Organised by Mark Grant, Richard Hepworth (Aberdeen), Clark Barwick (Edinburgh), Brendan Owens, Andy Wand (Glasgow)

### Next meeting

STS 20, University of Aberdeen, Monday September 10th, 2018
Everything takes place in the Fraser Noble Building

• 13:00 - 13:30
Lunch* in the Maths - Engineering Common Room
• 13:30 - 14:30 (in FN11)
Gregory Stevenson (Glasgow)
Singularity and cosingularity categories for cochains on BGI'll discuss joint work with John Greenlees aimed at developing a suitable notion of the bounded derived category' in the context of highly structured ring spectra. Having access to such a construction allows one to define categories measuring the failure of regularity and coregularity for these brave new rings and to develop a theory of Koszul duality. There are many examples, but one of particular interest is cochains on the classifying space of a group. I'll use this example, mostly for finite groups, to illustrate our construction and discuss what we do and don't know about the resulting categories in some special cases.
• 15:00 - 16:00 (in FN156)
Ehud Meir (Aberdeen)
Adams operations and finite groups representations

Let G be a finite group. By a theorem of Deligne the category of complex representations Rep-G determines G up to an isomorphism if one considers Rep-G as a symmetric monoidal category.
However, this is no longer true if one considers Rep-G only as a monoial category.

In this talk we will discuss the Adams operations in the context of finite groups representations. These are natural operations parametrized by the integers on the Grothendieck ring of Rep-G. These operations arise from the symmetric structure of the category.
I will show that all the odd Adams operations are in fact independent of the symmetric structure of Rep-G, and give some examples where this does not hold for the second Adams operation.
We will also discuss the possible symmetric structures on Rep-G, and the group of monoidal autoequivalences of Rep-G.

This talk is based on a joint work with Markus Szymik.

• 16:30 - 17:30 (in FN156)
Nina Otter (UCLA and MPI MiS Leipzig)
Homology theories for metric spaces

In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, an isometric invariant that encodes the “effective number of points” of the space, and is akin to the Euler characteristic of a space. In 2015 Hepworth and Willerton introduced a homology theory for metric spaces associated to finite graphs, called magnitude homology, which categorifies the magnitude of such spaces. Recently, Leinster and Shulman suggested a generalisation of magnitude homology for arbitrary metric spaces, which for finite spaces gives a categorification of the magnitude.

When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence.

In their paper Leinster and Shulman list a series of open questions, two of which are as follows:

1) Magnitude homology only “notices” whether the triangle inequality is a strict equality or not. Is there a “blurred” version that notices “approximate equalities”?

2) Almost everyone who encounters both magnitude homology and persistent homology feels that there should be some relationship between them. What is it?

In this talk I will first introduce persistent homology and magnitude homology, and attempt a first answer to these questions, which I believe are intertwined: it is the blurred version of magnitude homology that is related to persistent homology. Finally, I will discuss how ordinary and blurred version of magnitude homology differ in the limit: ordinary magnitude homology is trivial, while blurred magnitude homology coincides with Vietoris homology, and for compact metric spaces with Cech homology.

This talk is based on the preprint https://arxiv.org/abs/1807.01540

• 18:30 -
Dinner* in Union Square

For further information contact Richard Hepworth r.hepworth@abdn.ac.uk or Mark Grant mark.grant@abdn.ac.uk

* Please let Richard know by email (r.hepworth@abdn.ac.uk) if you would like to join us for lunch or for dinner.

### Past meetings

STS 19, Glasgow, Wednesday, 16th May 2017
12:15 - 17:00, Mathematics and Statistics Building: lunch in 110 Maths & Stats, coffee in Maths & Stats 3rd floor common room; all talks in 222 Kelvin Building

• 12:15 - 13:15 Lunch provided, location TBA
• 13:15 - 14:15 Anthony Conway (Durham)
Splitting numbers and signaturesThe splitting number of a link is the minimal number of crossing changes between different components required to convert it into a split union of its components. Since 2012, this invariant has been studied using various tools such as Khovanov homology, covering link calculus, the Alexander polynomial and Heegaard-Floer homology. After briefly reviewing some of these methods, we will show how (multivariable) signatures give strong lower bounds on the splitting number. This is joint work with David Cimasoni and Kleopatra Zacharova.
• 14:30 - 15:30 Viveka Erlandsson (Bristol)
Determining the shape of a billiard table from its bounces

Consider a billiard table shaped as a Euclidean polygon with labeled sides. A ball moving around on the table determines a bi-infinite “bounce sequence” by recording the labels of the sides it bounces off. We call the set of all possible bounce sequences the “bounce spectrum” of the table. In this talk I will explain why the bounce spectrum essentially determines the shape of the table: with the exception of a very small family (right-angled tables), if two tables have the same bounce spectrum, then they have to be related by a Euclidean similarity. The main ingredient in proving this fact is a technical result about Liouville currents for flat cone metrics.
This is joint work with Moon Duchin, Chris Leininger, and Chandrika Sadanand.

• 15:30 - 16:00 Coffee
• 16:00 - 17:00 Ana Lecuona (Glasgow)
Complexity and Casson-Gordon invariantsHomology groups provide bounds on the minimal number of handles needed in any handle decomposition of a manifold. We will use Casson-Gordon invariants to get better bounds in the case of 4-dimensional rational homology balls with boundary a given rational homology 3-sphere. This analysis can be used to understand the complexity of the discs associated to ribbon knots in S^3. This is a joint work with P. Aceto and M. Golla.

For further information contact Brendan Owens brendan.owens@glasgow.ac.uk or Andy Wand andy.wand@glasgow.ac.uk

STS 18, ICMS Edinburgh, Friday, 12th January 2018

• 13:30-14:30 Andy Baker (Glasgow)
A pack of Jokers and Kervaire invariant one elements

The original Joker J(1) is a very special A(1)-module (where A(1) is a certain subHopf algebra of the mod 2 Steenrod algebra) which represents the unique torsion element in the Picard group of the stable module category of A(1) (recent work shows that such exotic elements don't exist for A(2) and presumably for higher A(n)). Iterated doubling provides analogous A(n)-modules J(n). I will show that J(n) can be realised as the cohomology of a spectrum precisely when n=1,2,3. The constructions in these low cases rely on the fact that the first three

Kervaire invariant one elements theta_1=eta
2, theta_2=nu
2 and theta_3=sigma
2 lie in the Toda brackets <2,eta,2>, , which have trivial indeterminacy.

