I had the honour to sit down with Sir Michael Atiyah to discuss his recently presented proof of the Riemann Hypothesis at the Heidelberg Laureate Forum.
The Norwegian Academy of Science and Letters kindly provided me with a scholarship to attend the Abel Prize week in Oslo earlier this year where I interviewed the 2018 Abel Laureate Robert Langlands.
In the first of a series of videos documenting my experience, Robert describes how he came to do Mathematics at university…
Incredibly excited to announce that I’ve won an award courtesy of OxTALENT at the University of Oxford in the category of Outreach and Widening Participation. Thanks to you all for the support and here’s to many more exciting times for Tom Rocks Maths!
Outreach and Widening Participation
Outreach and widening participation activities deliver an important dimension of the University’s work in raising aspirations, promoting diversity and encouraging people from non-traditional backgrounds to enter higher education. This category awards staff and students who have made innovative use of technology to deliver exceptional widening participation activities and to support learners from disadvantaged backgrounds.
Tom Crawford (St Hugh’s College) for ‘Tom Rocks Maths’
Mathematician Thomas Hales explains the Honeycomb Conjecture in the context of bees. Hales proved that the hexagon tiling (hexagonal honeycomb) is the most efficient way to maximise area whilst minimising perimeter.
Produced by Tom Rocks Maths intern Joe Double, with assistance from Tom Crawford. Thanks to the Oxford University Society East Kent Branch for funding the placement and to the Isaac Newton Institute for arranging the interview.
We are now just over a week into the new year, how are those resolutions going? If like me you felt bad for (at least) a week’s worth of pigging out over the holiday season, then maybe you’ve been on a detox? They are a quick easy way to get healthy right? Yeah about that…
Most people think of a detox as some form of cleanse that removes various toxins from the body, usually after a period of excessive eating and drinking. While not completely incorrect, the use, or misuse, of the word nowadays is the result of very clever marketing. The correct definition of a detox is a medical procedure that removes dangerous and often life-threatening levels of drugs, alcohol and poisons from the body carried out by a trained medical professional in a hospital or clinic. Not quite the same as drinking some carrot juice then. What we’re doing is more of a ‘cleanse’.
The idea of cleansing the body isn’t new, but the way we do it has changed dramatically. Go back 100 years and we were using therapeutic vomiting, blood-letting and a process known as ‘smuding’ where smoke from burning sage is waved around the energy field of a person to destroy negative energy. These days it’s all intestinal cleaning, foot sponges which supposedly draw out toxins and coffee enemas (yes you read that correctly). The question is does detoxing (or cleansing) really do anything?
Let’s take a simple example: a master cleanse diet favoured by a number of Hollywood celebrities. Begin the day with a litre of warm salt water, consume 2 litres of a concoction of water, lemon juice, maple syrup and cayenne pepper throughout the day and finish with 250ml of laxative in the evening. Do this for 10 days and you will gain energy, lose weight and relieve symptoms of chronic conditions such as arthritis. Except you probably won’t. There are no data on this diet in the medical literature and similar studies on fasts and extremely low-calorie diets actually result in rapid weight gain following their completion. You will of course lose weight during the diet as you are only consuming 600 calories a day, but this is mostly due to fluid loss. Furthermore, the diet is lacking in protein, fatty acids and other essential nutrients, with the daily laxative regime likely to cause dehydration. Just this week doctors in the UK issued a health warning about the potential harms of undertaking a radical new year detox, highlighting the example of a 47-year-old woman admitted to intensive care following a detox diet of herbal remedies and water, that left her with dangerously low levels of salt in her body.
A search of medical literature using the words ‘detox’ and ‘clinical trial’ returns nothing. Quite simply there is no credible evidence to demonstrate that detox kits do anything at all. Perhaps most importantly they have not been shown to offer any of the supposed health benefits claimed by manufacturers and promoters. This may seem a little doom and gloom for the new year, but don’t worry, your body has got you covered. The human body is the best detoxing solution available and here’s how it works.
