I define what it means for a quantum
field theory to be asymptotically safe and
discuss possible applications to theories
of gravity and matter.
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
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.
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 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.
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.
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.
We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.