This raises the question of whether the next two known Kervaire elements theta_4 and theta_5 lead to some kind of interesting A(n)-modules. It is known that theta_4

is given by 4-fold brackets such as <2,sigma
2,2,sigma^2>.
• 14:40-15:40 Rachael Boyd (Aberdeen)
Homological stability for Artin monoids and groupsMany interesting sequences of groups satisfy the phenomenon known as "homological stability". Examples include the symmetric groups, braid groups, and mapping class groups of surfaces. I will provide an introduction to this topic and talk about my research on homological stability for certain families of Artin groups, which extends the known cases of stability for A_n, B_n and D_n to more general families of groups. I will explain the problem and the approach I have used, which exploits geometric properties of the Artin monoid.
• 15:40-16:20 Tea
• 16:20-17:20 Claudia Scheimbauer (Oxford)
Fully extended functorial field theories and dualizability in the higher Morita category

Atiyah and Segal's axiomatic approach to topological and conformal quantum field theories provided a beautiful link between the geometry of "spacetimes" (cobordisms) and algebraic structures. Combining this with the physical notion of "locality" led to the introduction of the language of higher categories into the topic.
Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms. After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualizablity in a factorization version’’ of the Morita category and using this, examples of low-dimensional field theories “relative” to their observables. An example will be given by polynomial differential operators, i.e. the Weyl algebra, in positive characteristic and its center. This is joint work with Owen Gwilliam.

STS 17, Aberdeen, Friday, 27th October 2017
11:45 - 17:15, Fraser Noble Building. First talk in FN2, all other talks in FN156

• 11:45 - 12:30 William Rushworth (University of Durham)
Doubled Khovanov homology

Virtual knot theory is an extension of classical knot theory which considers knots and links in equivalence classes of thickened orientable surfaces.
Khovanov homology is a powerful invariant of classical links, and it can be applied to virtual links using Z_2 coefficients. However, a number of problems arise when one attempts to use other coefficient rings. In this talk we describe doubled Khovanov homology: an extension of Khovanov homology to virtual links with arbitrary coefficients. Unlike other extensions of Khovanov homology, doubled Khovanov homology requires no new diagrammatics, as all the work is done algebraically. We shall describe the construction of the invariant as well as some of its applications, particularly those related to virtual knot concordance.

• 12:30 - 13:15 Lunch provided in Mathematics Common Room
• 13:15 - 14:00 Renee Hoekzema (University of Oxford)
Manifolds with odd Euler characteristic and higher orientability

Orientable manifolds have even Euler characteristic unless the dimension is a multiple of 4. I give a generalisation of this theorem: k-orientable manifolds have even Euler characteristic (and in fact vanishing top Wu class), unless their dimension is 2^{k+1}m for some m > 0. Here we call a manifold k-orientable if the i^{th} Stiefel-Whitney class vanishes for all i< 2^k. This theorem is strict for k=0,1,2,3, but whether there exist 4-orientable manifolds with an odd Euler characteristic is an open question.
An argument similar to Adams' work on the Hopf invariant one theorem yields that furthermore from k=4 on, m>1. This means that the lowest dimension in which we might hope to find a 4-orientable odd Euler characteristic manifold is 64. I present the results of calculations on the cohomology of the second Rosenfeld plane, a special 64-dimensional manifold with odd Euler characteristic.

• 14:15 - 15:00 Alan McLeay (University of Glasgow)
Mapping Class Groups, Covers, and Braids

The mapping class group of a surface is the group of isotopy classes of boundary preserving homeomorphisms of the surface. Given a finite sheeted covering space between surfaces we may ask what relationship, if any, exists between the mapping class groups of the two surfaces?
In joint work with Tyrone Ghaswala we investigate this question for surfaces with non-empty boundary. In this talk we will discuss a classical theorem of Birman and Hilden and lay the groundwork for a study of the evaluated Burau representation of braid groups.

• 15:00 - 15:30 Coffee
• 15:30 - 16:15 Tom Hockenhull (Imperial College London)
An equivalence between Heegaard Floer objectsBordered Heegaard Floer homology is a powerful invariant for three-manifolds with boundary, which generalises the Heegaard Floer homology for closed three-manifolds of Ozsváth and Szabó. I will show how, in the case of three-manifolds with torus boundary, this invariant is completely determined by well-understood polygon maps in Heegaard Floer homology.
• 16:30 - 17:15 Katie Vokes (University of Warwick)
Geometry of the separating curve graphThere are many graphs and complexes we can associate to a surface whose vertices are curves or collections of curves in the surface. A first example is the curve graph, which has a vertex for each isotopy class of curves, with edges corresponding to disjointness. Typically, these complexes are defined in such a way that they have a natural isometric action of the mapping class group, and they have proved to be useful tools in studying both geometric and algebraic properties of these groups. Masur and Minsky, in 2000, gave a distance estimate for the mapping class group in terms of a sum of certain projections to the curve graphs of subsurfaces. We will present a result which gives a similar distance estimate for the separating curve graph, making use of the concept of hierarchical hyperbolicity defined by Behrstock, Hagen and Sisto.

