Past

Abstract: Butcher’s B-series is a fundamental tool in analysis of numerical integration of differential equations. In the recent years algebraic and geometric understanding of B-series has developed dramatically. The interplay between geometry, algebra and computations reveals new mathematical landscapes with remarkable properties. 

The shuffle Hopf algebra,  which is fundamental in Lyons’s groundbreaking work on rough paths,  is based on Lie algebras without additional properties.  Pre-Lie algebras and the Connes-Kreimer Hopf algebra are providing algebraic descriptions of the geometry of Euclidean spaces. This is the foundation of B-series and was used elegantly in Gubinelli’s theory of Branched Rough Paths. 
Lie-Butcher theory combines Lie series with B-series in a unified algebraic structure based on post-Lie algebras and the MKW Hopf algebra, which is giving algebraic abstractions capturing the fundamental geometrical properties of Lie groups, homogeneous spaces and Klein geometries. 

In these talks we will give an introduction to these new algebraic structures. Building upon the works of Lyons, Gubinelli and Hairer-Kelly, we will present a new theory for rough paths on homogeneous spaces built upon the MKW Hopf algebra.

Joint work with: Charles Curry and Dominique Manchon

A generic surface homeomorphism (up to isotopy) is what we call it pseudo-Anosov. These maps come equipped with an algebraic integer that measures
how much the map stretches/shrinks in different direction, called the stretch factor. Given a surface homeomorsphism, one can ask if it is the lift (by a branched or unbranched cover) of another homeomorphism on a simpler surface possibly of small genus. Farb conjectured that if the algebraic degree of the stretch factor is bounded above, then the map can be obtained by lifting another homeomorphism on a surface of bounded genus.
This was known to be true for quadratic algebraic integers by a Theorem of Franks-Rykken. We construct counterexamples to Farb's conjecture.

Any four dimensional N=2 superconformal field theory (SCFT) contains a subsector of local operator which is isomorphic to a two dimensional chiral algebra.  If the 4d theory possesses N= 4 superconformal symmetry, the corresponding chiral algebra is an extension of the (small) N=4 super-Virasoro algebra.  In this talk I  will present some results on the classification of N=4 chiral algebras and discuss the conditions they should satisfy in order to correspond to a 4d theory. 
 

Hypoxia, i.e. a reduction in dissolved oxygen concentration below physiologically normal levels, has been identified as playing a critical role
in the progression of many types of disease and as a key determinant of the success of cancer treatment. It poses a particular challenge for treatments
such as radiotherapy, photodynamic and sonodynamic therapy which rely on the production of reactive oxygen species. Strategies for treating hypoxia have
included the development of hypoxia-selective drugs as well as methods for directly increasing blood oxygenation, e.g. hyperbaric oxygen therapy, pure
oxygen or carbogen breathing, ozone therapy, hydrogen peroxide injections and administration of suspensions of oxygen carrier liquids. To date, however,
these approaches have delivered limited success either due to lack of proven efficacy and/or unwanted side effects. Gas microbubbles, stabilised by a
biocompatible shell have been used as ultrasound contrast agents for several decades and have also been widely investigated as a means of promoting drug
delivery. This talk will present our recent research on the use of micro and nanobubbles to deliver both drug molecules and oxygen simultaneously to a
tumour to facilitate treatment.

Laura Capuano's talk 'Pell equations and continued fractions in number theory'

The classical Pell equation has an extraordinary long history and it is very useful in many different areas of number theory. For example, they given a way to write a prime congruent to 1 modulo 4 as a sum of two squares, or they can also be used to break RSA excryption when the decription key is too small. In this talk, I will present some properties of this wonderful equation and its relation with continued fractions. I will also treat the case of Pell equations in other contexts, such as the ring of polynomials, showing the differences with the classical case. 

