Forthcoming events in this series


Mon, 27 Jan 2020
15:45
L6

Commensurable coHopficity and hyperbolic groups

Daniel Woodhouse
(Oxford University)
Abstract


A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

Mon, 20 Jan 2020
15:45
L6

Algorithms for infinite linear groups: methods and applications

Alla Detinko
(Mathematics Dept., University of Hull)
Abstract

In this talk we will survey a novel domain of computational group theory: computing with linear groups over infinite fields.  We will provide an introduction to the area, and will discuss available methods and algorithms. Special consideration is given to algorithms for Zariski dense subgroups. This includes a computer realization of the strong approximation theorem, and algorithms for arithmetic groups. We illustrate applications of our methods to the solution of problems further afield by computer experimentation.

Fri, 10 Jan 2020
15:45
L6

TBA

Christian Blanchet
(Institut de Mathématiques de Jussieu (Paris 7))
Mon, 02 Dec 2019
15:45
L6

A cellular decomposition of the Fulton Mac Pherson operad

Paolo Salvatore
(University of Rome `Tor Vergata')
Abstract

We construct a cellular decomposition of the
Axelrod-Singer-Fulton-MacPherson compactification of the configuration
spaces in the plane, that is compatible with the operad composition.
Cells are indexed by trees with bi-coloured edges, and vertices are labelled by 
cells of the cacti operad. This answers positively a conjecture stated in 
2000 by Kontsevich and Soibelman.

Mon, 25 Nov 2019
15:45
L6

Irrationality and monodromy for cubic threefolds

Ivan Smith
(Cambridge)
Abstract

The homological monodromy of the universal family of cubic threefolds defines a representation of a certain Artin-type group into the symplectic group Sp(10;\Z). We use Thurston’s classification of surface automorphisms to prove this does not factor through the genus five mapping class group.  This gives a geometric group theory perspective on the well-known irrationality of cubic threefolds, as established by Clemens and Griffiths.
 

Mon, 18 Nov 2019
15:45
L6

On the smooth mapping class group of the 4-sphere

David Gay
(University of Georgia/MPIM Bonn)
Abstract

The smooth mapping class group of the 4-sphere is pi_0 of the space of orientation preserving self-diffeomorphisms of S^4. At the moment we have no idea whether this group is trivial or not. Watanabe has shown that higher homotopy groups can be nontrivial. Inspired by Watanabe's constructions, we'll look for interesting self-diffeomorphisms of S^4. Most of the talk will be an outline for a program to find a nice geometric generating set for this mapping class group; a few small steps in the program are actually theorems. The point of finding generators is that if they are explicit enough then you have a hope of either showing that they are all trivial or finding an invariant that is well adapted to obstructing triviality of these generators.

Mon, 11 Nov 2019
15:45
L6

The Witt vectors with coefficients

Emanuele Dotto
(University of Warwick)
Abstract

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.

Mon, 04 Nov 2019
15:45
L6

The Euler characteristic of Out(F_n) and renormalized topological field theory

Michael Borinsky
(Nikhef)
Abstract

I will report on recent joint work with Karen Vogtmann on the Euler characteristic of $Out(F_n)$ and the moduli space of graphs. A similar study has been performed in the seminal 1986 work of Harer and Zagier on the Euler characteristic of the mapping class group and the moduli space of curves. I will review a topological field theory proof, due to Kontsevich, of Harer and Zagier´s result and illustrate how an analogous `renormalized` topological field theory argument can be applied to $Out(F_n)$.

Mon, 28 Oct 2019
15:45
L6

Towards Higher Morse-Cerf Theory: Classifying Constructible Bundles on R^n

Christoph Dorn
(Oxford)
Abstract

We present a programme towards a combinatorial language for higher (stratified) Morse-Cerf theory. Our starting point will be the interpretation of a Morse function as a constructible bundle (of manifolds) over R^1. Generalising this, we describe a surprising combinatorial classification of constructible bundles on flag foliated R^n (the latter structure of a "flag foliation” is needed for us to capture the notions of "singularities of higher Morse-Cerf functions" independently of differentiable structure). We remark that flag foliations can also be seen to provide a notion of directed topology and in this sense higher Morse-Cerf singularities are closely related to coherences in higher category theory. The main result we will present is the algorithmic decidability of existence of mutual refinements of constructible bundles. Using this result, we discuss how "combinatorial stratified higher Morse-Cerf theory" opens up novel paths to the computational treatment of interesting questions in manifold topology.