This is a special STS organized by Rachael Boyd, with all talks being given by finishing graduate students. For further information contact Rachael Boyd r01rjb14@abdn.ac.uk, Mark Grant mark.grant@abdn.ac.uk or Richard Hepworth r.hepworth@abdn.ac.uk

STS 16, Glasgow, Wednesday, 3rd May 2017
12:15 - 17:15, Mathematics and Statistics Building: All talks in TBA

• 12:15 - 13:15 Lunch provided, location TBA
• 13:15 - 14:00 Joan Licata (ANU)
Morse Structures on Open BooksEvery contact 3-manifold is locally contactomorphic to the standard contact R^3, but this fact does not necessarily produce large charts that cover the manifold efficiently. I'll describe joint work with Dave Gay and Dan Mathews which uses an open book decomposition of a contact manifold to produce a particularly efficient collection of such contactomorphisms, together with simple combinatorial data describing how to reconstruct the contact 3-manifold from these charts. We use this construction to define front projections for Legendrian knots and links in arbitrary contact 3-manifolds, with or without convex boundary. This approach generalises existing constructions of front projections for Legendrian knots in S^3 and universally tight lens spaces.
• 14:15 - 15:00 Geoffrey Powell (Angers)
Algebraic infinite deloopingAbstract TBA.
• 15:00 - 15:30 Coffee
• 15:30 - 16:15 Nick Kuhn (UVa)
Hurewicz maps for infinite loopspaces

In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n-sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1,2,4, 8. This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace.

I have been studying such Hurewicz maps for generalized homology theories by relating the Adams filtration of the domain to an "augmentation ideal" filtration of the range. When specialized to ordinary mod p homology, my general results have some tidy consequences, with examples including including Milnor's theorem, a variant with ko replaced by tmf, a new proof of irreducibility results of Steve Wilson, and a new source of minimal atomic complexes as studied by Andy Baker and Peter May.

• 16:30 - 17:15 Tony Licata (ANU)
Braid groups and the categorified root latticeCoxeter groups (e.g. the symmetric group) can be studied using tools of finite-dimensional linear algebra because they have faithful finite dimensional representations. At least conjecturally, the braid groups of arbitrary Coxeter groups can be studied in a somewhat similar manner, using a faithful "finite-dimensional " 2-representation instead of an ordinary representation. The goal of this talk will be to explain what this means (and what kind of structure one uncovers) in the case of the ordinary type A Artin braid group.

For further information contact Brendan Owens brendan.owens@glasgow.ac.uk or Andy Wand andy.wand@glasgow.ac.uk

STS 15, Aberdeen, Tuesday, 13th December 2016 Poster
13:30 - 17:30, Institute of Mathematics, Fraser Noble Building, all talks in FN 156

• 13:00 - 13:30 Lunch in the Mathematics Common Room
• 13:30 - 14:30 Brendan Owens (Glasgow)
“Ribbon surfaces and concordance invariants”Abstract: In 1978, Gordon and Litherland described a pairing on the first homology of a spanning surface of a link in the 3-sphere, and used it to give a formula for the link signature. In 2015, Greene used Gordon-Litherland pairings to characterise alternating links. In this talk I will describe a Gordon-Litherland-type pairing for ribbon-immersed surfaces. I will discuss applications, including a proof of an identity between link signature and an Ozsvath-Szabó correction term for a class of links, and a possible application to closed nonorientable surfaces in the 4-sphere. This is joint work in progress with Greene and Strle.
• 14:45 - 15:45 Andy Wand (Glasgow)
“Filtering the Heegaard Floer contact invariant”The modern development of contact geometry in 3 dimensions has seen several (due to Giroux, Wendl, Latschev and Wendl, Hutchings, and others) invariants of contact structures meant in some sense to measure non-(Stein/symplectic)-fillability of the structure. We will discuss a new approach which uses Heegaard Floer homology to define an invariant with a similar aim, but which has several desirable properties lacking in earlier approaches. This is joint with joint work with Kutluhan, Matic, and Van Horn-Morris.
• 16:15 - 17:15 Diarmuid Crowley (Aberdeen)
“On the kappa-classes and a theorem of Madsen”The kappa-classes lie in the mod 2 cohomology of the space SG and correspond to the Kervaire invariant when evaluated on the homotopy groups of SG. The space SG is an infinite loop space and in his PhD Thesis Madsen proved that the kappa-classes cannot be delooped three times, whereas Madsen and Milgram proved these classes deloop twice. I will discuss an observation relating to Madsen’s theorem and applications.

STS 14, Edinburgh, Friday, 30th September 2016 (ICMS, 10.00-17.30)
Special seminar in honour of Andrew Ranicki on the occasion of his retirement from the Chair of Algebraic Surgery at the University of Edinburgh.

• 9:30 - 10:15 Coffee
• 10:15 - 11:15 Michael Atiyah (Edinburgh)
“The Kervaire invariant”Abstract. Andrew Ranicki and I, in collaboration with Jurgen Berndt of KCL, have been exploring, for some years, the tempting possibility that the famous Freudenthal-Tits magic square might shed light on the work of Hopkins et al on the Kervaire invariant 1 problem. Recently we have made encouraging progress which I will report on in my lecture.
• 11:30 - 12:30 Graeme Segal (Oxford)
“Poincaré duality”Andrew Ranicki has created an algebraic theory of surgery. I shall reflect on the place of this in the wider realm of topology, and why its use of chain complexes is so effective. One can contrast the problem of looking for locally well-behaved spaces in a homotopy type with the problem of looking for manifolds in the homotopy type: the local property of being a manifold is to a surprising extent encoded in the global property of Poincaré duality. I shall also discuss the conceptual change made by Ranicki’s new view of the surgery exact sequence, and how his assembly’ perspective relates to the role of the fundamental group in homotopy theory more generally.
• 14:00 - 15:00 Stefan Friedl (Regensburg)
“Polytope invariants of spaces”

We assign a formal difference of polytopes to L^2-acyclic spaces and groups and study their properties.
This is based on joint work with Jae Choon Cha, Florian Funke, Wolfgang Lueck, Kevin Schreve and Stephan Tillmann.

• 15:15 - 16:15 Tara Brendle (Glasgow)
“The integral Burau representation of braid groups”The Burau representation plays a key role in the classical theory of braid groups. When we let the complex parameter t take the value -1, we obtain a symplectic representation of the braid group, known as the integral Burau representation. In this talk we will give a survey of recent work with Margalit and others on braid congruence subgroups, that is, on the preimages in the braid group of the principal congruence subgroups of the symplectic groups. This survey will include a discussion of the kernel of the integral Burau representation itself, as well as connections with a wide variety of areas, including algebraic geometry and number theory.
• 16:15 - 16:45 Coffee
• 16:45 - 17:45 Erik Pedersen (Copenhagen)
“On Ranicki's work”The talk will give a number of examples of what Algebraic Surgery can be used for.

