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Forthcoming events in this series
15:45
Geometry and topology of data sets
Abstract
Coarse geometry provides a very useful organising point of view on the study
of geometry and analysis of discrete metric spaces, and has been very
successful in the context of geometric group theory and its applications. On
the other hand, the work of Carlsson, Ghrist and others on persistent
homology has paved the way for applications of topological methods to the
study of broadly understood data sets. This talk will provide an
introduction to this fascinating topic and will give an overview of possible
interactions between the two.
Stochastic Algebraic Topology
Abstract
Topological spaces and manifolds are commonly used to model configuration
spaces of systems of various nature. However, many practical tasks, such as
those dealing with the modelling, control and design of large systems, lead
to topological problems having mixed topological-probabilistic character
when spaces and manifolds depend on many random parameters.
In my talk I will describe several models of stochastic algebraic topology.
I will also mention some open problems and results known so far.
The Classification of Rational SubTangle Adjacencies, with Applications to Complex Nucleoprotein Assemblies.
Abstract
Many proteins cleave and reseal DNA molecules in precisely orchestrated
ways. Modelling these reactions has often relied on the axis of the DNA
double helix
being circular, so these cut-and-seal mechanisms can be
tracked by corresponding changes in the knot type of the DNA axis.
However, when the DNA molecule is linear, or the
protein action does not manifest itself as a change in knot type, or the
knots types are not 4-plats, these knot theoretic models are less germane.
We thus give a taxonomy of local DNA axis configurations. More precisely, we
characterise
all rational tangles obtained from a given rational tangle via a rational
subtangle
replacement (RSR). This builds on work of Berge and Gabai.
We further determine the sites for these RSR of distance greater than 1.
Finally, we classify all knots in lens spaces whose exteriors are
generalised Seifert fibered spaces and their lens space surgeries, extending work of
Darcy-Sumners.
Biologically then, this classification is endowed with a distance that
determines how many protein reactions
of a particular type (corresponding to steps of a specified size) are
needed to proceed from one local conformation to another.
We conclude by discussing a variety of biological applications of this
categorisation.
Joint work with Ken Baker
Curve complexes on nonorientable surfaces
Abstract
The curve complex on an orientable surface, introduced by William Harvey about 30 years ago, is the abstract simplicial complex whose vertices are isotopy classes of simple close curves. A set of vertices forms a simplex if they can be represented by pairwise disjoint elements. The mapping class group of S acts on this complex in a natural way, inducing a homomorphism from the mapping class group to the group of automorphisms of the curve complex. A remarkable theorem of Nikolai V. Ivanov says that this natural homomorphism is an isomorphism. From this fact, some algebraic properties of the mapping class group has been proved. In the last twenty years, this result has been extended in various directions. In the joint work with Ferihe Atalan, we have proved the corresponding theorem for non-orientable surfaces: the natural map from the mapping class group of a nonorientable surface to the automorphism group of the curve compex is an isomorphism. I will discuss the proof of this theorem and possible applications to the structure of the mapping class groups.
Deformations of algebras and their diagrams
Rigidity of manifolds without non-positive curvature
Abstract
In this talk I describe some results obtained in collaboration with
J.F. Lafont and A. Sisto, which concern rigidity theorems for a class of
manifolds which are ``mostly'' non-positively curved, but may not support
any actual non-positively curved metric.
More precisely, we define a class of manifolds which contains
non-positively curved examples.
Building on techniques coming from geometric group theory, we show
that smooth rigidity holds within our class of manifolds
(in fact, they are also topologically rigid - i.e. they satisfy the Borel
conjecture - but this fact won't be discussed in my talk).
We also discuss some results concerning the quasi-isometry type of the
fundamental groups
of mostly non-positively curved manifolds.
RAAGs in Ham
Abstract
I will explain how to embed arbitrary RAAGs (Right Angled
Artin Groups) in Ham (the group of hamiltonian symplectomorphisms of
the 2-sphere). The proof is combination of topology, geometry and
analysis: We will start with embeddings of RAAGs in the mapping class
groups of hyperbolic surfaces (topology), then will promote these
embeddings to faithful hamiltonian actions on the 2-sphere (hyperbolic
geometry and analysis).
Surfaces of large genus
Abstract
Surfaces of large genus are intriguing objects. Their geometry
has been studied by finding geometric properties that hold for all
surfaces of the same genus, and by finding families of surfaces with
unexpected or extreme geometric behavior. A classical example of this is
the size of systoles where on the one hand Gromov showed that there exists
a universal constant $C$ such that any (orientable) surface of genus $g$
with area normalized to $g$ has a homotopically non-trivial loop (a
systole) of length less than $C log(g)$. On the other hand, Buser and
Sarnak constructed a family of hyperbolic surfaces where the systole
roughly grows like $log(g)$. Another important example, in particular for
the study of hyperbolic surfaces and the related study of Teichmüller
spaces, is the study of short pants decompositions, first studied by Bers.
