Wednesday, 6 January 2021

Rethinking defects in patterns

Social distancing is integral to our lives these days, but distancing also underpins the ordered patterns and arrangements we see all around us in Nature. Oxford Mathematician Priya Subramanian studies the defects in such patterns and shows how they relate to the underlying pattern, i.e. to the distancing itself.

"Nature has many examples of situations where individuals in a dense group have to balance between short range repulsion (e.g., competition for resources) and long range attraction (e.g. safety in numbers). This naturally leads to an optimal length scale for separation between neighbours. If we think of each agent as a circle with this optimal length as the diameter, the most efficient packing is in a hexagonal arrangement. So, it is unsurprising that we find these group of gannets nesting on a beach in Muriwai, New Zealand forming a largely hexagonal pattern.


Figure 1: When neighbours keep their distance: gannets nesting at Muriwai Beach, New Zealand. The image on the left shows the overall hexagonal ordering of the birds while that on the right shows a penta-hepta defect (PHD). The green (pink) markers identify gannets with five (seven) neighbours instead of the usual six. Photo credit: Barbora Knobloch. Adapted from [1].

Patterns are rarely perfect and defects arise due to many factors such as boundary conditions (e.g. cliff edge), fluctuations in the background (e.g. unevenness in the ground), etc. A generic defect that arises in such hexagonal patterns is highlighted in the right panel of the picture of the gannets. Normally each gannet will have six neighbours, but here we see that the gannet marked with a green dot has only five neighbours while the gannet marked with a pink marker has seven. Such a structure consisting of a bound state with one location having five neighbours and another location seven neighbours, instead of the usual six, is called a penta-hepta defect (PHD). Hexagonal arrangements are found in many areas of physics, from patterns formed in heated fluids, to self-assembled crystals formed in both hard materials (e.g. graphene) and soft materials (e.g. star-shaped polymers [2]). Equally prevalent in all such hexagonal arrangements is the possibility for PHDs.

Traditionally pattern formation techniques used to investigate defects use an amplitude-phase formulation, where a periodic pattern has a homogenous amplitude and a varying phase. The topological charge of a defect is calculated by integrating the phase around any closed curve enclosing the defect, and this quantity does not change when the defect moves. Topological defects [3] are associated with zeros of the amplitude (where the phase becomes undefined): these defects have non-zero topological charge and so they can only be eliminated or healed by interacting with another topological defect with opposite charge. On the other hand, non-topological defects, such as PHDs, have a well-defined phase everywhere (implying zero topological charge) and so were thought to be able to heal by themselves. However, if the defect has an internal structure it may persist as a result of frustration and get locked/pinned to the background periodic state: the gannets in the PHD in Figure 1 could re-arrange themselves to remove the defect, but to do so would involve all nearby gannets moving a little bit, so in practice they don't.


Figure 2: Coexisting equilibria with penta-hepta defects separating regions of hexagons with different orientations in the SH23 system [1]. All three states are dynamically metastable. 

We explore such non-topological defects in the prototypical pattern forming Swift-Hohenberg model in our recent work [1], by adopting a different point of view and thinking of defects as spatially localised structures [4]. We focus on grain boundaries separating two-dimensional hexagon crystals at different orientations (shown in Figure 2): these grain boundaries are closed curves containing a ring of PHDs. Even with the parameters all the same, the model has many different stable configurations of these grain boundaries, and solution branches connected to each of these states form isolas that span a wide range of the model parameters, opening up multiple interesting questions about such defect states. Our results will also be applicable to understanding the role of grain boundaries in two dimensional solids such as graphene [5], in which defects play a crucial role near phase transition, i.e., melting [6], and in determining bulk properties of a material."


[1] Snaking without subcriticality: grain boundaries as non-topological defects, P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, arXiv:2011.08536, 2020.