Website (including registration)
Poster.
Photos.
For further information contact Diarmuid Crowley dcrowley@abdn.ac.uk or Richard Hepworth r.hepworth@abdn.ac.uk

STS 13, Glasgow, Monday, 16th May 2016 Poster
12:15 - 17:00, Mathematics Building: All talks in 516

• 12:15 - 13:15 Lunch in the Mathematics Common Room
• 13:15 - 14:14 Diarmuid Crowley (Aberdeen)
Obstructions to Stein fillings of almost contact manifolds

An almost contact structure on a (2q+1)-manifold M is a reduction of its structure group of M to unitary group U(q). A special class of almost contact structure arise when M is the boundary of a Stein domain.

In earlier work, we showed how Eliashberg's h-principle for Stein domains leads to a bordism-theoretic characterisation of Stein fillable almost contact manifolds. Building on this, we define a new and subtle invariant of almost contact manifolds which is an obstruction to Stein fillability, even after the addition of any almost contact homotopy sphere.

As an example, we show that S∧6 x S∧7 admits almost contact structures which are not Stein-fillable, even after the addition of any almost contact homotopy sphere.

This is part of joint work with Jonathan Bowden and Andras Stipsicz.

• 14:30 - 15:30 Matthias Nagel (CIRGET/UQaM)
Representation varieties and essential surfaces

We recall Culler-Shalen's construction of essential surfaces in a 3-manifold,
which uses the representation variety of SL(2,C).
After generalising this construction to SL(n,C) representations, we explain
how all essential surfaces can be obtained from it.

Based on joint work with Stefan Friedl and Takahiro Kitayama.

• 16:00 - 17:00 Mark Powell (UQaM)
Gropes and metrics on the knot concordance set

It was posited by Cochran, Harvey and Leidy that knot concordance ought to exhibit some kind of fractal structure.
A grope is a special type of 2-complex built as a union of surfaces with boundary, that approximates a disc. We will associate a rational number to a grope, that measures its failure to be a disc. By considering embeddings of these objects in 4-space, we will define a pseudo-metric on the set of concordance classes of knots. In the talk, which is based on joint work with Tim Cochran and Shelly Harvey, I will define these notions, I will discuss the interesting properties that our metric possesses, and I will discuss how these give evidence towards the existence of a fractal structure.

For further information contact Brendan Owens brendan.owens@glasgow.ac.uk or Liam Watson liam.watson@glasgow.ac.uk

STS 12, Edinburgh, Friday, 11th March 2016 (ICMS)

• 14:00 - 15:00 Michael Weiss (Muenster)
Rational Pontryagin classes of fiber bundles with fiber a euclidean space

The i-th Pontryagin characteristic class of a vector bundle is a class in the integral cohomology of degree 4i of the base space. If deRham cohomology is used, and the base space is a smooth manifold, and the vector bundle is a smooth vector bundle
equipped with a connection, then the Pontryagin classes admit a description in terms of the curvature of that connection (Chern-Weil theory). A key step in the development of surgery theory, accomplished by Thom and Novikov, was to show that the Pontryagin classes can also be defined for a fiber bundle with fiber homeomorphic to a euclidean space (without specified vector space structures on the fibers), provided that cohomology with rational coefficients is used. Such fiber bundles arise naturally in manifold topology as tangent bundles of topological manifolds. Understanding their classifying space(s) is therefore also essential when it comes to classifying smooth structures on a topological manifold of dimension >4.

The Pontryagin classes are stable characteristic classes: they do not change when a vector bundle (or fiber bundle with fiber homeomorphic to a euclidean space) is replaced by its Whitney sum with a trivial line bundle. Nevertheless a few interesting things can be said about the Pontryagin classes of vector bundles with a specified fiber dimension n. For example, the Pontryagin classes of such a vector bundle vanish in cohomology dimensions greater than 2n. If n is even and the vector bundle is oriented, then the Pontryagin class in degree 2n is the square of the Euler class in degree n.

For many years I tried to show that these relations or vanishing results are also satisfied in the absence of vector space structures, i.e., in the case of fiber bundles with fiber homeomorphic to a euclidean space and their rational Pontryagin classes. Only about 3 years ago I began to understand that this is not the case. The counterexamples are based on a method from 1960s differential topology called plumbing, enhanced with more recent results from parameterized surgery and functor calculus.

• 15.10-16.10 Carmen Rovi (MPIM, Bonn)
The signature mod 8 of a fibrationIn this talk we shall be concerned with the residues modulo 4 and 8 of the signature σ(M) in Z of an oriented 4k-dimensional geometric Poincaré complex M^{4k}. The precise relation between the signature modulo 8, the Brown-Kervaire invariant was worked out by Morita some 40 years ago. We shall discuss how the relation between these invariants and the Arf invariant can be applied to the study of the signature modulo 8 of a fibration. In particular it had been proved by Meyer in 1973 that a surface bundle has signature divisible by 4. This was generalized to higher dimensions by Hambleton, Korzeniewski and Ranicki in 2007. I will explain two results from my thesis concerning the signature modulo 8 of a fibration: firstly under what conditions can we guarantee divisibility of the signature by 8 and secondly what invariant detects non-divisibility by 8 in general.
• 16.10-16.40 Tea
• 16.40-17.40 Oscar Randal-Williams (Cambridge)
Realising characteristic numbers of fibre bundlesI will first give an introduction to my ongoing work with S. Galatius concerning the cohomology of moduli spaces of manifolds. I will then explain how this may be used to address certain realisation questions for fibre bundles: when is there a fibre bundle F -> E -> B with F, E, and B having prescribed characteristic numbers? In particular, I will show how one can deduce the existence of fibre bundles on which the signature is not multiplicative modulo 8.

