# Scottish Topology Seminar

Expenses form (UK payments) Email to Liam.Watson@glasgow.ac.uk

Expenses form (foreign payments) Email to Liam.Watson@glasgow.ac.uk

--!!!Forthcoming Meetings:
**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.

- 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.

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

--!!!Past Meetings:
**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 manifoldsAn 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 surfacesWe 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 setIt 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 spaceThe 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 surfacesLet 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 homologyThere 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-theoryT-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.

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.

*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 <www.moffatt.tc/publications>):

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-manifoldsTo 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. - 16.30-17.30 Spiros Adams-Florou (Glasgow)

Simplicially controlled algebraWhen 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 symmetriesTwo 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 homologiesKnot 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

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!

For more information, one can look at

http://perso.ens-lyon.fr/ghys/index.html

http://perso.ens-lyon.fr/ghys/articles/lorenzparadigm-english.pdf

http://perso.ens-lyon.fr/ghys/articles/lorenzparadigme.pdf

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.

The complete film can be downloaded here:

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-‐manifoldsA 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 - Pouya Adrom (Glasgow)

*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*

### About

The Scottish Topology Seminar based at the School of Mathematics in Edinburgh was the first topology seminar in Scotland, initiated by Elmer Rees in 1981. Now the seminar is a joint venture between the universities of Aberdeen, Edinburgh and Glasgow, with at least three meetings per year. The organisers are:

- Diarmuid Crowley (Aberdeen)
- Richard Hepworth (Aberdeen)
- Brendan Owens (Glasgow)
- Andrew Ranicki (Edinburgh)
- Liam Watson (Glasgow)

The Scottish Topology logo was designed by Simon Willerton.

The Scottish Topology Seminar is supported by the Glasgow Mathematical Journal Trust.