[2] Two-dimensional crystals of star polymers: a tale of tails, I. Bos, P. van der Scheer, W. G. Ellenbroek and J. Sprakel, Soft Matter, 15, 615-622, 2019

[3] The topological theory of defects in ordered media, N. D. Mermin, Rev. Mod. Phys., 51, 591-648, 1979.

[4] Spatial localisation in dissipative systems, E. Knobloch, Annu. Rev. Condens. Matter Phys., 6, 325-359, 2015. 

[5] Energetics and structure of grain boundary triple junctions in graphene, P. Hirvonen, Z. Fan, M. M. Ervasti, A. Harju, K. R. Elder and T. Ala-Nissila, Sci. Rep., 7, 1-14, 2017.

[6] Melting of graphene: from two to one dimension, K. V. Zakharchenko, A, Fasolino, J. H. Los and M. I . Katsnelson, J. Phys.: Condens. Matter, 23, 202202, 2011. 

Friday, 1 January 2021

The launch of the Oxford Online Maths Club

Happy New Year! 2021 has a lot to make up for after 2020, so we're starting with a bang with the launch of the Oxford Online Maths Club, a new weekly maths livestream from Oxford Mathematics.

The Club provides free super-curricular maths for ages 16-18. It is aimed at people about to start a maths degree at university or about to apply for one. We'll be livestreaming one hour of maths problems, puzzles, mini-lectures, and Q&A, and we'll be exploring links between A level maths and university maths with help from our Admissions Coordinator James Munro and our current Oxford Mathematics students. And you get to ask questions and share thoughts and feelings with like-minded mathematicians. 

In a nutshell, it’s free, interactive, casual, and relaxed, with an emphasis on problem-solving techniques, building fluency, and looking ahead at links to university maths. The Club follows in the footsteps of James's hugely popular weekly MAT (Mathematics Admissions Test) sessions where he went thorough entrance problems and took live questions.

Whether you're the only person you know interested in maths, or you're an entire sixth-form maths club looking for more content, we're here for you in 2021! Join us every Thursday 16:30 starting this Thursday, 7 January. 

Friday, 18 December 2020

Peter Michael Neumann OBE (28 December 1940 - 18 December 2020)

We are very sad to hear the news of the death of Peter Neumann earlier today. Peter was the son of the mathematicians Bernhard Neumann and Hanna Neumann and, after gaining a B.A. from The Queen's College, Oxford in 1963, obtained his D.Phil from Oxford University in 1966.

Peter was a Tutorial Fellow at The Queen's College, Oxford and a lecturer in the Mathematical Institute in Oxford, retiring in 2008. His work was in the field of group theory. He is also known for solving Alhazen's problem in 1997. In 2011 he published a book on the short-lived French mathematician Évariste Galois.

In 1987 Peter won the Lester R. Ford Award of the Mathematical Association of America for his review of Harold Edwards' book Galois Theory. In 2003, the London Mathematical Society awarded him the Senior Whitehead Prize. He was the first Chairman of the United Kingdom Mathematics Trust, from October 1996 to April 2004 and was appointed Officer of the Order of the British Empire (OBE) in the 2008 New Year Honours. Peter was President of the Mathematical Association from 2015-2016.

Tuesday, 15 December 2020
Sunday, 13 December 2020

The Oxford Mathematics E-Newsletter - our quarterly round-up of our greatest hits

The Oxford Mathematics e-newsletter for December is out. Produced each quarter, it's a sort of 'Now That's What I Call Maths,' pulling together our greatest hits of the last few months in one place.

It's for anyone who wants a flavour of what we do - research, online teaching, public lectures, having a laugh.

And it's COVID-lite. Click here.

Saturday, 12 December 2020

Full 2nd Year Oxford Mathematics Undergraduate course publicly available for the first time

Over the past few weeks we have made 7 undergraduate lectures publicly available, sampling a range of topics from Geometry to Differential Equations. Today & over the next 2 weeks for the first time we're showing a full course on our YouTube Channel. Ben Green's 2nd Year 'Metric Spaces' (the first half of the Metric Spaces and Complex Analysis course)' gets to grips with the concept of distance. 