Let’s pretend I’m one of these mysterious ‘toxins’ trying to enter your body, cleverly concealed in your favourite alcoholic beverage. First up, I pass through the stomach and into the intestines, where I am confronted with lymph nodes called Peyer’s patches. These guys screen out parasites and other foreign substances before they are absorbed into the blood along with nutrients. Well, what if I disguise myself as a nutrient? Now I’m in the blood and ready to do some damage. But what’s this? An army of cells and molecules sent by the immune system are here to fight me off. They are specifically designed to recognise foreign substances and eliminate them from the body, making me target number one. Okay, well, suppose I somehow survive the onslaught from the army of cells, things must surely start to look up? Afraid not. The blood is passed through the liver where proteins called metallothioneins act to neutralise harmful metals and enzymes process drugs. The job of the liver is basically to break down anything that can cause harm to your body, which is bad news for me as a ‘toxin’. If by some miracle there is anything left of me after the liver has worked its magic then I will enter into the kidneys. These are the body’s natural filtration system and remove any waste substances that remain. I think it’s safe to say I’m done for…
If after all of that you still think the so-called ‘toxins’ have a chance, then by all means please do try that coffee enema, but if it were up to me I’d trust my body. The best new year detox plan is simply to concentrate on giving your body what it needs to do its job. That means a healthy diet, regular exercise and sufficient sleep. It might be less exciting than that Colon Detox Pro yoga session you had planned, but it also might actually work…
You can listen to the full interview for the Naked Scientists here.
The author H. P. Lovecraft often described his fictional alien worlds as having ‘Non-Euclidean Geometry’, but what exactly is this? And would it really break our brains?
Produced by Tom Rocks Maths intern Joe Double, with assistance from Tom Crawford. Thanks to the Oxford University Society East Kent Branch for funding the placement.
Esther Lafferty meets Dr Tom Crawford in the surprisingly large and leafy grounds of St Hugh’s College Oxford as the leaves begin to fall from the trees. It’s a far cry from the northern town of Warrington where he grew up.
Tom is a lecturer in maths at St Hugh’s, where, defying all ‘mathematics lecturer’ stereotypes with his football fanaticism, piercings, tattoos, and wannabe rock musician attitude, he makes maths understandable, relevant and fun.
‘It was always maths that kept me captivated,’ he explains, ‘ever since I was seven or eight. I remember clearly a moment in school where we’d been taught long multiplication and set a series of questions in the textbook: I did them all and then kept going right to the end of the book because I was enjoying it so much! It was a bit of a surprise to my teacher because I could be naughty in class during other subjects, messing around once I’d finished whatever task we’d been set, but I’ve loved numbers for as long as I can remember and I still find the same satisfaction in them now. There’s such a clarity with numbers – there’s a right or else it’s wrong. In English or History you can write an essay packed with opinion and interpretation and however fascinating it might be, there are lots of grey areas, whereas maths is very black and white. I like that.’
‘My parents both left school at sixteen for various reasons but they appreciated the value of education. My mum worked in a bank so she perhaps had an underlying interest in numbers but it wasn’t something I was aware of. I went to the local school and was lucky enough to be one of the clever children but it wasn’t until I got my GCSE results [10 A*s] that the idea of Oxford or Cambridge was suggested to me. I would never have thought to consider it otherwise.
‘I remember coming down for an interview in Oxford, at St John’s, arriving late on a Sunday night and the following morning I took a stroll around the college grounds – I could feel the history and traditions in the old buildings and it was awesome. I really wanted to be part of everything it represented. I thought it would be so cool to study here so I was very excited when I was offered a place to read maths.
‘Studying in Oxford I found I was most interested in applied maths, the maths that underpins physics and engineering for example. ‘Pure’ maths can be very abstract whereas I prefer to be able to visualise the problems I am trying to solve and then when you work out the answer, there’s a sudden feeling when you just know it’s right.’
In his second year, Tom became interested in outreach work, volunteering to take the excitement of maths into secondary schools under the tutelage of Prof Marcus Du Sautoy OBE as one of Marcus’s Marvellous Mathematicians (or M3), a group who work to increase the public understanding of science.
‘I went to China one summer to teach sixth formers and it was great to have the freedom to talk about so many different topics. I spent another summer in an actuary’s office because I was told that was the way to make real money out of maths – it was a starkly different experience. I realised I was not at all cut out for a suit and a screen!’ Tom smiles. ‘I am a real people-person and get a real buzz from showing everyone and anyone that you can enjoy maths, and that it is interesting and relevant. I love the subject so much and I think numbers get a bad press for being dull and difficult and yet they underpin pretty much everything in the whole universe. They can explain almost everything and you’ll find maths in topics from the weather to the dinosaurs.