Mon, 21 Oct 2019
15:45
L6

Lower bounds on the tunnel number of composite spatial theta graphs

Scott Taylor
(Colby College)
Abstract

The tunnel number of a graph embedded in a 3-dimensional manifold is the fewest number of arcs needed so that the union of the graph with the arcs has handlebody exterior. The behavior of tunnel number with respect to connected sum of knots can vary dramatically, depending on the knots involved. However, a classical theorem of Scharlemann and Schultens says that the tunnel number of a composite knot is at least the number of factors. For theta graphs, trivalent vertex sum is the operation which most closely resembles the connected sum of knots. The analogous theorem of Scharlemann and Schultens no longer holds, however. I will provide a sharp lower bound for the tunnel number of composite theta graphs, using recent work on a new knot invariant which is additive under connected sum and trivalent vertex sum. This is joint work with Maggy Tomova.

Mon, 14 Oct 2019
15:45
L6

Uryson width and volume

Panos Papasoglu
(Oxford)
Abstract

I will give a brief survey of some problems in curvature free geometry and sketch

a new proof of the following:

Theorem (Guth). There is some $\delta (n)>0$ such that if $(M^n,g)$ is a closed aspherical Riemannian manifold and $V(R)$ is the volume of the largest ball of radius $R$ in the universal cover of $M$, then $V(R)\geq \delta(n)R^n$ for all $R$.

I will also discuss some recent related questions and results.

Mon, 07 Oct 2019
15:45
L6

Action rigidity for free products of hyperbolic manifold groups

Emily Stark
(University of Utah)
Abstract

The relationship between the large-scale geometry of a group and its algebraic structure can be studied via three notions: a group's quasi-isometry class, a group's abstract commensurability class, and geometric actions on proper geodesic metric spaces. A common model geometry for groups G and G' is a proper geodesic metric space on which G and G' act geometrically. A group G is action rigid if every group G' that has a common model geometry with G is abstractly commensurable to G. For example, a closed hyperbolic n-manifold group is not action rigid for all n at least three. In contrast, we show that free products of closed hyperbolic manifold groups are action rigid. Consequently, we obtain the first examples of Gromov hyperbolic groups that are quasi-isometric but do not virtually have a common model geometry. This is joint work with Daniel Woodhouse.

Mon, 24 Jun 2019
15:45
L6

Derived modular functors

Lukas Jannik Woike
(Hamburg)
Abstract

 For a semisimple modular tensor category the Reshetikhin-Turaev construction yields an extended three-dimensional topological field theory and hence by restriction a modular functor. By work of Lyubachenko-Majid the construction of a modular functor from a modular tensor category remains possible in the non-semisimple case. We explain that the latter construction is the shadow of a derived modular functor featuring homotopy coherent mapping class group actions on chain complex valued conformal blocks and a version of factorization and self-sewing via homotopy coends. On the torus we find a derived version of the Verlinde algebra, an algebra over the little disk operad (or more generally a little bundles algebra in the case of equivariant field theories). The concepts will be illustrated for modules over the Drinfeld double of a finite group in finite characteristic. This is joint work with Christoph Schweigert (Hamburg).

Thu, 20 Jun 2019

09:30 - 10:00
N3.12

From knots to homotopy theory

Markus Szymik
(NTNU)
Further Information

Note: unusual time!

Abstract

Knots and their groups are a traditional topic of geometric topology. In this talk, I will explain how aspects of the subject can be approached as a homotopy theorist, rephrasing old results and leading to new ones. Part of this reports on joint work with Tyler Lawson.

Mon, 17 Jun 2019
15:45
L6

The Teichmüller TQFT volume conjecture for twist knots

Fathi Ben Aribi
(Geneva)
Abstract

(joint work with E. Piguet-Nakazawa)

In 2014, Andersen and Kashaev defined an infinite-dimensional TQFT from quantum Teichmüller theory. This Teichmüller TQFT is an invariant of triangulated 3-manifolds, in particular knot complements.

The associated volume conjecture states that the Teichmüller TQFT of an hyperbolic knot complement contains the volume of the knot as a certain asymptotical coefficient, and Andersen-Kashaev proved this conjecture for the first two hyperbolic knots.

