Wednesday, 27 March 2019 
In this collaboration with researchers from the Umeå University and the University of Zurich, Renaud Lambiotte from Oxford Mathematics explores the use of higherorder networks to analyse complex data.
"Network science provides powerful analytical and computational methods to describe the behaviour of complex systems. From a networks viewpoint, the system is seen as a collection of elements interacting through pairwise connections. Canonical examples include social networks, neuronal networks or the Web. Importantly, elements often interact directly with a relatively small number of other elements, while they may influence large parts of the system indirectly via chains of direct interactions. In other words, networks allow for a sparse architecture together with global connectivity. Compared with meanfield approaches, network models often have greater explanatory power because they account for the nonrandom topologies of reallife systems. However, new forms of highdimensional and timeresolved data have now also shed light on the limitations of these models.
Rich data indicate who interacts with whom, but also what different types of interactions exist, when and in which order they occur, and whether interactions involve pairs or larger sets of nodes. In other words, they provide us with information on higherorder dependencies, which lay beyond the reach of models based on pairwise links. In this perspective published in Nature Physics, the authors review recent advances in the development of higherorder network models, which account for different types of higherorder dependencies in complex data. They focus in detail on models where chains of interactions are more than a combination of links, and when higherorder Markov models are required to reproduce realistic chains. As an illustration, the Figure shows (click to enlarge) chains of citations between journals as produced by standard network models (left) versus empirical data and higherorder models (right). As a next step, the authors illustrate how to generalise fundamental network science methods, namely community detection, node ranking and the modelling of dynamical processes. They conclude by discussing challenges in developing optimal higherorder models that take advantage of rich data on higherorder dependencies while avoiding the risk of overfitting."

Friday, 22 March 2019 
Oxford Mathematician Benjamin Walker talks about his work on the automatic identification of flagella from images, opening up a world of datadriven analysis.
"Across the University of Oxford alone there are millions of images of microorganisms propelling themselves along with flagella, slender taillike structures like those found on the sperm cells of most mammals. Studying the detailed motion of flagella is of intense scientific interest, not least amongst researchers in the fields of fertility medicine and human pathogens. Previously, flagella in each of these countless images would need to be traced (often by hand on a graphics tablet) by an unfortunate researcher before they can analyse the detailed motion of the fagellum. With even the moresophisticated 'pointandclick' computerised methods, in general this still requires vast amounts of time and laborious effort. With artificial intelligence capable of accurately detecting cats in videos on YouTube, and a lot more besides, it seems that automation of this somewhatsimple task might be easily accomplished using the powerful computing resources available today.
In fact, we needn't turn to complex algorithms or simulated intelligence at all to automate our identification and remove the need for a patient researcher. Instead we look at something fundamental in the biology of the flagellum, persistent across a wide range of organisms. Eukaryotic flagella, like those of mammalian sperm and the human parasites Trypanosomatidae, are all made up of the same structural backbone, referred to as the 9+2 axoneme, which gives them a welldefined and consistent width when viewed through typical microscopes. If we compare this to the rest of the cell, for example for the Trypanosome shown below in (a), the flagellum stands out as by far the largest portion of the microorganism that is a consistent thickness.
The width of the organism is something that can be easily quantified via a medial axis transform. The transform takes a binary representation of the whole cell (b), obtainable automatically, and reduces it down to a skeleton along the approximate midline of the cell (c). We then colour each point on this skeleton by the distance to the edge of the cell, with the wider parts corresponding to darker sections of the skeleton. Tracing out the skeleton from left to right, we can record the width of the cell at each point, as shown on the graph to the left below.
The flagellum can be easily identified by eye as the large region of approximately constant width. The histogram to the right in fact shows something more, that the flagellar width (approximately 6 pixels here) corresponds to the mode of all the cell widths. This observation leads to a remarkably simple scheme for automatically identifying flagella: take a medial axis transform, and keep only the regions corresponding to the modal width. We have tested this simple method on a sample of 150,000 images of a Trypanosome, orders of magnitude more data than previously analysed for this organism, and accurately captured the flagella of over 120 different cells, enabling populationlevel studies of flagellar movement.
Thus, with only a simple image transform and exploiting a conserved structural feature of flagella, this automated methodology could enable rapid quantification of large image datasets, freeing up researchers to analyse data rather than trace it out."

