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Thursday, 22 October 2020

Oxford Mathematics Online Public Lecture: David Sumpter - How Learning Ten Equations Can Improve Your Life

Oxford Mathematics Online Public Lecture
David Sumpter: How Learning Ten Equations Can Improve Your Life

Wednesday 28 October 2020
5.00-6.00pm

Is there a secret formula for becoming rich? Or for happiness? Or for becoming popular? Or for self-confidence and good judgement? David Sumpter answer these questions with an emphatic ‘Yes!' All YOU need are The Ten Equations.

In this lecture David will reveal three of these: the confidence equation that helps gamblers know when they have a winning strategy; the influencer equation that shapes our social interactions; and the learning equation that YouTube used to get us addicted to their videos. A small group of mathematicians have used these equations to revolutionise our world. Now you can use them too to better manage your time and make money, have a more balanced approach to your popularity and even to become a nicer person.

To order the book 'The Ten Equations That Rule the World' signed by David Sumpter from Blackwell's Bookshop, email oxford@Blackwells.co.uk by 15 November and they will provide you with all the information you need.

David Sumpter is Professor of Applied Mathematics at the University of Uppsala, Sweden.

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The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Monday, 19 October 2020

Heather Harrington wins Leverhulme Prize

Oxford Mathematician Heather Harrington has won one of this year's prestigious Philip Leverhulme Prizes. The award recognises the achievement of outstanding researchers whose work has already attracted international recognition and whose future career is exceptionally promising.

Heather certainly fits that bill. She has already won the Whitehead and Adams prizes for her work which covers a range of topics in applied mathematics, including algebraic systems biology, inverse problems, computational biology, and information processing in biological and chemical systems. Heather is the Co-Director of the Centre for Topological Data Analysis in Oxford.

Heather said of the award: I'm really humbled and honoured to have received this prize. My research is only possible through extensive collaborative networks, and I'm very grateful to my collaborators. I am hoping the prize funds can go towards exploring new research ideas as well as supporting students interested in research careers at the interface between pure and applied mathematics.

Each of the 30 Prize Winners receives £100,000 which can be used over two or three years to advance their research.

Thursday, 15 October 2020

Why Study Mathematics? and The Wonder Book of Geometry - two new books from Oxford Mathematicians

Vicky Neale and David Acheson are two of Oxford Mathematics's most engaging and accessible speakers and writers, and they both have new books to prove it.

In Why Study Mathematics?  Vicky describes the experience of studying mathematics at University as well as discussing the many benefits of a mathematics degree. Nick Higham, Royal Society Research Professor, University of Manchester, says, "I recommend it to all prospective maths students and their parents."

In The Wonder Book of Geometry, this week's New Scientist 'Don't Miss' Book of the Week, David take us on a geometrical tour from Ancient Greece to the present day. Packed with illustrations, the book demonstrates why geometry is the spirit of mathematics.

Both books have a simple message: mathematics is dazzling; and it is also invaluable. 

Tuesday, 13 October 2020

Modelling Removal of Sulphur Dioxide from Flue Gas

Oxford Mathematician Kristian Kiradjiev talks about his DPhil research, supervised by Chris Breward and Ian Griffiths in collaboration with W. L. Gore and Associates, Inc., on modelling filtration devices for removal of sulphur dioxide from flue gas.

"In the drive to protect the environment, reducing the concentrations of harmful chemicals that are released into the atmosphere has become a priority for industry. One key example is the removal of sulphur dioxide from flue (exhaust) gas. In order to remove sulphur dioxide from a stream of flue gas, a purification device, designed by W. L. Gore and Associates, Inc. (famous for GORE-Tex), which contains special thin porous sheets can be used (see figure 1). Gas containing sulphur dioxide is passed down the outside of the sheets and enters the pores by diffusion. Inside the pores, there are microscopic pellets held together by a network of fibres where a chemical reaction turns gaseous sulphur dioxide into liquid sulphuric acid. The accumulation of liquid limits the transport of sulphur dioxide to the reaction surface resulting in a loss of removal efficiency (see figure 2). My DPhil project, in collaboration with Gore, aimed to model the operation of their device and understand the effect of key material and operational parameters on the creation, motion, and accumulation of the liquid sulphuric acid within the porous medium, in order to inform Gore on an optimal filter design.