Noemi Picco's talk 'Cortical neurogenesis: how humans (and mathematicians) can do more than macaque, with less'

The cerebral cortex is perhaps the crowning achievement of evolution and is the region of the brain that distinguishes us from other species. Studying the developmental programmes that generate cortices of different sizes and neuron counts, is the key to understanding both brain evolution and disease. I will show what mathematical modeling has to say about cortex evolution, when data resolution is poor. I will then discuss why humans are so special in the way they create their cortex, and how we are just like everybody else in many other aspects of brain development.

The fourth QBIOX Colloquium will take place in the Mathematical Institute on Friday 10th November (5th week) and feature talks from Professor Paul Riley (Department of Pathology, Anatomy and Genetics / BHF Oxbridge Centre for Regenerative Medicine, https://www.dpag.ox.ac.uk/research/riley-group) and Professor Eleanor Stride (Institute of Biomedical Engineering, http://www.ibme.ox.ac.uk/research/non-invasive-therapy-drug-delivery/pe…).

1600-1645 - Paul Riley, "Enroute to mending broken hearts".
1645-1730 - Eleanor Stride, "Reducing tissue hypoxia for cancer therapy".
1730-1800 - Networking and refreshments.

We very much hope to see you there. As ever, tickets are not necessary, but registering to attend will help us with numbers for catering.
Please see the following link for further details and a link to register.
https://www.eventbrite.co.uk/e/qbiox-colloquium-michelmas-term-2017-tic…

Abstracts
Paul Riley - "En route to mending broken hearts".
We adopt the paradigm of understanding how the heart develops during pregnancy as a first principal to inform on adult heart repair and regeneration. Our target for cell-based repair is the epicardium and epicardium-derived cells (EPDCs) which line the outside of the forming heart and contribute vascular endothelial and smooth muscle cells to the coronary vasculature, interstitial fibroblasts and cardiomyocytes. The epicardium can also act as a source of signals to condition the growth of the underlying embryonic heart muscle. In the adult heart, whilst the epicardium is retained, it is effectively quiescent. We have sought to extrapolate the developmental potential of the epicardium to the adult heart following injury by stimulating dormant epicardial cells to give rise to new muscle and vasculature. In parallel, we seek to modulate the local environment into which the new cells emerge: a cytotoxic mixture of inflammation and fibrosis which prevents cell engraftment and integration with survived heart tissue. To this end we manipulate the lymphatic vessels in the heart given that, elsewhere in the body, the lymphatics survey the immune system and modulate inflammation at peripheral injury sites. We recently described the development of the cardiac lymphatic vasculature and revealed in the adult heart that they undergo increased vessel sprouting (lymphangiogenesis) in response to injury, to improve function, remodelling and fibrosis. We are currently investigating whether increased lymphangiogenesis functions to clear immune cells and constrain the reparative response for optimal healing. 

Eleanor Stride - "Reducing tissue hypoxia for cancer therapy"
Hypoxia, i.e. a reduction in dissolved oxygen concentration below physiologically normal levels, has been identified as playing a critical role in the progression of many types of disease and as a key determinant of the success of cancer treatment. It poses a particular challenge for treatments such as radiotherapy, photodynamic and sonodynamic therapy which rely on the production of reactive oxygen species. Strategies for treating hypoxia have included the development of hypoxia-selective drugs as well as methods for directly increasing blood oxygenation, e.g. hyperbaric oxygen therapy, pure oxygen or carbogen breathing, ozone therapy, hydrogen peroxide injections and administration of suspensions of oxygen carrier liquids. To date, however, these approaches have delivered limited success either due to lack of proven efficacy and/or unwanted side effects. Gas microbubbles, stabilised by a biocompatible shell have been used as ultrasound contrast agents for several decades and have also been widely investigated as a means of promoting drug delivery. This talk will present our recent research on the use of micro and nanobubbles to deliver both drug molecules and oxygen simultaneously to a tumour to facilitate treatment.