Thursday, 21 March 2019 
If you are Interested in postgraduate study and curious about what it would be like to do research at Oxford we are delighted to announce that Oxford Mathematics is taking part in the UNIQ+ pilot programme, a sixweek summer school encouraging access to postgraduate study from underrepresented groups in UK universities.
UNIQ+ is free to take part in and includes a £2,500 stipend, plus free accommodation in an Oxford college. It will give you the opportunity to experience postgraduate research at Oxford by carrying out a research project, and a chance to meet our staff and student community.
UNIQ+ will run from 1 July to 9 August 2019.
To see if you qualify for UNIQ+, and to apply, click here.

Wednesday, 20 March 2019 
Oxford Mathematician Vinayak Abrol talks about his and colleagues's work on using mathematical tools to provide insights in to deep learning.
Why it Matters!
From 4.5 billion photos uploaded to Facebook/Instagram per day, 400 hr of video uploaded to YouTube per minute, to the 320GB the large Hadron collider records per second, we know it is the age of big data. Indeed, last year the amount of data existing worldwide is estimated to have reached 2.5 quintillion bytes and while only 30% of this data is described as useful, if analysed only 1% actually is. However, the boom in big data has brought success in many fields such as weather forecasting, image recognition and language translation.
So how do we deal with this big data challenge? To fundamentally understand realworld data, we need to look into the intersection of mathematics, signal processing and machine learning, and combine tools from these areas. One such emerging field of study is 'deep learning' under the broader category of 'artificial intelligence'. There have been remarkable advances in deep learning over the decade, and it has found uses in many daytoday applications such as on smartphones where we have automatic face detection, recognition, tagging in photos or speech recognition for voice search and setting up reminders; and in homes predictive analysis and resource planning using the Internet of Things or smart home sensors. This shift in paradigm is due to the fact that in many tasks machines are efficient, can work continuously and perform better than humans. However, recently deep neural networks have been found to be vulnerable. For instance, they can be fooled by welldesigned examples, called adversarials, which is one of the major risks for applying deep neural networks in, say safetycritical scenarios. In addition most such systems are like a black box without any intuitive explanation, and with little or no innate knowledge about human psychology, meaning that the huge buzz about automation due to artificial intelligence is yet to be justified. Hence, understanding how these networks are vulnerable to attacks is attracting great attention, for example in biometric access to banking systems.
We try to present a different way of approaching this problem. We know deep networks work well in many applications but they can't abstractly reason and generalise about the world or automate ordinary human activities. For instance, a robot which can pick up a bottle, can pick up a cup only if it is retrained (not the case with humans). Hence, we need to understand the limits of such learning systems. Any other questions are meaningless if our resulting system appears to be solving the problem but is actually not. This is dubbed as the 'Clever Hans effect'. So, instead, we are interested in: 1) Does the system actually address the problem? 2) What is the system actually learning? 3) How can we make it address the actual problem?
General Methodology
In Oxford Mathematics, this work is funded under the OxfordEmirates Data Science Initiative, headed by Prof. Peter Grindrod. I am being supervised by Prof. Jared Tanner, and within the Data Science Research Group we are working to advance our understanding about deep learning approaches. Although, many recent studies have been proposed to explain and interpret deep networks, currently there is no unified coherent framework for understanding their insights. Our work here aims to mathematically study the reasons why deep architectures are working well, what they are learning and are they really addressing the problem they are built for. We aim to develop a theoretical understanding of deep architectures, which in general processes the data with a cascade of linear filters, nonlinearities, and contraction mapping via existing well understood mathematical tools. In particular, we focus on the underlying invariants learned by the network such as multiscale contractions, hierarchical structures etc. Our earliest research explored the key concept of sparsity, i.e., the low complexity of highdimensional data when represented in a suitable domain. In particular we are interested in the learning of representation systems (dictionaries) providing compact (sparse) descriptions for high dimensional data from a theoretical and algorithmic point of view, and in applying these systems to data processing and analysis tasks. Another direction is to employ tools from random matrix theory (RTM) that allow us to compute an approximation of such learning systems under a set of simplifying assumptions. Our previous studies have revealed that the representations obtained at different intermediate stages of a deep network have complimentary information and sparsity plays a very important role in the overall learning objective. Infact the so called nonlinearities and pooling operations which seems complicated can be explained via constrained matrix factorization problems along with hierarchical/sparsity aware signal processing. In terms of the choice of underlying building blocks of such systems such as convolutional networks (popular for image applications), these can be analysed via convolutional matrix factorization and wavelet analysis, and fully connected networks via discrete Weiner and Volterra Series analysis. Overall our main tools come from sparse approximation, RTM, geometric functional analysis, harmonic analysis and optimisation.