Figure 1: Three modules of Gore's device

 


Figure 2. Schematic diagram of the operating mechanism of the purification device

We started by exploring the problem on the microscale around a single catalytic pellet. We analysed how liquid is generated around it and how it spreads along fibres that are connected to the pellet. We then derived a model that upscales the local complex behaviour of the microstructure of the filter sheets, using the theory of homogenisation, to a device-scale model that describes how the liquid sulphuric acid and the gaseous sulphur dioxide are transported within the filter [1].

To give you a flavour of the kind of results we obtained, in the plot below (figure 3), we show a temporal profile of the sulphur dioxide concentration at the outlet of the filter channels, which is normally one of the most important quantities regulated by the plant, for different gas speeds. We see that, increasing the gas speed increases the concentration of sulphur dioxide, since there is less residence time for the gas in the filter. Thus, if the concentration should not exceed a given threshold, we can calculate the maximum speed the gas can be flowed at to achieve this goal. 


Figure 3. Temporal profile of the concentration of sulphur dioxide at the outlet of the filter channels for different gas speeds, where both time and outlet concentration are normalised.

Sulphuric acid is hygroscopic, which means it has a high affinity to water, and so we also developed a separate model that describes this behaviour [3]. The results were validated through a set of experiments performed at Gore.

The models we derived provide the basis for exploration of the performance of these so-called reactive filters and for optimisation to minimise the amount of sulphur dioxide released by the device into the atmosphere, while ensuring longevity. The mathematical framework we developed is widely applicable to other industrial processes and catalytic systems, such as carbon capture and storage, purification in fuel cells, and removal of nitric oxides, where similar types of reaction take place."

[1] K. B. Kiradjiev, C. J. W. Breward, I. M. Griffiths, D. W. Schwendeman. A Homogenised Model for a Reactive Filter. SIAM J. Appl. Math., 2020 (in print).

Tuesday, 6 October 2020

Roger Penrose joint winner of the 2020 Nobel Prize for Physics

The 2020 Nobel Prize for Physics has been awarded to Roger Penrose, Reinhard Genzel and Andrea Ghez for their work on black holes. Oxford Mathematician Penrose is cited “for the discovery that black hole formation is a robust prediction of the general theory of relativity.”

Mike Giles, Head of the Mathematical Institute in Oxford, said "We are absolutely delighted for Roger - it is a wonderful recognition of his ground-breaking contributions to mathematical physics."

Roger himself said: "It is a huge honour to receive this Prize. 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."

Sir Roger Penrose is famous for his many contributions to the mathematical physics of general relativity and cosmology. In 1965 with his ground-breaking paper "Gravitational Collapse and Space-Time Singularities" he predicted the existence of black holes, astronomical objects so dense that the geometry of space-time becomes singular inside them and not even light can escape their gravitational attraction. This remains, to this day, one of the most astonishing consequences of Einstein's theory of General Relativity, and we now see that they do exist in nature.

Roger also pioneered the development of the mathematical theory that describes the structure of space-time and, together with Stephen Hawking, he developed singularity theorems that form the basis of the modern theory of black holes.  In parallel, he developed twistor theory as an approach to the quantization of space-time and gravity. It has since become a powerful tool across mathematics and has more recently impacted on physics in the form of 'twistor-string theory' as a tool for calculating scattering amplitudes for collider experiments. It is still actively pursued as an approach to quantum gravity.

He has made many other scientific contributions that, despite their recreational origin, have nevertheless had Nobel prize winning impact.  His quasi-periodic tilings have a crystallographically forbidden 5-fold symmetry. These have not only inspired much mathematical research, but were subsequently discovered by Schechtman in 1984 to be realised in quasi-crystals that can be made in the laboratory. Roger Penrose, together with his father, was the originator of Escher's famous and iconic impossible pictures. His theory of spin networks in his Adam's prize essay has become one of the pillars of 'loop quantum gravity' and now has a worldwide following. Amongst his most cited papers is the theory of generalised inverses of matrices that have applications from statistics through to engineering.

Another particularly influential strand has been his work on the foundations of quantum mechanics, both on realistic models of wave function collapse, and on time asymmetry therein and its relation to that in thermodynamics and in the big bang versus gravitational collapse. His early work in the ‘70s and ‘80s laid the foundations of what is now a worldwide endeavour.

Last, but not least, his books on popular science have provided a benchmark for how to engage with the layperson without trivialising the science.

Roger Penrose is Emeritus Rouse Ball Professor of Mathematics and a fellow of Wadham College in Oxford.

