Monday, 16 October 2017

Exploring Steiner Chains with Möbius Transformations

Oxford Mathematician Kristian Kiradjiev has been awarded the Institute of Mathematics and its Applications (IMA) Early Career Mathematicians Catherine Richards Prize 2017 for his article on 'Exploring Steiner Chains with Möbius Transformations.' Here he explains his work.

"A beautiful example that one encounters in the geometry of circles is the so-called Steiner chains, named after the Swiss mathematician Jacob Steiner. A Steiner chain is defined as a chain of $n$ circles, each tangent to the previous one and the next one, and also to two given non-intersecting circles, which we will call bounding circles. We focus exclusively on Steiner chains, one of whose bounding circles lies within the other, although the concept also exists when the bounding circles are disjoint. We also define a closed Steiner chain to be such that the first and last circles of the chain are tangent to each other. For the sake of simplicity, we will limit ourselves to simple closed chains, i.e. wrapping only once around the inner bounding circle, although, again, one can have multi-cyclic closed chains, which wrap around several times before touching the first circle.

One of the main results, concerning Steiner chains is known as Steiner's porism. A porism is a mathematical proposition, which nowadays usually refers to a statement that is either not true, or is true and holds for an infinite number of values, provided a certain condition is satisfied (cf. Poncelet's porism). Steiner's porism states that given two bounding circles and the first circle from the chain, if a Steiner chain exists, then there are infinitely many of them, irrespective of the position of the first circle.

A lot of other fascinating properties have been discovered. For example, it is known that the centres of the circles in the chain lie either on an ellipse (or circle) when one of the bounding circles lies within the other, or on a hyperbola if not. Also, the points of tangency between the circles in the chain happen to lie on a circle. More interestingly, using inversion (and the inversive distance invariant), a feasibility criterion has been established for whether a closed Steiner chain is supported for a given $n$ and a pair of bounding circles. The problem I considered is somewhat the opposite: given $n$ positive numbers, does there exist a pair of bounding circles such that we can arrange $n$ circles with radii the given $n$ numbers in a simple closed Steiner chain between these bounding circles? This can also be reformulated as a geometrical problem of inscribing and circumscribing circles around a chain of touching circles with given radii. In order to answer this question, I essentially relied on the concept of Möbius transformations, which are conformal maps (i.e. preserve angles) in the extended complex plane $\widetilde{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$ of the form \begin{equation*} f(z)=\frac{az+b}{cz+d}, \label{eq:1} \end{equation*} where $z\in\widetilde{\mathbb{C}}$, and $a,b,c,d\in\mathbb{C}$ with $ad-bc\neq 0$ (so as to avoid constant maps). We note that translation, scaling, and inversion are particular cases of Möbius transformations. What is really useful about them, is that they map circlines (circles or lines) to circlines. As a result, I managed to derive a set of criteria on the $n$ radii for when they can form a Steiner chain. In addition, if such a chain exists, I gave a method how to construct it.

There is a number of generalisations, of which the Soddy's hexlet, which can be thought of as a 3D analogue of a Steiner chain, is a beautiful example and is such that the envelope of the touching spheres is the Dupin cyclide, an inversion of the torus."

The three rotating figures above show: a simple closed Steiner chain with 7 circles, Soddy’s Hexlet and Dupin Cyclide.

Friday, 13 October 2017

The importance of teaching - mathematicians scoop teaching awards

The importance of a University's teaching may seem a given, but it has received additional scrutiny in the last twelve months via the Government's Teaching Excellence Framework (TEF) and more widely as part of a debate on what Universities should offer their students. Oxford has annual teaching awards, voted by its most demanding assessors, namely its students, and this year plenty of mathematicians - Faculty, Postdocs and Graduate students - featured in those awards. Here is a list of the winners, all of whom demonstrate that we are both a research and teaching University and that the two are inseparable.

Prof. Dan Ciubotaru - MPLS Individual Teaching Award for Excellence in Teaching 
Dr Derek Goldrei,  Prof. Alex Scott, Dr David Seifert, Dr Phil Trinh, Prof. Andy Wathen - Departmental Teaching Award
Jamie Beacom, James Kwiecinski, Chris Nicholls, Lindon Roberts - Departmental Tutor/TA Teaching Award

Friday, 6 October 2017

Stephen Hawking's Oxford Mathematics Public Lecture on 27th October to be streamed live

In recognition of a lifetime's contribution across the mathematical sciences, we are initiating a series of annual Public Lectures in honour of Roger Penrose. The first lecture will be given by his long-time collaborator and friend Stephen Hawking on 27th October @5pm.

You will find the live podcast here (and also via the University of Oxford Facebook page).

Thursday, 5 October 2017

Frances Kirwan elected 20th Savilian Professor

Dame Frances Kirwan has been elected to the Savilian Professorship at the University of Oxford. Frances will be the 20th holder of the Savilian Chair (founded in 1619), and is the first woman to be elected to any of the historic chairs in mathematics.

Frances has received many honours including being elected a Fellow of the Royal Society in 2001 (only the third female mathematician to attain this honour), and President of the London Mathematical Society from 2003-2005 (only the second female ever elected).