Wednesday, 20 March 2019 
Oxford Mathematician Ric Wade explains how rightangled Artin groups, once neglected, are now central figures in lowdimension topology.
"Rightangled Artin groups (colloquially known as a RAAGs or 'rags') form a family of finitely presented discrete groups that generalise freeabelian groups (where all the generators commute) and free groups (where no generators commute). Every finite graph $\Gamma$ defines a finitely presented group $A_\Gamma$ generated by the vertex set, with the restriction that $vw=wv$ if two vertices $v$ and $w$ are connected by an edge (in other words, adjacent vertices commute). If the graph is complete (so that any two vertices are connected by an edge) then $A_\Gamma$ is a freeabelian group, and if the graph $\Gamma$ has no edges, then $A_\Gamma$ is a free group. Loops and multiple edges between vertices do not affect the group that we get using this construction, so in graphtheoretic language we can assume that $\Gamma$ is simple.
Figure 1: A graph $\Gamma$ and a topological space $X$ such that $\pi_1(X)=A_\Gamma$.
One example is given by the graph $\Gamma$ which consists of three edges joined in a line. In this case, the group $A_\Gamma$ that we get is obtained by taking the fundamental group of three tori, where the middle tours has its meridian and equator glued to the equators of the other two tori (see Figure 1).
Possibly because the definition of these groups sounds quite contrived, rightangled Artin groups saw relatively little interest until the 21st century. However, they have become central figures in recent progress in lowdimension topology. I can highlight two reasons for this:
 RAAGs have a nice simple combinatorial description, which is helpful for proving grouptheoretic results.
 RAAGs have a surprisingly diverse subgroup structure, which means that many groups that are harder to understand can be embedded in these 'simpler' groups.
The flagship result with respect to this viewpoint is a theorem of Agol (building upon work of Wise, KahnMarkovic, Sageev, and others), which states that the fundamental group of every closed hyperbolic 3manifold has a finite index subgroup which embeds into a rightangled Artin group in a particularly nice way. This was a key step in the resolution of Thurston's virtual fibering conjecture, which was one of the major open problems in the area not to fall to Perelman's solution of the Poincare conjecture.
I'm interested in automorphism groups of rightangled Artin groups: the groups of their selfsymmetries. Going back to thinking about RAAGs interpolating between free and freeabelian groups, these automorphism groups should have behaviour similar to $GL(n,\mathbb{Z})=Aut(\mathbb{Z}^n)$ from the freeabelian side, and automorphism groups of free groups on the free side. In practice, such interpolation results are much harder to obtain. In recent work with Matthew Day at the University of Arkansas, we have been studying the finiteness properties of these automorphism groups. Roughly speaking, this is the search for nice topological spaces (specifically, classifying spaces) that describe $Aut(A_\Gamma)$ for an arbitrary RAAG. For $GL(n,\mathbb{Z})$ the construction of such spaces goes back to work of Borel and Serre, who used deformation spaces of tori to build classifying spaces of certain finite index subgroups of $GL(n,\mathbb{Z})$ known as congruence subgroups. A classifying space for a finiteindex subgroup of $Aut(F_n)$ was built by Culler and Vogtmann, using a deformation space of metric graphs (these appear as tropical curves in the world of tropical geometry). In a paper due to appear in the Journal of Topology, we prove the following theorem:
Theorem 1 (DayW). For any graph $\Gamma$, the group $Aut(A_\Gamma)$ has a finiteindex subgroup with a classifying space given by a finite cell complex.
Our classifying space is built by carefully splicing together deformation spaces of tori, deformation spaces of graphs, and classifying spaces for rightangled Artin groups called Salvetti complexes. This theorem gives restrictions on the algebraic invariants of $Aut(A_\Gamma)$; in particular it says that the cohomology ring $H^*(Aut(A_\Gamma);\mathbb{Q})$ is finitely generated as a $\mathbb{Q}$vector space. Our construction is inductive rather than completely concrete, however the construction does allow for some calculations (via spectral sequence arguments, for instance). The next step for us in this programme is to give an algorithm to find the virtual cohomological dimension of any automorphism group $Aut(A_\Gamma)$, and we hope that other problems involving these automorphism groups can be understood through these classifying spaces as well."

Thursday, 14 March 2019 
Oxford Mathematician Kristian Kiradjiev has won the Gold Award in the Mathematical Sciences category at this year’s STEM for Britain at the House of Commons on 13th March. This prestigious competition provides an opportunity for researchers to communicate their research to parliamentarians.
Kristian’s poster covered his research into the mathematical modelling of fluegas purification and the removal of toxic chemicals from the gas.
As reported last week, Kristian was one of three Oxford Mathematicians presenting in the Commons.