The talk will discuss two ideas on how to further the understanding of
surfaces of large genus. The first part will be about joint results with
F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of
pants decompositions and related questions. In particular we investigate
how to find short pants decompositions on punctured spheres, and how to
find families of homologically independent short curves. The second part,
joint with L. Guth and R. Young, will be about how to construct surfaces
with large pants decompositions using random constructions.
A sampler of (algebraic) quantum field theory
Abstract
Conformal nets are an example of this kind of structure. Given a conformal net, one can assigns a von Neumann algebra to any 1-dimensional manifold, and (at least conjecturally) a Hilbert space to any 2-dimensional Riemann surfaces.
I will start by explaining what conformal nets are. I will then give some examples of conformal net: the ones associated to loop groups of compact Lie groups. Finally, I will present a new proof of a celebrated result of Kawahigashi, Longo, and
Mueger:
The representation category of a conformal net (subject to appropriate finiteness conditions) is a modular tensor category.
All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.
Generic conformal dimension estimates for random groups
Abstract
What is a random group? What does it look like? In Gromov's few relator
and density models (with density < 1/2) a random group is a hyperbolic
group whose boundary at infinity is homeomorphic to a Menger curve.
Pansu's conformal dimension is an invariant of the boundary of a
hyperbolic group which can capture more information than just the
topology. I will discuss some new bounds on the conformal dimension of the
boundary of a small cancellation group, and apply them in the context of
random few relator groups, and random groups at densities less than 1/24.
$L^p$ cohomology and pinching
Abstract
We prove that no Riemannian manifold quasiisometric to
complex hyperbolic plane can have a better curvature pinching. The proof
uses cup-products in $L^p$-cohomology.
The fundamental group of $\text{ Hom}(\bb Z^k,G)$
Abstract
Let $G $ be a compact Lie group, and consider the variety $\text {Hom} (\bb Z^k,G)$
of representations of the rank $k$ abelian free group $\bb Z^k$ into $G$. We prove
that the fundamental group of $\text {Hom} (\bb Z^k,G) $ is naturally isomorphic to direct
product of $k$ copies of the fundamental group of $G$. This is joint work with
Jose Manuel Gomez and Juan Souto.
Analogues of Euler characteristic
Abstract
There is a close but underexploited analogy between the Euler characteristic
of a topological space and the cardinality of a set. I will give a quite
general definition of the "magnitude" of a mathematical structure, framed
categorically. From this single definition can be derived many
cardinality-like invariants (some old, some new): the Euler characteristic
of a manifold or orbifold, the Euler characteristic of a category, the
magnitude of a metric space, the Euler characteristic of a Koszul algebra,
and others. A conjecture states that this purely categorical definition
also produces the classical invariants of integral geometry: volume, surface
area, perimeter, .... No specialist knowledge will be assumed.
03:45
Discrete differential geometry: constant mean curvature surfaces
Abstract
The is the second part of the Analysis and Geometry Seminar today.
15:45
Curve complex projections and the mapping class group
Abstract
Abstract: We will explain a certain natural way to project elements of
the mapping class to simple closed curves on subsurfaces. Generalizing
a coordinate system on hyperbolic space, we will use these projections
to describe a way to characterize elements of the mapping class group
in terms of these projections. This point of view is useful in several
applications; time permitting we shall discuss how we have used this
to prove the Rapid Decay property for the mapping class group. This
talk will include joint work with Kleiner, Minksy, and Mosher.
15:45
15:45
Link Invariants Given by Homotopy Groups
Abstract
In this talk, we introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover all homotopy groups of any dimensional spheres can be obtained from the geometric Massey products on certain links.
15:45
Surface quotients of hyperbolic buildings
Abstract
Bourdon's building is a negatively curved 2-complex built out of hyperbolic right-angled polygons. Its automorphism group is large (uncountable) and remarkably rich. We study, and mostly answer, the question of when there is a discrete subgroup of the automorphism group such that the quotient is a closed surface of genus g. This involves some fun elementary combinatorics, but quickly leads to open questions in group theory and number theory. This is joint work with David Futer.
15:45
Higher string topology
Abstract
The talk will begin with a brief account of the construction of string topology operations. I will point out some mysteries with the formulation of these operations, such as the role of (moduli) of surfaces, and pose some questions. The remainder of the talk will address these issues. In particular, I will sketch some ideas for a higher-dimensional version of string topology. For instance, (1) I will describe an E_{d+1} algebra structure on the (shifted) homology of the free mapping space H_*(Map(S^d,M^n)) and (2) I will outline how to obtain operations H_*(Map(P,M)) -> H_*(Map(Q,M)) indexes by a moduli space of zero-surgery data on a smooth d-manifold P with resulting surgered manifold Q.
15:45
On spaces of homomorphisms and spaces of representations
Abstract
The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations.
These spaces of representations admit the structure of a simplicial space at the heart of the work here.
Features of geometric realizations will be developed.
What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group ?
This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.