In this talk I will present the construction of the Teichmüller TQFT and how we approached this volume conjecture for the infinite family of twist knots, by constructing new geometric triangulations of the knot complements.

No prerequisites in Quantum Topology are needed.

Mon, 10 Jun 2019
17:00
L6

Curve complexes of Artin groups and Borel-Serre bordifications of hyperplane arrangement complements

Michael Davis
(Ohio State University)
Abstract

This is a report on work in progress with Jingyin Huang. The complement of an arrangement of linear hyperplanes in a complex vector space has a natural “Borel-Serre bordification” as a smooth manifold with corners. Its universal cover is analogous to the Borel-Serre bordification of an arithmetic lattice acting on a symmetric space as well as to the Harvey bordification of Teichmuller space. In the first case the boundary of this bordification is homotopy equivalent to a spherical building; in the second case it is homotopy equivalent to curve complex of the surface. In the case of a reflection arrangement the boundary of its universal cover is the “curve complex” of the corresponding spherical Artin group. By definition this is the simplicial complex of all conjugates of proper, irreducible, spherical parabolic subgroups in the Artin group. A cohomological method is used to show that the curve complex of a spherical Artin group has the homotopy type of a wedge of spheres.

Mon, 10 Jun 2019
15:45
L6

Unitary group integrals, surfaces, and mapping class groups

Michael Magee
(Durham University)
Abstract


For any word w in a free group of rank r>0, and any compact group G, w induces a `word map' from G^r to G by substitutions of elements of G for the letters of w. We may also choose the r elements of G independently with respect to Haar measure on G, and then apply the word map. This gives a random element of G whose distribution depends on w. An interesting observation is that this distribution doesn't change if we change w by an automorphism of the free group. It is a wide open question whether the measures induced by w on compact groups determine w up to automorphisms.
My talk will be mostly about the case G = U(n), the n by n complex unitary matrices. The technical tool we use is a precise formula for the moments of the distribution induced by w on U(n). In the formula, there is a surprising appearance of concepts from infinite group theory, more specifically, Euler characteristics of mapping class groups of surfaces. I'll explain how our formula allows us to make progress on the question described above.
This is joint work with Doron Puder (Tel Aviv).
 

Mon, 03 Jun 2019
15:45
L6

The Tits alternative for two-dimensional Artin groups

Alexandre Martin
(Heriot Watt University)
Abstract

A group is said to satisfy the Tits Alternative if its finitely generated subgroups exhibit a striking dichotomy: they are either "big" (they contain a non-abelian free subgroup) or "small" (they are virtually soluble). Many groups of geometric interest have been shown to satisfy the Tits Alternative: linear groups, mapping class groups of hyperbolic surfaces, etc. In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Mon, 27 May 2019
15:45
L6

Secondary invariants and mock modularity

Theo Johnson-Freyd
(Perimeter Institute for Theoretical Physics)
Abstract

A two-dimensional, minimally Supersymmetric Quantum Field Theory is "nullhomotopic" if it can be deformed to one with spontaneous supersymmetry breaking, including along deformations that are allowed to "flow up" along RG flow lines. SQFTs modulo nullhomotopic SQFTs form a graded abelian group $SQFT_\bullet$. There are many SQFTs with nonzero index; these are definitely not nullhomotopic, and indeed represent nontorision classes in $SQFT_\bullet$. But relations to topological modular forms suggests that $SQFT_\bullet$ also has rich torsion. Based on an analysis of mock modularity and holomorphic anomalies, I will describe explicitly a "secondary invariant" of SQFTs and use it to show that a certain element of $SQFT_3$ has exact order $24$. This work is joint with D. Gaiotto and E. Witten.

Mon, 20 May 2019
15:45
L6

Rational cobordisms and integral homology

Paolo Aceto
(Oxford)
Abstract

We prove that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in
any other element in the same class. As a consequence we show that several natural maps to
the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility
condition between the determinants of certain 2-bridge knots and other knots in the same
concordance class. This is joint work with Daniele Celoria and JungHwan Park.

Mon, 13 May 2019
15:45
L6

On operads with homological stability

Tom Zeman
(Oxford)
Abstract

In a recent paper, Basterra, Bobkova, Ponto, Tillmann and Yeakel defined
topological operads with homological stability (OHS) and proved that the
group completion of an algebra over an OHS is weakly equivalent to an
infinite loop space.