Frances' specialisation is algebraic and symplectic geometry, notably moduli spaces in algebraic geometry, geometric invariant theory (GIT), and the link between GIT and moment maps in symplectic geometry.


Wednesday, 4 October 2017

Oxford Mathematics London Public Lecture - Andrew Wiles

Oxford Mathematics in partnership with the Science Museum is delighted to announce its first Public Lecture in London. World-renowned mathematician Andrew Wiles will be our speaker. Andrew will be talking about his current work and will also be in conversation with mathematician and broadcaster Hannah Fry after the lecture. Attendance is free.

28th November, 6.30pm, Science Museum, London, SW7 2DD

Please email to attend.

Tuesday, 3 October 2017

Per-Gunnar Martinsson awarded the 2017 Germund Dahlquist Prize

Oxford Mathematician Per-Gunnar Martinsson has been awarded the 2017 Germund Dahlquist Prize by the Society for Industrial and Applied Mathematics. The Germund Dahlquist Prize is awarded for original contributions to fields associated with Germund Dahlquist, especially the numerical solution of differential equations and numerical methods for scientific computing.

The prize honors Martinsson for fundamental contributions to numerical analysis and scientific computing that are making a significant impact in data science applications. Specific contributions include his development of linear time algorithms for dense matrix operations related to multidimensional elliptic PDEs and integral equations; and he has made deep and innovative contributions to the development of probabilistic algorithms for the rapid solution of certain classes of large-scale linear algebra problems. 
Per-Gunnar is currently Professor of Numerical Analysis at the University of Oxford. Hear more from him in this Q & A.

Monday, 2 October 2017

QBIOX - a new network bringing together expertise in quantitative biology

QBIOX – Quantitative Biology in Oxford – is a new network that brings together biomedical and physical scientists from across the University who share a commitment to making biology and medicine quantitative. A wide range of bioscience research fields are interested in the behaviour of populations of cells: how they work individually and collectively, how they interact with their environment, how they repair themselves and what happens when these mechanisms go wrong. At the cell and tissue levels, similar processes are at work in areas as diverse as developmental biology, regenerative medicine and cancer, which means that common tools can be brought to bear on them.

QBIOX’s focus is on mechanistic modelling: using maths to model biological processes and refining those models in order to answer a particular biological question. Researchers now have access to more data than ever before, and using the data effectively requires a joined-up approach. It is this challenge that has encouraged Professors Ruth Baker, Helen Byrne and Sarah Waters from the Mathematical Institute to set up QBIOX. The aim is to help researchers with the necessary depth and range of specialist knowledge to open up new collaborations, and share expertise and knowledge, in order to bring about a step-change in understanding in these areas. In regenerative medicine, for example, QBIOX has brought together a team of people from across the sciences and medical sciences in Oxford who are working on problems at the level of basic stem cell science right through to translational medicine that will have real impacts on patients.

A look at the list of QBIOX collaborators demonstrates that Oxford researchers from a wide range of backgrounds are already involved: from maths, statistics, physics, computer science and engineering, through to pathology, oncology, cardiology and infectious disease. QBIOX is encouraging any University researcher with an interest in quantitative biology to join the network. It runs a programme of activities to catalyse interactions between members. For example, QBIOX’s termly colloquia offer opportunities for academics to showcase research that is of interest to network members, and there are regular smaller meetings that look in detail at specific topics. QBIOX also has funding for researchers who would like to run small meetings to scope out the potential for using theoretical and experimental techniques to tackle new problems in the biosciences.

The QBIOX website has details of all the activities run by the network, as well as relevant events taking place across the University. If you have events you would like to feature here, just complete the contact form. You can also sign up to be a collaborator and to receive QBIOX’s termly newsletter.

Sunday, 1 October 2017

Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials

Oxford Mathematician Dmitry Belyaev is interested in the interface between analysis and probability. Here he discusses his latest work.

"There are two areas of mathematics that clearly have nothing to do with each other: projective geometry and conformally invariant critical models of statistical physics. It turns out that the situation is not as simple as it looks and these two areas might be connected.

We start with projective geometry. Let $g(x):\mathbb{R}^{m+1} \to \mathbb{R}$ be a homogeneous polynomial of degree $n$ in $m + 1$ variables. Although the values of the polynomial are not well defined in homogeneous coordinates $[x_0 : x_1 : \dotsm : x_m]$, but the zero locus, the set where $g([x_0 : x_1 : \dotsm : x_m]) = 0$, is well defined. The set $S = \{x ∈ \mathbb{PR}^m : g(x) = 0\}$ is a projective variety.