For further information contact Andrew Ranicki a.ranicki@ed.ac.uk

STS 11, Glasgow, Monday 7th December 2015 Poster
12:15 - 17:00, Mathematics Building: All talks in 416

• 12:15 - 13:15 Lunch in the Mathematics Common Room
• 13:15 - 14:15 Clark Barwick (Glasgow/MIT)
Transfers in equivariant stable homotopy theoryIn this talk, I will explain how to model the seemingly very delicate topological act of stabilization with respect to representation spheres of groups with purely algebraic structures - Mackey functors - and to rewire the whole of equivariant homotopy theory accordingly. This has two benefits: (1) Stripping out these structures permits us to get extremely refined information about – and universal characterizations of – the basic constructions of equivariant stable homotopy theory. (2) At the same time, we are now able to untether equivariant stable homotopy theory from the world of groups; this opens the door to many more interesting structures and many more interactions with other areas.
• 14:30 - 15:30 Andrew Lobb (Durham)
A stable homotopy type for colored Khovanov cohomologyLipshitz and Sarkar recently gave a new knot invariant. This takes the form of a stable homotopy type whose cohomology recovers Khovanov cohomology. Khovanov cohomology is a categorification of the Jones polynomial which arises from the ("quantized") fundamental representation of sl(2). According to Reshetikhin-Turaev, if one takes any semisimple Lie algebra and an irreducible representation, then there is an associated knot polynomial. The colored Jones polynomials arise from the other irreducible representations of sl(2), and they also admit categorifications - the colored Khovanov cohomologies. On the other hand, the polynomials arising from the fundamental representations of sl(n) are categorified by Khovanov-Rozansky cohomology. We discuss the construction and computations of a /putative/ Khovanov-Rozansky stable homotopy type (joint work with Dan Jones and Dirk Schuetz) and the construction and computation of a colored Khovanov stable homotopy type (joint work with Patrick Orson and Dirk Schuetz). Knowledge of Khovanov cohomology will not be assumed.
• 16:00 - 17:00 Stefan Schwede (Bonn)
Equivariant bordism from the global perspectiveGlobal homotopy theory is, informally speaking, equivariant homotopy theory in which all compact Lie groups acts at once on a space or a spectrum, in a compatible way. In this talk I will advertise a rigorous and reasonably simple formalism to make this precise, using orthogonal specctra. I will then illustrate the formalism by a geometrically motivated example, namely equivariant bordism of smooth manifolds.
• Dinner to follow.
• To be followed by 3 further days of Workshop on equivariant stable homotopy theory and parametrized higher category theory, organised by Andy Baker.

For further information contact Brendan Owens brendan.owens@glasgow.ac.uk or Liam Watson liam.watson@glasgow.ac.uk

STS 10, Aberdeen, Thursday 10th September 2015 Poster
13:30 - 17:30, Institute of Mathematics, Fraser Noble Building, all talks in FN 156

• 12:30 - 13:30 Lunch in the Mathematics Common Room
• 13:30 - 14:30 Jarek Kedra (Aberdeen)
Braids and the complexity of diffeomorphisms of surfaces

Let S be an oriented surface equipped with an area form. A function H:S --> R defines a vector field X by the formula area(X,-)=dH. The flow of this vector field preserves the area and a diffeomorphism obtained as an element of such flow is called autonomous. The flow lines of X are contained in the level sets of the function H and that is why autonomous diffeomorphisms are easy to understand (draw) in terms of the function H.

Every area preserving diffeomorphism isotopic to the identity is a product of a number of autonomous ones. The number of factors can be thought of as a measure of a complexity of a given diffeomorphism. The main result of the talk is to show that there are arbitrarily complex diffeomorphisms (in the above sense).

For the proof I will construct a function F:Diff(S,area) --> R whose value F(g) will bound below the number of autonomous diffeomorphisms necessary to represent a given diffeomorphism g. The function F will be constructed from braids obtained by evaluating an isotopy from the identity to g at a collection of points of S.

• 15:00 - 16:00 Alexander Berglund (Stockholm)
Automorphisms of high dimensional manifolds and graph homology

There is a classical programme for understanding diffeomorphisms of high dimensional manifolds whereby one studies, in turn, the monoid of homotopy automorphisms, the block diffeomorphism group, and finally the diffeomorphism group. The difference in each step is measured by, respectively, the surgery exact sequence and, in a range, Waldhausen's algebraic K-theory of spaces.

In recent joint work with Ib Madsen, we calculated the rational cohomology of the block diffeomorphism group of the g-fold connected sum of S x S minus a disk (2d>4), in a stable range (S the d-sphere). Our result is expressed in terms of a certain decorated graph complex, which, quite surprisingly, is related to the "hairy graph complex" introduced by Conant-Kassabov-Vogtmann in the study of automorphism groups of free groups. An immediate corollary is that the cohomology of the block diffeomorphism group is much larger than that of the diffeomorphism group. We also have conjectures for what graph homology classes correspond to the generalized Miller-Morita-Mumford classes.

• 16:30 - 17:30 Thomas Schick (Göttingen)
Topological T-duality and twisted K-theory

T-Duality for phycisists is a (conjectured) equivalence of certain models of string theory. This involves a lot of data. We as topologists will concentrate on the role of the topological data.

This means:
we have a topological space E, given as principal k-torus bundle over a base space B (for physicists, E is a "background space-time compactified along k torus directions). Moreover, E comes with a twist t for topological K-theory (for physicists, this is a background field).

T-duality means now that we look for another k-torus bundle E' over B with twist t' for topological K-theory (a "dual space-time with background field). The main topological consequence of the duality is that the t-twisted K-theory of E has to be isomorphic to the t'-twisted K-theory of E' (in physics, these groups correspond to certain "charges", and equivalent theories in particular have the same set of charges).

In the talk, we will describe a precise mathematical setup for the following:
a) twisted K-theory which has many appearences in topology beyond T-duality;
b) topological T-duality as a relation betwenn (E,t) and (E',t'). In our precise mathematical setup we will address question of "esistence of the dual", "uniqueness of the dual";
c) in the special case k=1 we will describe topological T-duality as a transformation producing (E',t') from (E,t).

