In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.

Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.

During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".

A 3-manifold fibers over the circle if it can be identified with the mapping torus of a surface homeomorphism. If the surface is compact with non-empty boundary then the corresponding 3-manifold group is free-by-cyclic, and the action of the cyclic group on the free group is induced by the surface homeomorphism. Although most free-by-cyclic groups do not arise as fundamental groups of 3-manifolds which fiber over the circle, there is a strong analogy between the two families.

In this talk I will discuss how dynamical properties of the monodromy affect the geometry/algebra of the corresponding mapping torus. We will see how the same 3-manifold or group can admit multiple fiberings and what properties of the monodromy are known to be preserved under different fiberings.

This talk will be an introduction to L^2 homology, which is roughly "square-summable" homology. We begin by defining the L^2 homology of a G-CW complex (a CW complex with a cellular G-action), and we will discuss some applications of these invariants to group theory and topology. We will then focus on a criterion of Wise, which proves the vanishing of the 2nd L^2 Betti number in combinatorial CW-complexes with elementary methods. If time permits, we will also introduce Wise's energy criterion.
 

We will discuss a generalisation of hyperbolic groups, from the group actions point of view. By studying torsion, we will see how this can help to answer questions about ordinary hyperbolic groups.

The asymptotic cone of a metric space X is what you see when you "look at X from infinitely far away". The asymptotic cone therefore captures much of the large scale geometry of the metric space. Furthermore, the construction often produces a smooth space from a discrete one, allowing us to apply the techniques of calculus. Notably, Gromov used asymptotic cones in his proof that finitely generated groups of polynomial growth are virtually nilpotent.

In the talk I will define asymptotic cones using the language of ultrafilters and ultralimits. We will then look at the particular cases of asymptotic cones of virtually nilpotent groups and hyperbolic metric spaces. At the end, we will prove a result of Gromov which relates the fundamental group of the asymptotic cone to the filling order of the underlying metric space.

A major tool used to understand manifolds is understanding how different measures of complexity relate to one another. One particularly combinatorial measure of the complexity of a 3-manifold M is the minimal number of tetrahedra in a simplicial complex homeomorphic to M, called the triangulation complexity of M. A natural question is whether we can relate this with more geometric measures of the complexity of a manifold, especially understanding these relationships as combinatorial complexity grows.

In the case when the manifold fibres over the circle, a recent theorem of Marc Lackenby and Jessica Purcell gives both an upper and lower bound on the triangulation complexity in terms of a geometric invariant of the gluing map (its translation length in the triangulation graph). We will discuss this result as well as a new result concerning what happens when we alter the gluing map by a Dehn twist.

In this session, William Durham from the Bank of England will give a presentation about working as a mathematician in the BoE, and will give advice on interviewing for non-academic jobs. He has previously provided mock interviews in our department for jobs aimed at mathematicians with PhDs, and is happy to conduct some mock interviews (remotely, of course) for individuals as well.

Please email Helen McGregor (@email) by Monday 22 February if you might be interested in having a mock interview with William Durham on 5 March.
 

Biological systems provide an inspiration for creating a new paradigm
for materials synthesis. What would it take to enable inanimate material
to acquire the properties of living things? A key difference between
living and synthetic materials is that the former are programmed to
behave as they do, through interactions, energy consumption and so
forth. The nature of the program is the result of billions of years of
evolution. Understanding and emulating this program in materials that
are synthesizable in the lab is a grand challenge. At its core is an
optimization problem: how do we choose the properties of material
components that we can create in the lab to carry out complex reactions?
I will discuss our (not-yet-terribly-successful efforts)  to date to
address this problem, by designing both equiliibrium and kinetic 
properties of materials, using a combination of statistical mechanics,
kinetic modeling and ideas from machine learning.

The pandemic has had a deleterious influence on the Hollywood film
industry.  Fortunately,  however, the thin film industry continues to
flourish.  A host of effects are responsible for confined liquids
exhibiting properties that differ from their bulk counterparts. For
example, the dominant polarization and surface forces across a layered
system can control the material behavior on length scales vastly larger
than the film thickness.  This basic class of phenomena, wherein
volume-volume interactions create large pressures, are at play in,
amongst many other settings, wetting, biomaterials, ceramics, colloids,
and tribology.  When the films so created involve phase change and are
present in disequilibrium, the forces can be so large that they destroy
the setting that allowed them to form in the first place. I will
describe the connection between such films in a semi-traditional wetting
dynamics geometry and active brownian dynamics.  I then explore their
power to explain a wide range of processes from materials- to astro- to
geo-science.