Forthcoming events in this series
15:30
Triangulated surfaces in triangulated categories
Abstract
Given a triangulated category A, equipped with a differential
Z/2-graded enhancement, and a triangulated oriented marked surface S, we
explain how to define a space X(S,A) which classifies systems of exact
triangles in A parametrized by the triangles of S. The space X(S,A) is
independent, up to essentially unique Morita equivalence, of the choice of
triangulation and is therefore acted upon by the mapping class group of the
surface. We can describe the space X(S,A) as a mapping space Map(F(S),A),
where F(S) is the universal differential Z/2-graded category of exact
triangles parametrized by S. It turns out that F(S) is a purely topological
variant of the Fukaya category of S. Our construction of F(S) can then be
regarded as implementing a 2-dimensional instance of Kontsevich's proposal
on localizing the Fukaya category along a singular Lagrangian spine. As we
will see, these results arise as applications of a general theory of cyclic
2-Segal spaces.
This talk is based on joint work with Mikhail Kapranov.
15:30
Spectral sequences from Khovanov homology
Abstract
There are various Floer-theoretical invariants of links and 3-manifolds
which take the form of homology groups which are the E_infinity page of
spectral sequences starting from Khovanov homology. We shall discuss recent
work, joint with Raphael Zentner, and work in progress, joint with John
Baldwin and Matthew Hedden, in investigating and exploiting these spectral
sequences.
15:30
Rational tangles and the colored HOMFLY polynomial
Abstract
The HOMFLY polynomial is an invariant of knots in S^3 which can be
extended to an invariant of tangles in B^3. I'll give a geometrical
description of this invariant for rational tangles, and
explain how this description extends to a more general invariant
(the lambda^k colored HOMFLY polynomial of a rational tangle). I'll then
use this description to sketch a proof of a conjecture of Gukov and Stosic
about the colored HOMFLY homology of rational knots.
Parts of this are joint work with Paul Wedrich and Mihaljo Cevic.
15:30
Poincare Koszul duality and factorization homology
Abstract
Factorization homology is an invariant of an n-manifold M together with an n-disk algebra A. Should M be
a circle, this recovers the Hochschild complex of A; should A be a commutative algebra, this recovers the
homology of M with coefficients in A. In general, factorization homology retains more information about
a manifold than its underlying homotopy type.
In this talk we will lift Poincare' duality to factorization homology as it intertwines with Koszul
duality for n-disk algebras -- all terms will be explained. We will point out a number of consequences
of this duality, which concern manifold invariants as well as algebra invariants.
This is a report on joint work with John Francis.
On Sofic Groups
Abstract
The class of sofic groups was introduced by Gromov in 1999. It
includes all residually finite and all amenable groups. In fact, no group has been proved
not to be sofic, so it remains possible that all groups are sofic. Their
defining property is that, roughly speaking, for any finite subset F of
the group G, there is a map from G to a finite symmetric group, which is
approximates to an injective homomorphism on F. The widespread interest in
these group results partly from their connections with other branches of
mathematics, including dynamical systems. In the talk, we will concentrate
on their definition and algebraic properties.
Derived A-infinity algebras from the point of view of operads
Abstract
A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the more general notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general commutative ground ring.
Operads provide a very nice way of saying what A-infinity algebras are - they are described by a kind of free resolution of a strictly associative structure. I will explain the analogous result for derived A_infinity algebras - these are obtained in the same manner from a strictly associative structure with an extra differential.
This is joint work with Muriel Livernet and Constanze Roitzheim.
Fibering 5-manifolds with fundamental group Z over the circle
Abstract
In this talk I will introduce my joint work with Kreck on a classification of
certain 5-manifolds with fundamental group Z. This result can be interpreted as a
generalization of the classical Browder-Levine's fibering theorem to dimension 5.
Metric aspects of generalized Baumslag-Solitar groups
Abstract
A generalized Baumslag-Solitar group is a group G acting co-compactly on a tree X, with all vertex- and edge stabilizers isomorphic to the free abelian group of rank n. We will discuss the $L^p$-metric and $L^p$-equivariant compression of G, and also the quasi-isometric embeddability of G in a finite product of binary trees. Complete results are obtained when either $n=1$, or the quotient graph $G\X$ is either a tree or homotopic to a circle. This is joint work with Yves Cornulier.
The moduli space of topological realisations of an unstable coalgebra
Abstract
The mod p homology of a space is an unstable coalgebra over the Steenrod algebra at the prime p. This talk will be about the classical problem of realising an unstable coalgebra as the homology of a space. More generally, one can consider the moduli space of all such topological realisations and ask for a description of its homotopy type. I will discuss an obstruction theory which describes this moduli space in terms of the Andr\'{e}-Quillen cohomology of the unstable coalgebra. This is joint work with G. Biedermann and M. Stelzer.