Friday, 8 March 2019 
Oxford Mathematics Visiting Fellow and Reader in Applied Mathematics at the University of Bath, Apala Majumdar has been awarded the 2019 FDM Everywoman in Tech Academic Award. This is awarded to a woman in academia who has made an outstanding contribution to technology and science and whose work has made or has the potential to make a significant longterm impact in STEM.
Apala is an applied mathematician researching fundamental mathematical theories in material science. She specialises in Liquid Crystals and has published over 40 papers to date. Moreover, Apala works to inspire female researchers globally through mentorship and is deeply committed to teaching and training young people.
Apala was nominated by Oxford Mathematician and Director of the Oxford Centre for Industrial and Applied Mathematics (OCIAM), Alain Goriely, who said: “I cannot think of a more deserving candidate for an academic award for young women who are inspiring other female researchers around the world. Apala has singlehandedly built an international network spanning four continents, making her one of the world leaders in her field and most internationally recognised of her generation."
The FDM Tech Awards take place in the week of International Women’s Day and celebrate 50 of the most talented individuals shaking up the tech industry.

Wednesday, 6 March 2019 
Three Oxford Mathematicians, Kristian Kiradjiev, Liam Brown and Tom Crawford are to present their research in Parliament at this year’s STEM for Britain competition at the House of Commons on 13th March. This prestigious competition provides an opportunity for researchers to communicate their research to parliamentarians.
Kristian’s poster covers his research into the mathematical modelling of fluegas purification, Liam's poster researches computational models of cancer immunotherapy while Tom is researching the spread of pollution in the ocean.
Judged by leading academics, the gold medalist receives £2,000, while silver and bronze receive £1,250 and £750 respectively.

Thursday, 28 February 2019 
Oxford Mathematics' Heather Harrington is the joint winner of the 2019 Adams Prize. The prize is one of the University of Cambridge's oldest and most prestigious prizes. Named after the mathematician John Couch Adams and endowed by members of St John's College, it commemorates Adams's role in the discovery of the planet Neptune. Previous prizewinners include James Clerk Maxwell, Roger Penrose and Stephen Hawking.
This year's Prize has been awarded for achievements in the field of The Mathematics of Networks. Heather's work uses mathematical and statistical techniques including numerical algebraic geometry, Bayesian statistics, network science and optimisation, in order to solve interdisciplinary problems. She is the CoDirector of the recently established Centre for Topological Data Analysis.

Tuesday, 26 February 2019 
We’re all familiar with liquid droplets moving under gravity (especially if you live somewhere as rainy as Oxford). However, emerging applications such as labonachip technologies require precise control of extremely small droplets; on these scales, the forces associated with surface tension become dominant over gravity, and it is therefore not practical to rely on the weight of the drops for motion. Many active processes (requiring external energy inputs), such as those involving the use of temperature gradients, electric fields, and mechanical actuation, have been used successfully to move small droplets. Recently, however, there has been increasing interest in passive processes, which do not require external driving. One example of this is durotaxis, in which droplets spontaneously move in response to rigidity gradients (similar to the active motion of biological cells, which generally move to stiffer regions of a deformable substrate). Here, the suffix ‘taxis’ refers to the selfpropulsive nature of the motion. In a recent study, Oxford Mathematicians Alex Bradley, Finn Box, Ian Hewitt and Dominic Vella introduced another such mechanism; Bendotaxis is selfpropelled droplet motion in response to bending. What is particularly interesting is that the motion occurs in the same direction, regardless of whether the drop has an affinity to (referred to as ‘wetting’) the channel walls or not (‘nonwetting’), which is atypical for droplet physics.
A small drop confined to a channel exerts a force on the walls, as a result of surface tension; this force pulls the walls together when the drop wets them, and pushes them apart otherwise. By manipulating the geometry of the channel (leaving one end free, and clamping the other end), the deformation that results from this surface tension force is asymmetric—it creates a tapering in the channel. The drop subsequently moves in response to this tapering, which is towards the free end in both the wetting and nonwetting cases.
Using a combination of scaling arguments and numerical solutions to a mathematical model of the problem, the team were able to verify that it is indeed the capillary induced elastic deformation of the channel that drives the experimentally observed motion. This model allowed them to understand the dynamic nature of bendotaxis, and predict the motion of drops in these deformable channels. In particular, they identified several interesting features of the motion; counterintuitively, it is predicted (and observed) that the time taken for a drop to move along the channel decreases as it increases in length. However, relatively long channels are susceptible to ‘trapping’, whereby the force exerted by the drop is sufficient to bring the channel walls into contact. It is hoped that understanding the motion will pave the way for its application on a variety of scales  for example, drug delivery on a laboratoryscale, and selfcleaning surfaces on a microscale.