We adopt the paradigm of understanding how the heart develops during pregnancy as a first principal to inform on adult heart repair and regeneration. Our target for cell-based repair is the epicardium and epicardium-derived cells (EPDCs) which line the outside of the forming heart and contribute vascular endothelial and smooth muscle cells to the coronary vasculature, interstitial fibroblasts and cardiomyocytes. The epicardium can also act as a source of signals to condition the growth of the underlying embryonic heart muscle. In the adult heart, whilst the epicardium is retained, it is effectively quiescent. We have sought to extrapolate the developmental potential of the epicardium to the adult heart following injury by stimulating dormant epicardial cells to give rise to new muscle and vasculature. In parallel, we seek to modulate the local environment into which the new cells emerge: a cytotoxic mixture of inflammation and fibrosis which prevents cell engraftment and integration with survived heart tissue. To this end we manipulate the lymphatic vessels in the heart given that, elsewhere in the body, the lymphatics survey the immune system and modulate inflammation at peripheral injury sites. We recently described the development of the cardiac lymphatic vasculature and revealed in the adult heart that they undergo increased vessel sprouting (lymphangiogenesis) in response to injury, to improve function, remodelling and fibrosis. We are currently investigating whether increased lymphangiogenesis functions to clear immune cells and constrain the reparative response for optimal healing.

Topologists have the Steenrod squares, a collection of additive homomorphisms on the Z/2 cohomology of a space M. They can be defined axiomatically and are often be regarded as algebraic operations on cohomology groups (for many purposes). However, Betz and Cohen showed that they could be viewed geometrically. 

Symplectic geometers have quantum cohomology, which on a symplectic manifold M is a deformation of singular cohomology using holomorphic spheres.

The geometric definition of the Steenrod square extends to quantum cohomology. This talk will describe the Steenrod square and quantum cohomology in terms of the intersection product, and then give a description of this quantum Steenrod square by putting these both together. We will describe some properties of the quantum squares, such as the quantum Cartan formula, and perform calculations in certain cases.

A famous theorem due to Erdős and Kac states that the number of prime divisors of an integer N behaves like a normal distribution. In this talk we consider analogues of this result in the setting of arithmetic geometry, and obtain probability distributions for questions related to local solubility of algebraic varieties. This is joint work with Efthymios Sofos.

The statistical physics governing phase-ordering dynamics following a symmetry breaking rst-order phase transition is an area of active research. The Coarsening/Ageing of the ensemble of phase domains, wherein irreversible annihilation or joining of domains yields a growing characteristic domain length, is an omniprescent feature whose universal characteristics one would wish to understand. Driven kinetic Ising models and growing nano-faceted crystals are theoretically important examples of such Coarsening (Ageing) Dynamical Systems (CDS), since they additionally break thermodynamic uctuation-dissipation relations. Power-laws for the growth in time of the characteristic size of domains, and a concomitant scale-invariance of associated length distributions, have so frequently been empirically observed that their presence has acquired the status of a principle; the so-called Dynamic-Scaling Hypothesis. But the dynamical symmetries of a given CDS- its Coarsening Group G - may include more than the global spatio-temporal scalings underlying the Dynamic Scaling Hypothesis. In this talk, I will present a recently developed theoretical framework (Ref.[1]) that shows how the symmetry group G of a Coarsening (ageing) Dynamical System necessarily yields G-equivariance (covariance) of its universal statistical observables. We exhibit this theory for a variety of model systems, of both thermodynamic and driven type, with symmetries that may also be Emergent (Ref. [2,3]) and/or Hidden. We will close with a magical theoretical coarsening law that combines Lorentzian and Parabolic symmetries!