In 2018 Roger Penrose gave an Oxford Mathematics Public Lecture where he outlined his latest thinking on Cosmology and in an interview with Hannah Fry talked about his career and how he wasn't always so far ahead of the game, especially when it came to arithmetic

 

Wednesday, 30 September 2020

Oxford Mathematics Online Public Lecture: Tim Harford - How to Make the World Add up. Thursday 8 October 5-6pm

Oxford Mathematics Online Public Lecture:Tim Harford - How to Make the World Add up

Thursday 8 October 2020
5.00-6.00pm

When was the last time you read a grand statement, accompanied by a large number, and wondered whether it could really be true?

Statistics are vital in helping us tell stories – we see them in the papers, on social media, and we hear them used in everyday conversation – and yet we doubt them more than ever. But numbers, in the right hands, have the power to change the world for the better. Contrary to popular belief, good statistics are not a trick, although they are a kind of magic. Good statistics are like a telescope for an astronomer, or a microscope for a bacteriologist. If we are willing to let them, good statistics help us see things about the world around us and about ourselves.

Tim Harford is a senior columnist for the Financial Times, the presenter of Radio 4’s More or Less and is a visiting fellow at Nuffield College, Oxford. His books include The Fifty Things that Made the Modern Economy, Messy, and The Undercover Economist.

To order a personalised copy of Tim's book email oxford@blackwells.co.uk, providing your name and contact phone number/email and the personalisation you would like. You can then pick up from 16/10 or contact Blackwell's on 01865 792792 from that date to pay and have it sent.

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The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Friday, 25 September 2020

Lifting the Newfoundland Travel Ban - a Story of Mathematics, Coronavirus and the Law

When Oxford Mathematician Alain Goriely was approached by his collaborator Ellen Kuhl from Stanford University to work on a travel restriction issue in Newfoundland he started a Coronavirus journey that ended up in the Canadian Supreme Court.

"The island of Newfoundland is part of the Canadian province of Newfoundland and Labrador. Following a travel ban on May 5, 2020, this Atlantic province enjoyed the rather exceptional and enviable position of having the Coronavirus pandemic under control. By July 3, 2020, it had a cumulative number of 261 cases, with 258 recovered, 3 deaths, and no new cases for 36 days. The same day, the Atlantic Bubble opened to allow air travel between the four Atlantic Provinces - Newfoundland and Labrador, Nova Scotia, New Brunswick, and Prince Edward Island - with no quarantine requirements for travellers. With respect to COVID, the inhabitants of the province are in a dangerous position as they have the highest rates of obesity, metabolic disease, and cancer nationally, and an unhealthy lifestyle with the highest rate of cigarette smoking among all provinces. Despite its success in eliminating the virus, the government found itself in a precarious position. Its travel ban, Bill 38, was being challenged by a Halifax resident who was denied entry for her mother’s funeral in the Spring and the lawsuit was further supported by the Canadian Civil Liberties Association. They were seeking a declaration from the provincial Supreme Court in St John’s that the travel ban was unconstitutional, a decision that could apply to the entire country. Determined to keep control of its borders, the Office of the Attorney-General reached out to Ellen. Would her models be applicable to this situation? What would happen during gradual or full reopening under perfect or imperfect quarantine conditions?

Ellen and I had been talking about a hypothetical problem like this one. If the virus is eliminated from a region, can it come back, like a boomerang, when restrictions are eased? Newfoundland seemed to be the perfect case study for us, and with the help of her outstanding Postdoc, Kevin Linka and Dr Proton Rahman, a clinical epidemiologist and Professor of Medicine at Memorial University of Newfoundland, we jumped at the opportunity to test some of our ideas. Soon, we converged on a network model where each node represents a US state or a Canadian province. On each node, we ran a local Suscetible-Exposed-Infected-Recovered epidemiological model and modelled air traffic by a graph Laplacian-type transport process as commonly done for network transport. Parameters were estimated by Bayesian inference with Markov-chain Monte Carlo sampling using a Student’s t-distribution for the likelihood between the reported cumulative case numbers and the simulated cumulative case numbers.

Conceptually, the model is quite simple. I have a natural preference for parsimony when it comes to modelling complex phenomena as the assumptions are completely known and in full display. This is a personal choice and the outcomes of such models should be seen as estimates rather than a hardcore forecast. What we found is quite interesting. Using air traffic information from the previous 15 months, we showed that opening Newfoundland to the Atlantic provinces or the rest of Canada would have negligible effects on the evolution of the disease as prevalence dropped considerably in Canada. Yet, opening the airports to the USA would lead to 2-5 infected passengers entering the island a week, with as many as 1-2 asymptomatic travellers. Without an air-tight quarantine system, the disease would reach 0.1% of the Newfoundland population within 1 to 2 months.