Exact Lagrangian immersions in Euclidean space
Abstract
Exact Lagrangian immersions are governed by an h-principle, whilst exact Lagrangian
embeddings are well-known to be constrained by strong rigidity theorems coming from
holomorphic curve theory. We consider exact Lagrangian immersions in Euclidean space with a
prescribed number of double points, and find that the borderline between flexibility and
rigidity is more delicate than had been imagined. The main result obtains constraints on such
immersions with exactly one double point which go beyond the usual setting of Morse or Floer
theory. This is joint work with Tobias Ekholm, and in part with Ekholm, Eliashberg and Murphy.
Metric Geometry of Mapping Class and Relatively Hyperbolic Groups
Abstract
We prove that quasi-trees of spaces satisfying the axiomatisation given by Bestvina, Bromberg and Fujiwara are quasi-isometric to tree-graded spaces in the sense of Dru\c{t}u and Sapir. We then present a technique for obtaining `good' embeddings of such spaces into $\ell^p$ spaces, and show how results of Bestvina-Bromberg-Fujiwara and Mackay-Sisto allow us to better understand the metric geometry of such groups.
Orthogonal Calculus and Model Categories.
Abstract
Orthogonal calculus is a calculus of functors, inspired by Goodwillie calculus. It takes as input a functor from finite dimensional inner product spaces to topological spaces and as output gives a tower of approximations by well-behaved functors. The output captures a lot of important homotopical information and is an important tool for calculations.
In this talk I will report on joint work with Peter Oman in which we use model categories to improve the foundations of orthogonal calculus. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. The classification of n-homogeneous functors in terms of spectra with O(n)-action can then be phrased as a zig-zag of Quillen equivalences.
The complexity of group presentations, manifolds, and the Andrews-Curtis conjecture
Abstract
Quasi-hyperbolic planes in hyperbolic and relatively hyperbolic groups
Abstract
In 2005, Bonk and Kleiner showed that a hyperbolic group admits a
quasi-isometrically embedded copy of the hyperbolic plane if and only if the
group is not virtually free. This answered a question of Papasoglu. I will
discuss a generalisation of their result to certain relatively hyperbolic
groups (joint work with Alessandro Sisto). Key tools involved are new
existence results for quasi-circles, and a better understanding of the
geometry of boundaries of relatively hyperbolic groups.
Coarse median spaces
Abstract
By a "coarse median" we mean a ternary operation on a path metric space, satisfying certain conditions which generalise those of a median algebra. It can be interpreted as a kind of non-positive curvature condition, and is applicable, for example to finitely generated groups. It is a consequence of work of Behrstock and Minsky, for example, that the mapping class group of a surface satisfies this condition. We aim to give some examples, results and applications concerning this notion.
Balloons and Hoops and their Universal Finite Type Invariant, BF Theory, and an Ultimate Alexander Invariant
Abstract
Balloons are two-dimensional spheres. Hoops are one dimensional loops. Knotted Balloons and Hoops (KBH) in 4-space behave much like the first and second fundamental groups of a topological space - hoops can be composed like in π1, balloons like in π2, and hoops "act" on balloons as π1 acts on π2. We will observe that ordinary knots and tangles in 3-space map into KBH in 4-space and become amalgams of both balloons and hoops.
We give an ansatz for a tree and wheel (that is, free-Lie and cyclic word) -valued invariant ζ of KBHs in terms of the said compositions and action and we explain its relationship with finite type invariants. We speculate that ζ is a complete evaluation of the BF topological quantum field theory in 4D, though we are not sure what that means. We show that a certain "reduction and repackaging" of ζ is an "ultimate Alexander invariant" that contains the Alexander polynomial (multivariable, if you wish), has extremely good composition properties, is evaluated in a topologically meaningful way, and is least-wasteful in a computational sense. If you believe in categorification, that's a wonderful playground.
For further information see http://www.math.toronto.edu/~drorbn/Talks/Oxford-130121/
Automorphisms of relatively hyperbolic groups and McCool groups
Abstract
We define a McCool group of G as the group of outer automorphisms of G acting as a conjugation on a given family of subgroups. We will explain that these groups appear naturally in the description of many natural groups of automorphisms. On the other hand, McCool groups of a toral relatively hyperbolic group have strong finiteness properties: they have a finite index subgroup with a finite classifying space. Moreover, they satisfy a chain condition that has several other applications.
This is a joint work with Gilbert Levitt.