• 6:15 Dinner at Goulash restaurant (17 Adelphi Ln)

For further information contact Diarmuid Crowley dcrowley@abdn.ac.uk or Richard Hepworth r.hepworth@abdn.ac.uk

STS 9, ICMS, 15 South College Street, Edinburgh, Thursday, 14 May Hydrodynamics and Topology: Meeting in honour of Keith Moffatt's 80th birthday 2-6PM.

2.00-2.50 Gunnar Hornig (Dundee) Magnetic Helicity: applications and generalisations

Magnetic helicity is an integral that measures the averaged pairwise linkage of field lines in a magnetic field. In a seminal paper Keith Moffatt JFM,1969 introduced and interpreted this quantity. It turned out that the magnetic helicity integral is preserved to a high accuracy in many astrophysical and technical plasmas and hence it has been widely used to help to understand the evolution of magnetic fields in plasmas. One of the most successful applications of magnetic helicity is the prediction of the final state of the magnetic field after a turbulent relaxation in a Reversed Field Pinch by J.B. Taylor. We will review the concepts behind these results and then discuss recent attempts to refine this theory using the notion of a field line helicity or generalised flux function. This concept can reveal additional information about the topology of a magnetic field which the total magnetic helicity does not capture. It thus helps to predict the evolution of more general classes of magnetic fields.
Moffatt, H. K. (1969). The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics, 35(01), 117–129.
Taylor, J. (2000). Relaxation revisited. Physics of Plasmas, 7(5), 1623–1629.
Yeates, A. R., & Hornig, G. (2013). Unique topological characterization of braided magnetic fields. Physics of Plasmas, 20(1), 012102. doi:10.1063/1.4773903

2.55-3.45 Etienne Ghys (ENS-Lyon) Is helicity a topological invariant?

3.45-4.15 Tea

4.15-5.05 Michael Proctor (DAMTP, Cambridge) Mean-field Electrodynamics in the nonlinear regime?Keith Moffatt has made seminal contributions to the theory of mean-field electrodynamics - the theory of magnetic field generation on scales large compared to those of the velocity fields that drive the dynamo. But there is another kind of dynamo, usually called a small-scale dynamo, which has magnetic field and velocity scales that are comparable. It is thus possible to have a state of essentially homogeneous MHD turbulence where the small scale magnetic field is dynamically active. What does it then mean to look for large scale magnetic instabilities? In this case the induction and momentum equations are on an equal footing, and the linear perturbation problem has to involve both the equations. The question then arises: can a coherent mean-field theory be constructed for the analysis of such long-wavelength modes? For relatively simple states the answer is in the affirmative, producing extended mean field equations with new coupling terms, providing a unification of the AKA instability of Frisch and the usual mean-field electrodynamics. However the validity of the ansatz seems to depend on properties of the basic state, and I will try to show by means of simple examples how things can go wrong. At present the question of whether the new mean field system is useful is open, pending more detailed numerical investigation.

5.10-6.00 Keith Moffatt (DAMTP, Cambridge) introduced by Michael Atiyah, Topological jumps in deforming soap films and in vortex dynamics

Suppose that a flexible circular wire is twisted and folded back on itself to form (nearly) the double cover of a circle, then dipped in soap solution in such a way as to create a soap film in the form of a Möbius strip. Suppose now that the wire is slowly untwisted and unfolded back towards its original circular form. At a certain critical stage in this process, the film jumps from the one-sided Möbius strip to a two-sided surface spanning the wire. We have analysed both experimentally and theoretically how this topological jump occurs. This involves consideration of the role of the finite cross-section of the wire, no matter how small this may be. The surface before the jump may be idealised as the (minimum area) incomplete 'Meeks surface', which becomes unstable at a critical value of its defining parameter. This topological jump is, in certain respects, analogous to the jump that occurs when a circular vortex (or magnetic flux) tube is twisted to the form of a figure-of-eight and forced to reconnect to form two separate tubes through viscous diffusion. This process will also be described, and it will be shown that helicity, a topological invariant of the ideal Euler equations, is no longer invariant during such a reconnection process. References (downloadable from ): Goldstein, R. E., McTavish, J., Moffatt, H. K. & Pesci, A. I. 2014 Boundary singularities produced by the motion of soap films. Proc. Natl. Acad. Sci. 111 (23), 8339-8344. Kimura, Y. & Moffatt, H.K. 2014 Reconnection of skewed vortices. J. Fluid Mech. 751, 329-345. Moffatt, H. K. 2014 Helicity and singular structures in fluid dynamics. Proc. Nat. Acad. Sci. 111 (10), 3663-3670.

For further information contact Andrew Ranicki a.ranicki@ed.ac.uk

STS 8, ICMS, 15 South College Street, Edinburgh, Thursday, 19th March
Poster

• 13:00 - 15:00, Saul Schleimer (Warwick), SMSTC Geometry/Topology guest lecture.
Recognizing three-manifolds

To the eyes of a topologist manifolds have no local properties: every point has a small neighborhood that looks like euclidean space. Accordingly, as initiated by Poincaré, the classification of manifolds is one of the central problems in topology. The homeomorphism problem'' is somewhat easier: given a pair of manifolds, we are asked to decide if they are homeomorphic.

These problems are solved for zero-, one-, and two-manifolds. Even better, the solutions are effective'': there are complete topological invariants that we can compute in polynomial time. On the other hand, in dimensions four and higher the homeomorphism problem is logically undecidable.

This leaves the provocative third dimension. Work of Haken, Rubenstein, Casson, Manning, Perelman, and others shows that these problems are decidable. Sometimes we can do better: for example, if one of the manifolds is the three-sphere then I showed that the homeomorphism problem lies in the complexity class NP. In joint work with Marc Lackenby, we show that recognizing spherical space forms also lies in NP. If time permits, we'll discuss the standing of the other seven Thurston geometries.