We can ask what a typical projective variety looks like. The answer to this question very much depends on the meaning of the word ‘typical’. One possibility is to define some ‘natural’ probability measure on the space of all homogeneous polynomials $g$ and treat ‘typical’ behaviour as almost sure behaviour with respect to this measure. Since the space of polynomials is too large, there is no canonical way to define the most natural uniform measure. Second best choice is a Gaussian measure. This still does not completely determine the measure, but there is one Gaussian measure which stands out: this is the only Gaussian measure which is the real trace of a complex Gaussian measure on space of homogeneous polynomials on $\mathbb{CP}^m$ which is invariant with respect to the unitary group. A random polynomial of degree n with respect to this measure could be written as

$$f_n(x) = f_{n;m}(x) = \sum_{|J|=n}\sqrt{\binom{n}{J}} a_J x^J,$$

where $J = (j_0, . . . , j_m)$ is the multi-index, $|J| = j_0 + \dotsb + j_m$, $\binom{n}{J} = \frac{n!}{j_0! \dotsb j_m!}$, and $\{a_J\}$ are i.i.d. standard Gaussian random variables. This random function is called the Kostlan ensemble or complex Fubini-Study ensemble. We can think that a ‘typical’ variety of degree $n$ is the nodal set of the Kostlan ensemble of degree $n$. We are mostly interested in the two-dimensional case $m = 2$.

It has been shown by V. Beffara and D. Gayet that there is Russo-Seymour-Welsh type estimate for Bargmann-Fock random function which is the scaling limit of the Kostlan ensemble. This means that if one fixes a nice domain with two marked boundary arcs, then the probability that there is a nodal line connecting two arcs inside the domain is bounded from below by a constant which depends on the shape of the domain, but not on its scale. These types of estimates first appeared in the study of critical percolation models and are a strong indication that the corresponding curves have conformally invariant scaling limits.

In the recent work with S. Muirhead and I. Wigman we have extended this result to the Kostlan ensemble on the sphere. Namely, we have obtained a lower bound on the probability to cross a domain which is uniform in the degree of the polynomial and in the scale of the domain. This suggests that large components of a ‘typical’ projective curve have a scaling limit which is conformally invariant and should be described by the Schramm-Loewner Evolution."

For a fuller explanation of Dmitry and colleagues' work please clck here.

Monday, 25 September 2017

Oxford Mathematics Research looks at Ricci Flow

As part of our series of research articles focusing on the rigour and intricacies of mathematics and its problems, Oxford Mathematician Andrew Dancer discusses his work on Ricci Flow.

"A sphere and an ellipsoid (rugby ball) are the same topologically, in that each can be continuously deformed into the other without tearing, but obviously they are not the same geometrically. We can see that the sphere is in some sense uniformly curved, while the curvature of the ellipsoid varies as we move around the surface. 

The mathematical gadget that encodes information about curvature, lengths, angles, volumes etc. is called a metric. This concept in fact makes sense not just for surfaces but in higher dimensions as well. The curvature is now not a single function but an object called the Riemann curvature tensor.

A fundamental question in geometry is whether a given manifold has a best or nicest metric. One popular candidate is the notion of an Einstein metric. The equations expressing the Einstein condition are a complicated nonlinear system of partial differential equations, and questions about existence and uniqueness of Einstein metrics in dimensions above three are still not well understood in general.

One strategy to study the existence of Einstein  metrics is via the Ricci flow. This is a nonlinear version of heat flow, whose fixed points correspond to Einstein metrics, rather as fixed points of heat flow correspond to harmonic functions (solutions of Laplace's equation). In good situations the Ricci flow may converge to an Einstein metric, but it is also possible for singularities to develop, arising from the nonlinear nature of the flow. I am particularly interested in so-called  soliton solutions of the heat flow, corresponding to metrics that evolve just by rescaling and coordinate changes under the flow. These give a natural generalisation of the Einstein condition, and are also very important in understanding singularities of the flow via a rescaling of 

In collaboration with Mckenzie Wang of McMaster University in Canada, I have produced new examples of solitons by assuming the existence of a large enough symmetry group to reduce the PDEs to ordinary differential equations. With my student Alejandro Betancourt de la Parra, we have even found some cases where the soliton equations may be solved explicitly, due to unexpected integrability structures in certain dimensions."

For fuller explanations of Andrew's work please click on the links below:

On Ricci solitons of cohomogeneity one

Some New Examples of Non-Kähler Ricci Solitons

A Hamiltonian approach to the cohomogeneity one Ricci soliton equations

Image above courtesy of Syafiq Johar

Wednesday, 20 September 2017

Oxford Mathematics Public Lectures for the Autumn (and a bit of Winter)

We have an exciting series of Oxford Mathematics Public Lectures this Autumn. Summary below and full details here. All will be podcast and on Facebook Live. We also have a London Lecture by Andrew Wiles on 28 November (details will follow separately). Please email to register for the lectures below.

Closing the Gap: the quest to understand prime numbers - Vicky Neale

18 October, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute, Oxford


Maths v Disease - Julia Gog

1 November, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute, Oxford


The Seduction of Curves: The Lines of Beauty That Connect Mathematics, Art and The Nude - Allan McRobie

13 November, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute, Oxford


Oxford Mathematics Christmas Public Lecture - Alex Bellos, title tbc

6 December, 5.00-6.00pm, Lecture Theatre 1, Mathematical Institute, Oxford


Please email to register