We are making these lectures available to give an insight in to life in Oxford Mathematics. All lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated worksheet. Course materials can be found here




Friday, 4 December 2020

Roger Penrose's Nobel Lecture and presentation of Prize

This Tuesday, 8th December, from 8am GMT onwards (repeated) you can watch 2020 Physics Laureate and Oxford Mathematician Roger Penrose's specially recorded Nobel Lecture in which he talks about the background to and genesis of his work on Black Holes which won him the prize; and also where our understanding of Black Holes is taking us. 

On the same day Roger will be presented with the Nobel diploma and medal at the Swedish Ambassador’s Residence in London and you can watch this as part of the Nobel Prize Awards Ceremony from 3.30pm GMT on Thursday 10 December. Watch both here

As Roger said on receiving the news of the award: "In 1964 the existence of Black Holes was not properly appreciated. Since then they have become of increased importance in our understanding of the Universe and I believe this could increase in unexpected ways in the future."

Roger Penrose is one of our greatest living scientists. His work on Black Holes provided the mathematical tools needed by experimentalists to go and find Black Holes. His fellow prize winners, Andrea Ghez and Reinhard Genzel went and did just that.

However, Roger's work has ranged much further than just the Universe, from twistor theory to quasi-periodic tiling, spin networks to impossible triangles, a range that perhaps might not be so encouraged in academia today.

Now in his 90th year Roger is still researching and writing. He will give an Oxford Mathematics Public Lecture in January 2021 to celebrate the Nobel Prize.

Photography below and above by Professor Alain Goriely.  Updated photographs further below of Roger receiving the Nobel Medal and Diploma from the Swedish Ambassador in London on 8 December.

Sunday, 22 November 2020

Our latest Online Student Lecture - 2nd Year Linear Algebra

The latest in our Autumn 2020 series of lectures is the first lecture in Alan Lauder's Second Year Linear Algebra Course. In this lecture Alan (with help from Cosi) explains to students how the course will unfold before going on to talk specifically about Vector Spaces and Linear Maps.

All lectures are followed by tutorials where students meet their tutor in pairs to go through the lecture and associated worksheet. The course materials and worksheets can be found here.

That's Cosi on the left.





Friday, 20 November 2020

Numerically solving parametric PDEs with deep learning to break the curse of dimensionality

Oxford Mathematician Markus Dablander talks about his collaboration with Julius Berner and Philipp Grohs from the University of Vienna. Together they developed a new deep-learning-based method for the computationally efficient solution of high-dimensional parametric Kolmogorov PDEs.

"Kolmogorov PDEs are linear parabolic partial differential equations of the form \begin{equation*} \label{kol_pde_intro} \tfrac{\partial u_\gamma }{\partial t} = \tfrac{1}{2} \text{Trace}\big(\sigma_\gamma [\sigma_\gamma ]^{*}\nabla_x^2 u_\gamma \big) + \langle \mu_\gamma , \nabla_x u_\gamma \rangle, \quad u_\gamma (x,0) = \varphi_\gamma(x). \end{equation*} The functions \begin{equation*}\varphi_\gamma : \mathbb{R}^d \rightarrow \mathbb{R} \quad \text{(initial condition)}, \quad \sigma_\gamma : \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}, \quad \mu_\gamma : \mathbb{R}^d \rightarrow \mathbb{R}^{d} \quad \text{(coefficient maps)}, \end{equation*} are continuous, and are implicitly determined by a real parameter vector $\gamma \in D $ whereby $D$ is a compact set in Euclidean space. Equations of this type represent a broad class of problems and frequently appear in practical applications in physics and financial engineering. In particular, the heat equation from physical modelling as well as the widely-known Black-Scholes equation from computational finance are important special cases of Kolmogorov PDEs.

