graph $G\X$ is either a tree or homotopic to a circle. This is joint work with Yves Cornulier.

The mod p homology of a space is an unstable coalgebra over the Steenrod algebra at the prime p. This talk will be about the classical problem of realising an unstable coalgebra as the homology of a space. More generally, one can consider the moduli space of all such topological realisations and ask for a description of its homotopy type. I will discuss an obstruction theory which describes this moduli space in terms of the Andr\'{e}-Quillen cohomology of the unstable coalgebra. This is joint work with G. Biedermann and M. Stelzer.

Exact Lagrangian immersions are governed by an h-principle, whilst exact Lagrangian

embeddings are well-known to be constrained by strong rigidity theorems coming from

holomorphic curve theory. We consider exact Lagrangian immersions in Euclidean space with a

prescribed number of double points, and find that the borderline between flexibility and

rigidity is more delicate than had been imagined. The main result obtains constraints on such

immersions with exactly one double point which go beyond the usual setting of Morse or Floer

theory. This is joint work with Tobias Ekholm, and in part with Ekholm, Eliashberg and Murphy.

We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.

Orthogonal calculus is a calculus of functors, inspired by Goodwillie calculus. It takes as input a functor from finite dimensional inner product  spaces to topological spaces and as output gives a tower of  approximations by well-behaved functors.  The output captures a lot of important homotopical information and is an important tool for calculations.

In this talk I will report on joint work with Peter Oman in which we use model categories to improve the foundations of orthogonal calculus. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer.  The classification of n-homogeneous functors in terms of spectra with O(n)-action can then be phrased as a zig-zag of Quillen equivalences.

In 2005, Bonk and Kleiner showed that a hyperbolic group admits a

quasi-isometrically embedded copy of the hyperbolic plane if and only if the

group is not virtually free. This answered a question of Papasoglu. I will

discuss a generalisation of their result to certain relatively hyperbolic

groups (joint work with Alessandro Sisto). Key tools involved are new

existence results for quasi-circles, and a better understanding of the

geometry of boundaries of relatively hyperbolic groups.

By a "coarse median" we mean a ternary operation on a path metric space, satisfying certain conditions which generalise those of a median algebra. It can be interpreted as a kind of non-positive curvature condition, and is applicable, for example to finitely generated groups. It is a consequence of work of Behrstock and Minsky, for example, that the mapping class group of a surface satisfies this condition. We aim to give some examples, results and applications concerning this notion.

Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second fundamental groups of a topological space - hoops can be composed like in π1, balloons like in π2, and hoops "act" on balloons as π1 acts on π2. We will observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops.

We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant ζ of KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.

For further information see http://www.math.toronto.edu/~drorbn/Talks/Oxford-130121/

We define a McCool group of G as the group of outer automorphisms of G acting as a conjugation on a given family of subgroups. We will explain that these groups appear naturally in the description of many natural groups of automorphisms. On the other hand, McCool groups of a toral relatively hyperbolic group have strong finiteness properties: they have a finite index subgroup with a finite classifying space. Moreover, they satisfy a chain condition that has several other applications.
This is a joint work with Gilbert Levitt.

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.