In the first week of August, evidence were presented to the court. The Chief Medical Officer of Health Dr. Janice Fitzgerald opened with the following quote: “In 1775 the American revolutionary Patrick Henry declared, ’Give me liberty or give me death.’ In this case, if the applicants’ remedy is granted, it will result in both.” The same week Proton testified in court about our model, its assumptions, and our findings. To my surprise, the scientists were heard and on 17 September, the judge rendered his verdict. In his ruling, Justice Burrage declared that ‘‘The upshot of the modelling ... is that the travel restriction is an effective measure at reducing the spread of COVID19 in Newfoundland and Labrador.” He concluded that yes, the ban was legal and justified. Having an impact on the lives of Newfoundlanders, however small, is a strange but rather pleasant feeling."

Read more about the case here.

Figures above:

1. Mobility modelling. Discrete graphs of the Atlantic Provinces of Canada and of North America with 4, 13, and 64 nodes that represent the main travel routes to Newfoundland and Labrador. Dark blue edges represent the connections from the Atlantic Provinces, light blue edges from the other Canadian provinces and territories, and red edges from the United States.

2. Estimated COVID-19 infectious travellers to Newfoundland and Labrador. Number of daily incoming air passengers from the Canadian provinces and territories and the United States that are infectious with COVID-19.

Thursday, 17 September 2020

Re-examining geometry: p-adic numbers and perfectoid spaces

A tower of modular curves

Oxford Mathematician Daniel Gulotta talks about his work on $p$-adic geometry and the Langlands program.

"Geometry is one of the more visceral areas of mathematics. Concepts like distance and curvature are things that we can actually see and feel.

Mathematicians like to question things that seem obvious. For example, what if distance did not work the way that we are used to? Normally, we would say that a fraction like $\frac{1}{65536}$ is very close to zero because the denominator is much larger than the numerator, and conversely $65536 = \frac{65536}{1}$ is very far from zero because the numerator is much larger than the denominator. In $p$-adic geometry, we instead choose a prime number $p$, and we say that a fraction $\frac{a}{b}$ is close to zero if $a$ is divisible by a large power of $p$. So for $p=2$, $65536 = \frac{2^{16}}{1}$ is actually $2$-adically very close to zero and $\frac{1}{65536} = \frac{1}{2^{16}}$ is $2$-adically very far from zero. It can be shown that the only 'well-behaved' notions of distance on the rational numbers are the usual one and the $p$-adic ones.

It is difficult to directly relate our intuition about the real world to this $p$-adic notion of distances. One of the challenges of $p$-adic geometry is to find ways of framing familiar concepts so that they still make sense in the $p$-adic world.

A common technique in geometry is to study a complicated space by chopping it into smaller, simpler pieces and studying how those pieces fit together. For example, a complicated surface could be divided into discs.

A surprising fact about $p$-adic geometry is that the $p$-adic disc is still a very complicated space - it loops back on itself in all sorts of ways. It turns out that the simpler spaces are actually these very large objects called perfectoid spaces. Whereas one can specify a point on the $p$-adic disc with a single $p$-adic coordinate (similar to how a point on the usual disc can be specified by a single complex number), a point on a perfectoid space might be specified by an infinite sequence of coordinates $(x_0,x_1,x_2,\dotsc)$ satisfying $x_i = x_{i+1}^p$ for all $i \ge 0$. So to understand $p$-adic spaces, in addition to cutting them into pieces, we also want to cover those pieces by perfectoid spaces.

I am particularly interested in applications of $p$-adic geometry to the Langlands program, which explores connections between number theory, geometry, and representation theory.  One way that geometry enters into the Langlands program is through Shimura varieties. These are geometric spaces whose symmetries have a particularly nice arithmetic description.  (Modular curves, a particular kind of Shimura variety, play a significant role in Andrew Wiles's famous proof of Fermat's last theorem.)

Along with Ana Caraiani and Christian Johansson, I have been studying the $p$-adic geometry of Shimura varieties. In particular, we have proved vanishing of certain compactly supported cohomology groups.  In plain terms, we have shown that these Shimura varieties do not have any loops of high dimensions.  This result has allowed us to improve on Scholze's results on the p-adic Langlands correspondence. The proof involves using techniques from geometric representation theory, namely Weyl groups and the Bruhat stratification, to divide the Shimura variety into pieces and then cover those pieces with perfectoid spaces."