Typically, one is interested in finding the (viscosity) solution \begin{equation*} u_\gamma : [v,w]^d \times [0,T] \rightarrow \mathbb{R} \end{equation*} of a given Kolmogorov PDE on a predefined space-time region of the form $[v,w]^{d} \times [0,T]$. In almost all cases, however, Kolmogorov PDEs cannot be solved explicitly. Furthermore, standard numerical solution algorithms for PDEs, in particular those based on a discretisation of the considered domain, are known to suffer from the so-called curse of dimensionality. This means that their computational cost grows exponentially in the dimension of the domain, which makes these techniques unusable in high dimensions. The development of new, computationally efficient methods for the numerical solution of Kolmogorov PDEs is therefore of high interest for applied scientists.

We were able to develop a novel deep learning algorithm capable of efficiently approximating the solutions $(u_\gamma )_{\gamma \in D}$ of a whole family of potentially very high-dimensional $\gamma$-parametrised Kolmogorov PDEs on a full space-time region. Specifically, our proposed method allows us to train a single deep neural network \begin{equation*} \Phi\colon D \times [v,w]^d \times [0,T] \rightarrow \mathbb{R} \end{equation*} to approximate the parametric solution map \begin{align*} \label{eq:gen_sol_map} \bar{u} : D \times [v,w]^d \times [0,T] \rightarrow \mathbb{R}, \quad (\gamma, x, t) \mapsto \bar{u}(\gamma, x, t) := u_\gamma (x,t), \end{align*} of a family of $\gamma$-parametrized Kolmogorov PDEs on the generalised domain $D \times [v,w]^d \times [0,T]$. The key idea of our novel algorithm is to reformulate the parametric solution map $\bar{u}$ of the $\gamma$-parametrized Kolmogorov PDE as the regression function of an appropriately chosen supervised statistical learning problem. This reformulation is based on an application of the so-called Feynman-Kac formula, which links the theory of linear parabolic PDEs to the theory of stochastic differential equations via $$ \bar{u}(\gamma,x,t) = \mathbb{E}[\varphi_{\gamma}(S_{\gamma,x,t})]. $$ Here $S_{\gamma,x,t}$ is the solution of an associated $\gamma$-parametrised stochastic differential equation with starting point $x$ at time $t$: $$ dS_{\gamma,x,t} = \mu_\gamma   (S_{\gamma,x,t}) dt + \sigma_\gamma  (S_{\gamma,x,t}) dB_t, \quad S_{\gamma,x,0} = x. $$ The resulting stochastic framework can be exploited to prove that $\bar{u}$ must be the solution of a specifically constructed statistical learning problem. Realisations of the predictor variable of this learning problem can be simulated by drawing uniformly distributed samples of the domain $(\gamma_i, x_i, t_i) \in D \times [v,w]^d \times [0,T]$ while realisations of the dependent target variable can be generated by simulating realisations $\varphi_{\gamma_i}(s_{\gamma_i, x_i, t_i})$ of $\varphi_{\gamma_i}(S_{\gamma_i, x_i, t_i})$. Thus, a potentially infinite amount of independent and identically distributed training points can be simulated by solving for $S_{\gamma_i, x_i, t_i}$ via simple standard numerical techniques such as the Euler-Maruyama scheme. The simulated predictor-target-pairs $((\gamma_i, x_i, t_i)  ,\varphi_{\gamma_i}(s_{\gamma_i, x_i, t_i}))$ can then be used to train the deep network $\Phi$ to approximate the solution $\bar{u}.$ An illustration of this training workflow is depicted in the figure below.


We mathematically investigated the approximation and generalisation errors of our constructed learning algorithm for important special cases and discovered that, remarkably, it does not suffer from the curse of dimensionality. In addition to our theoretical findings, we were able to also observe numerically that the computational cost of our our proposed technique does indeed grow polynomially rather than exponentially in the problem dimension; this makes very high-dimensional settings computationally accessible for our method.

Our work was accepted for publication at the 2020 Conference on Neural Information Processing Systems (NeurIPS). A preprint can be accessed here.