• 15.30 - 16.30, Maciej Borodzik (Warsaw)
Heegaard Floer homologies and unknotting sequences of torus knotsStudying the unknotting sequences of torus knots is the topological counterpart of studying adjacency of singularities of plane curves. I will discuss two obstructions for one torus knot to belong to a minimal unknotting sequence of another torus knot: one is d-invariants of large surgeries and the other is the Ozsvath-Szabo-Stipsicz Upsilon function. I will show that the second one can be deduced from the first one via the Fenchel--Legendre transform. Joint work with Matt Hedden and Charles Livingston.
• 16.40 - 17.40, Jacob Rasmussen (Cambridge)
L-space Filling SlopesAn intriguing conjecture of Boyer, Gordon, and Watson relates the Floer homology property of being an L-space with an algebraic condition (non left-orderability) on \pi_1. Boyer and Clay generalized this conjecture to manifolds with toroidal boundary by introducing the notion of "detected slopes." The simplest example to consider that of a homology S1xD2 which admits more than one L-space filling. I'll characterize the set of L-space filling slopes on such a manifold and discuss some applications to the work of Boyer and Clay. Joint work with Sarah Rasmussen.

For further information contact Andrew Ranicki a.ranicki@ed.ac.uk

STS 7, Aberdeen, Friday 28th November, 2014 Poster
13:30 - 17:30, Institute of Mathematics, Fraser Noble Building, all talks in FN 185

• 14.10-15.10 David Chataur (Lille)
Topology of complex projective varieties with isolated singularitiesI will explain a homotopical treatment of intersection cohomology recently developed in collaboration with Saralegui and Tanré, which associates a "perverse homotopy type" to every singular space. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. The perverse homotopy type of a complex projective variety with isolated singularities can be computed from the morphism of differential graded algebras induced by the inclusion of the link of the singularity into the regular part of the variety. I will show how, in this case, mixed Hodge theory allows us prove some intersection-formality results (work in progress with Joana Cirici).
• 15.20-16.20 Duncan McCoy (Glasgow)
Surgery and tangle replacement in alternating diagramsIt is conjectured that any knot with unknotting number one must have an unknotting crossing in a minimal diagram. Whilst still unresolved in general, this conjecture is now known to be true for alternating knots. The proof of these facts builds on the work of Greene, who showed that the Goeritz form of any alternating diagram of an unknotting number one knot must obey the 'changemaker' conditions. I will explain how these conditions allow you to identify unknotting crossings in the diagram. More generally, I will explain how to show that if the branched double cover of an alternating knot arises as non-integer surgery on a knot in the 3-sphere, then this surgery can be exhibited by tangle replacement in an alternating diagram.
Simplicially controlled algebra

When is a space homotopy equivalent to a manifold? This classic question in topology can often be tackled by imposing strong local conditions and proving a 'local to global' theorem. An important example of this is the alpha-approximation theorem of Chapman and Ferry.

This is an example of a typical strategy in controlled topology: show that a geometric obstruction can have a 'size' associated to it and that if its size is sufficiently small then the obstruction must vanish. Another theme in controlled topology is to define obstructions which live in 'geometric categories' which keep track of where in a space algebraic generators come from.

In this talk I will introduce the notion of simplicially controlled algebra, present some results concerning the detection of (homology) manifolds and, time permitting, mention how this approach fits into the surgery programme.

• 18:30-... Joint STS and SOAS diner
Nazma restaurant, 62 Bridge Street (next to the train station)

For further information contact Diarmuid Crowley dcrowley@abdn.ac.uk or Richard Hepworth r.hepworth@abdn.ac.uk

STS 6, Glasgow, Tuesday, 27th May, 2014 Poster

• 1:15-2:15 Michel Boileau (Marseille)
Commensurability of knot complements and hidden symmetries

Two knot complements are commensurable if they share a finite sheeted cover. For hyperbolic knots without hidden symmetries, commensurable knot complements are cyclically commensurable, which means that they have homeomorphic cyclic covers. This is no longer true for knot complements with hidden symmetries. To date, there are only three knots in S^3 which are known to admit hidden symmetries: the figure eight knot and the two commensurable dodecahedral knots. In this talk, I will discuss open questions and present new results in the case of small knots.
This is a joint work with Steve Boyer, Radu Cebanu and Genevieve Walsh.

• 2:30-3:30 Tim Riley (Cornell)
Hyperbolic groups, Cannon-Thurston maps, and hydraGroups are Gromov-hyperbolic when all geodesic triangles in their Cayley graphs are close to being tripods. Despite being tree-like in this manner, they can harbour extreme wildness in their subgroups. I will describe examples stemming from a re-imagining of Hercules' battle with the hydra, where wildness is found in properties of “Cannon-Thurston maps” between boundaries. Also, I will give examples where this map between boundaries fails to be defined.
• 4:00-5:00 Andras Stipsicz (Renyi Institute)
Knot Floer homologies

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.

For further information contact Brendan Owens brendan.owens@glasgow.ac.uk.

STS 5, Edinburgh, Thursday, 20th March, 2014
13:00 - 17:30

Etienne Ghys (ENS Lyon) "A chaotic afternoon with Etienne Ghys" The event will include a showing of the film "Chaos" directed by Ghys, and a QA session.Newhaven Room, ICMS, 15 South College Street, Edinburgh Programme

The programme will have two parts: a somewhat technical lecture, assuming that the audience knows what differential equations are, and a showing of a film on chaos theory produced by Ghys, which should be accessible to the layman.