References
[1] Lorentzian symmetry predicts universality beyond scaling laws, SJ Watson, EPL 118 (5), 56001, (Aug.2, 2017) Editor's Choice
[2] Emergent parabolic scaling of nano-faceting crystal growth Stephen J. Watson, Proc. R. Soc. A 471: 20140560 (2015)
[3] Scaling Theory and Morphometrics for a Coarsening Multiscale Surface, via a Principle of Maximal Dissipation", Stephen

This paper formulates an utility indifference pricing model for investors trading in a discrete time financial market under non-dominated model uncertainty.
The investors preferences are described by strictly increasing concave random functions defined on the positive axis. We prove that under suitable
conditions the multiple-priors utility indifference prices of a contingent claim converge to its multiple-priors superreplication price. We also
revisit the notion of certainty equivalent for random utility functions and establish its relation with the absolute risk aversion.

I will describe a simple macroscopic model describing the evolution of a cloud of particles confined in a magneto-optical trap. The behavior of the particles is mainly driven by self--consistent attractive forces. In contrast to the standard model of gravitational forces, the force field does not result from a potential; moreover, the nonlinear coupling is more singular than the coupling based on the Poisson equation.  In addition to existence of uniqueness results of the model PDE, I will discuss the convergence of the  particles description towards the solution of the PDE system in the mean field regime.

To fly is not to fall. How does an insect fly, why does it fly so well, and how can we infer its ‘thoughts’ from its flight dynamics?  We have been seeking  mechanistic explanations of the complex movement of insect flight. Starting from the Navier-Stokes equations governing the unsteady aerodynamics of flapping flight, a  theoretical framework for computing flight leads to new interpretations and predictions of the functions of an insect’s internal machinery that orchestrate its flight. The talk will discuss recent computational and experimental studies of the balancing act of dragonflies and fruit flies:  how a dragonfly recovers from falling upside-down and how a fly balances in air. In each case,  the physics of flight informs us about the neural feedback circuitries underlying their fast reflexes.

I will give a survey of known results about when two RAAGs are quasi-isometric, and will then describe a visual graph of groups decomposition of a RAAG (its JSJ tree of cylinders) that can often be used to determine whether or not two RAAGs are quasi-isometric.

Adaptive oblivious transfer (OT) is a protocol where a sender
initially commits to a database {M_i}_{i=1}^N . Then, a receiver can query the
sender up to k times with private indexes ρ_1, …, ρ_k so as to obtain
M_{ρ_1}, …, M_{ρ_k} and nothing else. Moreover, for each i ∈ [k], the receiver’s
choice ρ_i may depend on previously obtained messages {M_ρ_j}_{j<i} . Oblivious transfer
with access control (OT-AC) is a flavor of adaptive OT
where database records are protected by distinct access control policies
that specify which credentials a receiver should obtain in order to access
each M_i . So far, all known OT-AC protocols only support access policies
made of conjunctions or rely on ad hoc assumptions in pairing-friendly
groups (or both). In this paper, we provide an OT-AC protocol where access policies may consist of any branching program of polynomial length, which is sufficient to realize any access policy in NC^1. The security of
our protocol is proved under the Learning-with-Errors (LWE) and Short-
Integer-Solution (SIS) assumptions. As a result of independent interest,
we provide protocols for proving the correct evaluation of a committed
branching program on a committed input.

Joint work with Benoît Libert, San Ling, Khoa Nguyen and Huaxiong Wang.

By way of shameless advertising for a TCC course I hope to give next term on the theory of totally disconnected locally compact groups, I will present two interesting and illuminating examples of such groups: the full automorphism group of a regular tree, and Neretin's group of spheromorphisms
 

The talk is based on my joint work with Anand Pillay and Tomasz Rzepecki.

I will describe some connections between various objects from topological dynamics associated with a given first order theory and various Galois groups of this theory. One of the main corollaries is a natural presentation of the closure of the neutral element of the Lascar Galois group of any given theory $T$ (this closure is a group sometimes denoted by $Gal_0(T)$) as a quotient of a compact Hausdorff group by a dense subgroup.