Below is a picture of the Multilevel architecture we used for our deep network $\Phi$ (click to enlarge)."



Monday, 16 November 2020

Generalising Leighton's Graph Covering Theorem

Oxford Mathematician Daniel Woodhouse talks about the theorem that motivates much of his research.

"According to lore, denizens of the Prussian city of Königsberg would spend their Sunday afternoons wandering the streets. From this pastime came a puzzle - could a wanderer cross each of the city's seven bridges without crossing the same bridge twice? The path of the River Pregel divides the city into four distinct regions, including a central isolated island, so the question was more complicated than a matter of crossing back and forth from one side of a river to the other.


If a positive solution existed, then someone would surely have found it, so it could be guessed that the answer was negative. In principle, you could write down an exhaustive list of all possible sequences of bridge crossings, but this would be a very crude kind of proof. The problem eventually came to the attention of Leonhard Euler, who set about finding an explanation that offered more insight into the nature of the problem. In 1736 Euler wrote up his solution, providing arguments that not only applied to the bridges of Königsberg, but to any arrangement of rivers and bridges. Initially, it seems from his correspondence that Euler was inclined to dismiss the intellectual merits of the problem, not regarding it as mathematical at all. But in the introduction of his paper he declares it to belong to a kind of mathematics only previously speculated: the geometry of position ("Geometriam situs''). Today this kind of mathematics is better known as topology.


The kind of mathematics that Euler would have been familiar with - geometry - concerned notions of distance, area, and angles. Yet considering the problem of the bridges from a purely mathematical point of view, such quantities become irrelevant. The exact shape that the river takes and the distance between bridges are relevant to the individual planning on spending an afternoon walking over them, but less so to the mathematician. Indeed, the relevant information can be reduced to a list of the distinct land masses, and the bridges they connect. In modern terminology the problem can be redrawn as a graph:


Each land mass is now represented by a vertex and each bridge is now an edge joining the corresponding vertices. We might say that the graph is topologically equivalent to the original diagram of the river and the bridges. I have labelled the vertices according to how Euler labelled them in his previous diagram. I have also added arrows indicating an orientation of the edges. Graphs are now ubiquitous in mathematics and science, and topology is only one perspective of study. You should not equate the graph with the way that it has been drawn above. The same graph could be drawn in many ways:

Let us now move on from Euler's problem and take the topological perspective one step further. We imagine the wanderer of the city is now a wanderer of our graph instead. They travels from vertex to vertex via the edges. Suppose that our wander has no map of the world they are living in. That is to say the wanderer has no idea what graph they are wandering around; whichever vertex they have arrived at, all they can see are the edges immediately connected to the vertex on which they stand. They can remember the route that they travelled to get there, but no more. Frustratingly, since we are now in a graph, and not the city of Königsberg, there are no landmarks with which the wanderer can orient themselves. If they return to a vertex they have no way of knowing. As we look down on our wanderer we can see exactly where they are, while they remain completely unaware. We can also take a look at the graph as they imagine it:


In the wanderer's imagination, how do they envision the graph? Since they never know if they have arrived at a vertex they have previously visited (without doubling back on themselves) this graph cannot contain any closed cycles - a path the begins and ends at the same vertex without repeating any other edge or vertex. A graph that does not contain any closed cycles is called a tree.


This tree, let's call it $\widetilde X$, is labelled in the picture according to which edge in $X$ it really corresponds to. These labels are invisible to our wanderer - if the labels were known then they could deduce what graph they were wandering about on. There is one particularly important vertex in $X$ and $\widetilde{X}$: the point at which our wanderer begins their journey, which we call the basepoint. This mapping between the graph $X$ and the universal cover $\widetilde X$ is usually written as a function $\widetilde X \rightarrow X$. The important feature here is that given a path that our wanderer takes in $X$, we can track the path they imagine they are taking in $\widetilde X$. Conversely, given a path in $\widetilde X$, using the labelling that only we can see, we can deduce the path that our wanderer has taken in $X$.