1-3pm : A brief history of dynamics
According to Y. Ilyashenko, there are three main steps in the history of dynamical systems.
1- Newton : Given a differential equation, find its solutions!
2- Poincaré : Given a differential equation, say something about its solutions!
3- Smale : A differential equation is NOT given : say something about its solutions!
The goal of Etienne Ghys in this talk is to explain this joke. This will be an opportunity to discuss some fundamental
examples like periodic motions, quasi-periodic motions, Smale’s horseshoe and the famous
Lorenz butterfly, paradigmatic of chaos. More importantly, he will try to describe some of the current
conjectures. Unfortunately, one has to admit that this story, since Newton, is nothing more than a
succession of conjectures by great mathematicians, shown to be wrong by their successors.
Nevertheless, Ghys believes that we do understand the situation better than Newton!
http://perso.ens-lyon.fr/ghys/index.html

3-4pm : Tea/coffee break
4:00-5.30pm : A brief cinematic history of dynamics for the layman
In 2013 Jos Leys, Aurélien Alvarez and Etienne Ghys produced a film on chaos theory,
for the layman. Basically, this film tells the story of dynamics from Newton to current research,
explained in an elementary way. The total length of the film is about two hours,
so that it wouldn’t be reasonable to show it from A to Z. Instead, Etienne Ghys will
show some extracts, to explain the « making of », and discuss it with the audience.
http://www.chaos-math.org/en

Book tickets on Eventbrite

STS 4, Edinburgh, Thursday, 19th December, 2013
13:00 - 17:40, Newhaven Room, ICMS, 15 South College Street, Edinburgh

"From the history of topology" : four lectures.

• Jeremy Gray (Open & Warwick)
13.00-13.55 Poincaré and the study of surfacesThe study of surfaces was one of Henri Poincaré’s lifelong interests. He began in the early 1880s with the study of flows on surfaces, which he partly regarded as a preliminary to the study of celestial mechanics, and then switched to the study of complex differential equations and their connection to the study of complex (Riemann) surfaces. His discovery of the role of non-Euclidean geometry in the theory of Riemann surfaces led to a competition with the German mathematician Felix Klein, and to the conjecture of the uniformisation theorem, which was to resist proof for a further 25 years. Video
14.05-15.00 Poincaré and the creation of the theory of 3-manifoldsIn the opening years of the 20th century Poincaré was led to create a theory of three-dimensional manifolds, and to try to impose some order on a new subject in mathematics. How can three-dimensional manifolds be defined, and how can they be classified? Poincaré’s attempts to answer these questions led him to deepen the tools of algebraic topology and to pose – but, famously, not to answer – what became known as the Poincaré conjecture.
• June Barrow-Green (Open)
15.30-16.30 GD Birkhoff and the development of dynamical systems theoryIn October 1912, the young American mathematician GD Birkhoff 'astonished the mathematical world' by providing a proof of Poincaré's last geometric theorem. The theorem, which was connected to Poincaré's longstanding interest in the periodic solutions of the three-body problem, had been proposed by Poincaré only months before he died. Birkhoff continued to work on aspects of dynamical systems throughout his career, his aim being to create a general theory. Many of his ideas are contained in his book Dynamical Systems (1927), the first book to develop the qualitative theory of systems defined by differential equations and where he effectively 'created a new branch of mathematics separate from its roots in celestial mechanics and making broad use of topology'.
• Julia Collins (Edinburgh)
16.40-17.40 A Knot's Tale: the story of Peter Guthrie TaitPeter Guthrie Tait (1831 - 1901) was significantly less famous than his friends Maxwell and Kelvin, but unfairly so because he was an important and prolific mathematical physicist. He was Professor of Natural Philosophy at the University of Edinburgh from 1859, narrowly beating Maxwell to the post, and worked on a variety of topics including thermodynamics and the kinetic theory of gases. In a fantastic experiment involving smoke rings, Tait and Kelvin came up with a new atomic theory based around the idea of knots and links. This took on a mathematical life on its own, with Tait becoming one of the world's first topologists and inventing conjectures which remained unproven for over a hundred years.

Eventbrite booking form — tickets are free.

For further information contact Andrew Ranicki a.ranicki@ed.ac.uk

STS 3, Aberdeen, Friday, 20th November, 2013. Poster
12:30 - 17:00, Institute of Mathematics, Fraser Noble Building

• 13.15-14.15 Anne Thomas (Glasgow)
Infinite reduced words and the Tits Boundary of a Coxeter groupI will start by explaining the terms in my title. The main result is a theorem saying that the topology of the Tits boundary encodes a natural partial order on infinite reduced words in a Coxeter group. This project lies somewhere between algebraic combinatorics and geometric group theory, and is joint work with Thomas Lam.
• 14.30-15.30 Bernhard Hanke (Augsburg)
Fibre bundles over spheres and the space of positive scalar curvature metrics
• 16.00-17.00 Diarmuid Crowley (MPIM, Bonn)
Counting G_2 structures and smooth structures on 7-­‐manifolds

A G_2 structure on a 7-manifold M is a reduction of the structure
of the tangent bundle of M to the exceptional Lie group G_2.
G_2-structures are of interest in part because because they are an interesting topological trace left by a Riemannian 7-manifold with holonomy G_2.

Recently with Nordström we defined a new invariant of homotopy and diffeomorphisms for G_2-structures, showing that every spin 7-manifold admits
at least 24 deformation classes of G_2-structure. On the other hand, it is a classical result from the 1960s that a spin 7-manifold has at most 28 distinct smooth structures.

In this talk I report on further joint work with Nördstrom, where we show how counting G_2-structures and smooth structures on M are closely connected problems. Indeed, both have subtle and interesting answers related to the mapping class group of M and properties of the torsion linking form of M.

For further information contact Richard Hepworth r.hepworth@abdn.ac.uk

STS 2, Glasgow, Friday, 20th September, 2013. Poster

• Liam Watson (Glasgow)
Khovanov homology and the symmetry group of a knot
Video
• Stefan Friedl (Cologne)
Splittings of knot groups
Video
• Patrick Orson (Edinburgh)
Doubly slice knots and algebraic L-theory
Video
Modelling homotopy types by internal categories
Video
• Sophie Raynor (Aberdeen)
Towards a topological model of the brain
Video

STS 1, Edinburgh, Thursday, 21st March, 2013. Poster
ICMS, 15 South College Street, Edinburgh

• Sir Michael Atiyah (Edinburgh)
Geometry in the 21st Century
Video
• Marc Lackenby (Oxford)
Polynomial upper bounds on Reidemeister moves
Video
• Sir Michael Atiyah (Edinburgh)
50 years of index theory