For more on Daniel's work:

Wednesday, 16 September 2020

Frontiers of secrecy - the story of Eve, Alice and Bob

Oxford Mathematician Artur Ekert describes how his research in to using Quantum properties for cryptography led to some very strange results.

"The most secure methods of communication rely on pre-distributed, random and secret sequences of bits, known as cryptographic keys. Any two parties who share the key, we call them Alice and Bob (not their real names, of course), can then use it to communicate secretly. The key bits must be truly random, never reused, and securely delivered to Alice and Bob, which is by no means easy for Alice and Bob may be miles apart. Still, it can be done. About thirty years ago my work triggered an active field of research by showing that quantum entanglement and peculiar non-local quantum correlations can be used for secure key distribution. More recently, building upon this work, cryptologists started probing the ultimate limits of security and showed that what looks like an insane scenario is actually possible - devices of unknown or dubious provenance, even those that are manufactured by our enemies, can be safely used to generate secure keys.

In this more dramatic version, known as the device independent scenario, an omniscient adversary, called Eve, is in charge of the key distribution. She prepares two sealed and impregnable devices, and gives one to Alice and one to Bob. The inner working of the devices is unknown to Alice and Bob but they can take them to their respective locations and probe them with randomly and independently chosen binary inputs to which the devices respond with binary outputs. For Alice's and Bob's inputs and y their devices generate outputs a and b, respectively. If in the repetitive use the responses of the two devices show a certain pattern then the output bits can be turned into a secret key.

At first this narrative makes no sense. Surely, if Eve manufactured the two devices she must also have pre-programmed them, and hence she would know how they respond to all possible inputs, which, of course, renders the resulting keys insecure. Surprisingly enough, if Alice and Bob prepare their inputs freely, so that Eve does not know in advance which inputs will be chosen in each run, and the devices are kept separated and incommunicado, then some patterns of outputs cannot be pre-programmed. For example, requesting that in each run the outputs are identical for all inputs except when Alice prepares input 1 and Bob prepares input 1, in which case the outputs must be different, that is \[ x y=a\oplus b, \] defeats the coding prowess of any, no matter how powerful, Eve. The best Eve can do in this case is to have this request satisfied in 75% runs of the devices. Anything more than 75% indicates outputs that are not pre-programmed or pre-determined and hence unknown to any third party, Eve included. No classical devices can deliver such a correlated performance but quantum devices can, pushing the success rate to roughly 86%. Thus, if Eve is prepared to offer Alice and Bob quantum devices that beat the 75% classical limit she has to concede that at least some of the outputs bits will be unknown to her. Alice and Bob infer how much Eve knows about the output bits from the success rate and then, if Eve does not know too much, they can use conventional cryptographic tools to turn the partially secret bits into fewer secret ones, which will then form a cryptographic key.

The key distribution proceeds as follows. Alice and Bob run their devices choosing their inputs randomly, and independently from each other. In each run, once the outputs are registered, Alice and Bob communicate in public, with Eve listening, and reveal their inputs, but not the outputs. The repetitive use of the devices results in two binary output strings, one held by Alice and one by Bob. In a subsequent public communication, again with Eve listening, Alice and Bob agree on a random sample of the recorded runs for which they reveal the outputs. This allows them to estimate the success rate on the data in the fully disclosed runs. If the estimated success rate is below 75% the key distribution is abandoned, otherwise Alice and Bob turn the remaining undisclosed outputs into a secret key.

Needless to say, proving security under such weak assumptions, with all the nuts and bolts, is considerably more challenging than in the case of trusted devices. Between the two extremes - at the success rate of 75% Eve may know everything about the key and at 86% Eve knows nothing - lies an important uncharted twilight zone. Suppose the success rate is 80% then what? We must put an upper bound on Eve's knowledge for any intermediate success rate, as this determines how Alice and Bob turn their raw output strings into a shorter but secret key. This is not easy.

The important technical figure of merit to be evaluated here is called (just in case you want to impress your friends with the mathematical lingo) "the conditional smooth min-entropy", and it should be expressed as a function of the success rate. This quantity determines the length of the final key that can be distilled from a given raw output, but deriving it required considerable mathematical gymnastics. The early results provided an explicit expression for the asymptotic key rate (the ratio between the distillable key length and the length of raw output in the limit of a large number of runs), but the analysis applied only to cases where the devices behaved in the same way in each run, unaffected by the previous or subsequent runs. This is known as the independent and identically distributed (i.i.d.) assumption. At the time even the most fervent advocates of the device independent cryptography had to admit that the result, as neat as it was, had no direct bearing on the device independent scenario, for Eve can manufacture the devices as she sees fit, making successive outputs dependent on what happened in all the previous runs. More recently, the most general case has been finally worked out. It turns out that a realistic device independent scenario can be reduced to the i.i.d. case, showing that Eve cannot do better than making each run of the devices independent from and statistically identical to all other runs. This is of great comfort to both mathematicians and experimentalists for it shows that reasonable key rates can be achieved even in noisy implementations.