It is impossible for our wanderer to deduce what the graph $X$ is from $\widetilde X$. This is because $X$ isn't the only graph with universal cover $\widetilde X$. Setting aside Königsberg for a moment, consider the following graphs:


These graphs are obviously distinct, just by counting the vertices. But to a wanderer of these graphs, all they know is that whatever edge they choose to wander down, they will arrive at a vertex incident to precisely three edges. Wherever they go, the world to them looks exactly the same. There is no way for them to deduce if they are living in $X_1$, $X_2$, or $X_3$. Another way to say this is that $X_1$, $X_2$, and $X_3$ have the same universal cover $T_3$ - the 3-valent tree.


This isn't all that our wanderer could imagine. In principle they could try imagining that they are wandering around any graph we like. But that imagined graph should at least be consistent with what they actually experience. You can't arrive at a vertex with three incident edges but imagine there are four incident edges - that would present a contradiction to our wanderer and the illusion would be ruined. As a simple example, consider the reverse situation to the above: suppose our wanderer imagines they are navigating the Königsberg graph $X$, but in reality they are inside $\widetilde X$:


Provided that the basepoints match up as before, the wanderer would encounter no contradiction between what they are imagining and what they actually experience. But the situation would be particularly deceptive: as depicted above, our wanderer could imagine that they had walked a closed loop, when in fact they had not. This misconception is particularly bad for topologists, so we rule them out. Imagined graphs that do not cause the misconception are called covers, and there exist many of them aside from the universal cover. This definition for a cover is quite confusing, which is why we usually work with an equivalent and more practical condition in terms of labelling the edges of the cover according to the corresponding edges in the original graph. For example:


Note also that the cover given above is finite, while the universal cover is infinite. Unless the graph $X$ is itself a tree, the universal cover will always be infinite. Indeed, if $X$ is not a tree, then there is a closed cycle that our wanderer can walk around indefinitely, quite unaware that they are trapped in a closed loop.

A large part of my research is focused on the the following theorem:

Theorem: Let $X_1$ and $X_2$ be finite graphs that have the same universal cover. Then they have a common finite cover.

Here is a picture of two wanderers living on two graphs with the same universal cover, imagining that they are living on a common cover:


This theorem was conjectured in an 1980 paper by the computer scientist Dana Angluin. Angluin was interested in distributed systems, which you can imagine as being a graph with a computer at each vertex and each edge corresponding to a 'port' connecting the neighbouring computers, allowing them to communicate. If this picture seems familiar that is because the internet is an example. The first complete proof of this theorem was given in a 1982 paper by Frank Thomas Leighton. Leighton is also a computer scientist and most of his research in graph theory was closely related to similar applications in computer science. The kind of problems that concerned Angluin and Leighton more closely resemble the kind of problem that Euler was considering than my own research. On the strength of the insights graph theory offers, Leighton co-founded Akamai Technologies in 1998, and today they are one of the internet's largest content delivery networks. He also retains an affiliation as a professor of applied mathematics at MIT and I have been told he still teaches an occasional course (I have the same teaching obligations as a billionaire, it turns out).

My own research has involved applying Leighton's theorem to the field of geometry and topology. Leighton's original proof was very much the argument of a computer scientist, and finding a new proof that offered a different theoretical framework has allowed me to solve previously unapproachable problems. There are two goals: the first is to generalise the theorem to higher dimensional topological spaces; and the second is to apply the generalisations to topological spaces or groups that decompose into a graph like structure. For example, in a recent preprint with fellow Oxford Mathematician Sam Shepherd we are able to prove quasi-isometric rigidity for a large family of hyperbolic groups. There are close connections to the theory of $3$-manifolds, their JSJ decompositions, and the recent resolution of the virtual Haken conjecture. In particular I believe that the right setting to generalise Leighton's theorem is special cube complexes."