As an application, I will present a very general theorem concerning the complexity of bounded, invariant equivalence relations (whose classes are sometimes called strong types) in countable theories, generalizing a theorem of Kaplan, Miller and Simon concerning Borel cardinalities of Lascar strong types and also later extensions of this result to certain bounded, $F_\sigma$ equivalence relations (which were obtained in a paper of Kaplan and Miller and, independently, in a paper of Rzepecki and myself). The main point of our general theorem says that in a countable theory, any bounded, invariant equivalence relation defined
on the set of realizations of a single complete type over $\emptyset$ is type-definable if and only if it is smooth (in the sense of descriptive set theory). If time permits, I will very briefly mention more recent developments in this direction (also based on the results from the first paragraph) which will appear in my future paper with Rzepecki.
 

We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields, and their behaviour under tame base change. For a semiabelian variety, this behaviour is governed by a finite sequence of (a priori) real numbers between 0 and 1, called "jumps". The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions, and generalize Raynaud's description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence which decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

Classical algorithms for numerical integration (quadrature/cubature) proceed by approximating the integrand with a simple function (e.g. a polynomial), and integrate the approximant exactly. In high-dimensional integration, such methods quickly become infeasible due to the curse of dimensionality.

A common alternative is the Monte Carlo method (MC), which simply takes the average of random samples, improving the estimate as more and more samples are taken. The main issue with MC is its slow "sqrt(variance/#samples)" convergence, and various techniques have been proposed to reduce the variance.

In this work we reveal a numerical analyst's interpretation of MC: it approximates the integrand with a simple(st) function, and integrates that function exactly. This observation leads naturally to MC-like methods that combines MC with function approximation theory, including polynomial approximation and sparse grids. The resulting method can be regarded as another variance reduction technique for Monte Carlo.

Reed conjectured in 1998 that the chromatic number of a graph should be at most the average of the clique number (a trivial lower bound) and maximum degree plus one (a trivial upper bound); in support of this conjecture, Reed proved that the chromatic number is at most some nontrivial convex combination of these two quantities.  King and Reed later showed that a fraction of roughly 1/130000 away from the upper bound holds. Motivated by a paper by Bruhn and Joos, last year Bonamy, Perrett, and I proved that for large enough maximum degree, a fraction of 1/26 away from the upper bound holds. Then using new techniques, Delcourt and I showed that the list-coloring version holds; moreover, we improved the fraction for ordinary coloring to 1/13. Most recently, Kelly and I proved that a 'local' list version holds with a fraction of 1/52 wherein the degrees, list sizes, and clique sizes of vertices are allowed to vary.
 

We develop a general purpose solver for quadratic programs based on operator splitting. We introduce a novel splitting that requires the solution of a quasi-definite linear system with the same coefficient matrix in each iteration. The resulting algorithm is very robust, and once the initial factorization is carried out, division free; it also eliminates requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. Moreover, it is able to detect primal or dual infeasible problems providing infeasibility certificates. The method supports caching the factorization of the quasi-definite system and warm starting, making it efficient for solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint and is library-free. Numerical benchmarks on problems arising from several application domains show that OSQP is typically 10x faster than interior-point methods, especially when factorization caching or warm start is used.

This is joint work with Goran Banjac, Paul Goulart, Alberto Bemporad and Stephen Boyd
 

Identifying clusters or "communities" of densely connected nodes in networks is an active area of research, with relevance to many applications. Recent advances in the field have focused especially on temporal, multiplex, and other kinds of multilayer networks.

One method for detecting communities in multilayer networks is to maximise a generalised version of an objective function known as modularity. Writing down multilayer modularity requires the specification of two types of resolution parameters, and choosing appropriate values is crucial for uncovering meaningful community structure. In the simplest case, there are just two parameters, one controlling the sizes of detected communities, and the other influencing how much communities change from layer to layer. By establishing an equivalence between modularity optimisation and a multilayer maximum-likelihood approach to community detection, we are able to determine statistically optimal values for these two parameters. 

When applied to existing multilayer benchmarks, our optimized approach performs significantly better than using parameter choices guided by heuristics. We also apply the method to supermarket data, revealing changes in consumer behaviour over time.