One should perhaps mention that the device independent scenario does not have to involve untrusted devices. The protocol is more likely to offer an additional security layer to quantum cryptography with trusted devices, protecting against attacks that exploit unintentional flaws in the design. What is crucial here, however, is the assumption that Alice and Bob prepare their respective inputs randomly and independently from each other. If Alice's and Bob's choices were known in advance then Eve can easily pre-program the results and Alice and Bob would foolishly believe that they generated a secret key. However, as long as Alice's and Bob's choices are their own, unknown and unpredictable, then the most mind-boggling cryptographic scheme ever proposed works just fine and can see the light of the day sooner than many of us expected."

 

 

Monday, 14 September 2020

Local Topology for Anomaly Detection in Data

Oxford Mathematician Vidit Nanda discusses his recent work with colleagues Bernadette Stolz, Jared Tanner and Heather Harrington on detecting singularities in data.

"Fitting geometric models to high-dimensional point clouds plays an essential role in all sorts of tools in contemporary data analysis, from linear regression to deep neural networks. By far the most common and well-studied geometric models are manifolds. For instance, the plane, sphere and torus illustrated below are all two-dimensional manifolds that can be embedded in three-dimensional Euclidean space.

                                            

There is a local test which characterizes d-dimensional manifolds: around each point, can you find a small region which resembles standard d-dimensional Euclidean space? If the answer is yes, then you have a d-manifold on your hands. Thus, if we were to zoom in with a very high-powered microscope at any point on the sphere or torus, we would approximately see a plane. 

When confronting heterogeneous data which comes from several different sources or measurements, it may no longer be feasible to expect that a single underlying manifold will provide a good fit to all the data points. This is because the union of two d-dimensional manifolds (such as a sphere and a plane) along a shared sub-manifold (such as an equator) will not itself be a d-manifold. If you examine small neighbourhoods around points in the common sub-manifold, you can see that they will fail the local test for manifold-ness. For instance, if you zoom in on any point lying in the equator of the figure below, you will see two planes that intersect along a common line rather than a single plane.

                                                                    

Such non-manifold spaces which are built out of manifold pieces are called stratified spaces, and their non-manifold regions, such as the equator in the example above, are called singularities. The study of stratified spaces has been a long and fruitful enterprise across several disciplines in pure mathematics, including algebraic geometry, algebraic topology and representation theory. 

We have recently developed a framework to automatically detect singularities directly from data points even when none of the data points lie exactly on the singular regions. One key advantage is that we are now able to partition a heterogeneous dataset into separate clusters based on their intrinsic dimensionality. Thus, a dataset living on the plane-plus-sphere described above would be decomposed into five clusters, one of which lives near the one-dimensional singular equator, while the other four lie on various parts of the sphere or the plane. 

                                             

All five pieces are manifolds, so the standard manifold-fitting techniques which are pervasive in data science can be safely applied to them individually.

The key technique in this singularity-detection framework is persistent cohomology, which assigns a family of combinatorial invariants called barcodes, one for each dimension, to a collection of data points. If these points have been sampled densely from a d-dimensional sphere, then there is a prominent bar in the d-th persistent cohomology barcode and not much else.  Given each point p in the dataset, one examines the set of all annular neighbours - this consists of all data points q satisfying α < dist(p,q) < β for some small positive distances α and β, where dist denotes Euclidean distance. By the local manifold property, the central point p lies on a d-dimensional manifold, then for suitably small α < β the set of all annular neighbours will live approximately on a (d-1)-dimensional sphere, which can be detected accurately by the presence of a single dominant bar in the (d-1)-st persistent cohomology barcode. 

We tried this technique on a dataset whose points correspond to configurations of a molecule called cyclo-octane. The data consists of 5000 points in 24-dimensional space, and the points live on the intersection of two embedded surfaces along two circles. As expected, points lying near the two singular circles are easily identified by their local persistent cohomology barcodes. These special points are coloured red in the 2-dimensional projection of the data below."

                                                                       

 

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