In this talk, I shall outline a construction which to an algebra A over
an OHS associates a new infinite loop space. Under mild conditions on
the operad, this space is equivalent as an infinite loop space to the
group completion of A. This generalises a result of Wahl on the
equivalence of the two infinite loop space structures constructed by
Tillmann on the classifying space of the stable mapping class group. I
shall also talk about an application of this construction to stable
moduli spaces of high-dimensional manifolds in thesense of Galatius and
Randal-Williams.

Mon, 06 May 2019
15:45
L6

Holomorphic curves and Seiberg-Witten invariants for 4-dimensional cobordisms

Yi-Jen Lee
(The Chinese University of Hong Kong)
Abstract

We will discuss a variant of Taubes’s Seiberg-Witten to Gromov theorem in the context of a 4-manifold with cylindrical ends, equipped with a nontrivial harmonic 2-form. This harmonic 2-form is allowed to be asymptotic to 0 on some (but not all) of its ends, and may have nondegenerate zeros along 1-submanifolds. Corollaries include various positivity results; some simple special cases of these constitute a key ingredient in Kutluhan-Lee-Taubes’s proof of HM = HF (Monopole Floer homology equals Heegaard Floer homology). The aforementioned general theorem is motivated by (potential) extensions of the HM = HF and Lee-Taubes’s HM = PFH (Periodic Floer homology) theorems.

Mon, 29 Apr 2019
15:45
L6

Knots, SL_2(R) representations, and a total Lin invariant

Jacob Rasmussen
(Cambridge)
Abstract

X.S. Lin defined an invariant of knots in S^3 by counting represenations 
of the knot group into SU(2) with fixed meridinal holonomy. Lin's 
invariant was subsequently shown to coincide with the Levine-Tristam 
signature. I'll define an analogous total Lin invariant which counts 
repesentations into both SU(2) and SL_2(R). Unlike the SU(2) version, this 
invariant does not (as far as I know) coincide with other known 
invariants. I'll describe some applications to left-orderability of Dehn 
fillings and branched covers, as well as a curious connection with the 
Alexander polynomial. This is joint work with Nathan Dunfield.

Mon, 18 Mar 2019
15:45
C4

Algebraic cobordism categories and Grothendieck-Witt-theory

Fabian Hebestreit
(University of Bonn)
Abstract

I will explain how Lurie‘s approach to L-theory via Poincaré categories can be extended to yield cobordism categories of Poincaré objects à la Ranicki. These categories can be delooped by an iterated Q-construction and the resulting spectrum is a derived version of Grothendieck-Witt-theory.  Its homotopy type can be described in terms of K- and L-theory as conjectured by Hesselholt-Madsen. Furthermore, it has a clean universal property analogous to that of K-theory, localisation sequences in much greater generality than classical Grothendieck-Witt theory, gives a cycle description of Weiss-Williams‘ LA-theory and allows for maps from the geometric cobordism category, refining and unifying various known invariants.

All original material is joint work with B.Calmès, E.Dotto, Y.Harpaz, M.Land, K.Moi, D.Nardin, T.Nikolaus and W.Steimle.

Mon, 18 Mar 2019
14:15
C4

Invariants for sublinearly biLipschitz equivalence

Gabriel Pallier
(Université Paris-Sud 11)
Abstract


The large-scale features of groups and spaces are recorded by asymptotic invariants. Examples of asymptotic invariants are the asymptotic cone and, for hyperbolic groups, the Gromov boundary.
In his study of asymptotic cones of connected Lie groups, Yves Cornulier introduced a class of maps called sublinearly biLipschitz equivalences. Like the more traditionnal quasiisometries, sublinearly biLipschitz equivalences are biLipschitz on the large-scale, but unlike quasiisometries, they are generally not coarse. Sublinearly biLipschitz equivalences still induce biLipschitz homeomorphisms between asymptotic cones. In this talk, I will focus on Gromov-hyperbolic groups and show how the Gromov boundary can be used to produce invariants distinguishing them up to sublinearly biLipschitz equivalences when the asymptotic cones do not. I will especially give applications to the large-scale sublinear geometry of hyperbolic Lie groups.