Take something like the circus for example – hula-hoops spinning and circles in the ring, and then the trapeze is all about trigonometry: the lengths and angles of the triangle. Those sequinned trapeze artists are working out the distances and directions they need to leap as they traverse between trapezes and its maths that stops them plummeting to the floor!’
Having spent four years in Oxford Tom then spent five years at Cambridge University looking at the flow of river water when it enters the sea, researching the fluid dynamics of air, ice and water, and conducting fieldwork in the Antarctic confined to a boat for six weeks taking various measurements in sub-zero temperatures. You’d never expect a mathematician to be storm-chasing force 11 gales in a furry-hooded parka, but to get the data needed to help to improve our predictions of climate change, that was what had to be done!
Tom also spent a year as part of a production group known as the Naked Scientists, a team of scientists, doctors and communicators whose passion is to help the general public to understand and engage with the worlds of science, technology and medicine. The skills he obtained allowed him to kick-start his own maths communication programme Tom Rocks Maths, where he brings his own enthusiasm and inspiring ideas to a new generation alongside his lectureship in maths at St Hugh’s.
A keen footballer (and a massive Manchester United fan) it’s no surprise Tom has turned his thoughts to football and as part of IF Oxford, the science and ideas festival taking over Oxford city centre in October, Tom is presenting a free interactive talk (recommend for age twelve and over) on Maths versus Sport – covering how do you take the perfect penalty kick? What is the limit of human endurance – can we predict the fastest marathon time that will ever be achieved? And over a 2km race in a rowing eight, does the rotation of the earth really make a difference? Expect to be surprised by the answers.
Esther Lafferty, OX Magazine
The original article can be found here.
The second episode of season 2 of Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station. Featuring the numbers behind the sub 2-hour marathon world record attempt, P versus NP and the battle for control of the world, and the usual dose of Funbers with my super sweet 16. Plus, music from Blink 182, Billy Talent and Hollywood Undead. This is maths, but not as you know it…
Christmas stamps are sold with the following values 16p, 17p, 23p, 24p, 39p and 40p. You want to send a present which has a postage cost of £1.00. How many stamps do you need to buy to make the exact amount?
The year is 1888, and the infamous serial killer Jack the Ripper is haunting the streets of Whitechapel. As a detective in Victorian London, your mission is to track down this notorious criminal – but you have a problem. The only information that you have to go on is the map below, which shows the locations of crimes attributed to Jack. Based on this information alone, where on earth should you start looking?
The fact that Jack the Ripper was never caught suggests that the real Victorian detectives didn’t know the answer to this question any more than you do, and modern detectives are faced with the same problem when they are trying to track down serial offenders. Fortunately for us, there is a fascinating way in which we can apply maths to help us to catch these criminals – a technique known as geospatial profiling.
Geospatial profiling is the use of statistics to find patterns in the geographical locations of certain events. If we know the locations of the crimes committed by a serial offender, we can use geospatial profiling to work out their likely base location, or anchor point. This may be their home, place of work, or any other location of importance to them – meaning it’s a good place to start looking for clues!
Perhaps the simplest approach is to find the centre of minimum distance to the crime locations. That is, find the place which gives the overall shortest distance for the criminal to travel to commit their crimes. However, there are a couple of problems with this approach. Firstly, it doesn’t tend to consider criminal psychology and other important factors. For example, it might not be very sensible to assume that a criminal will commit crimes as close to home as they can! In fact, it is often the case that an offender will only commit crimes outside of a buffer zone around their base location. Secondly, this technique will provide us with a single point location, which is highly unlikely to exactly match the true anchor point. We would prefer to end up with a distribution of possible locations which we can use to identify the areas that have the highest probability of containing the anchor point, and are therefore the best places to search.
With this in mind, let’s call the anchor point of the criminal z. Our aim is then to find a probability distribution for z, which takes into account the locations of the crime scenes, so that we can work out where our criminal is most likely to be. In order to do this, we will need two things.
- A prior distribution for z. This is just a function which defines our best guess at what z might be, before we have used any of our information about the crime locations. The prior distribution is usually based off data from previous offenders whose location was successfully determined, but it’s usually not hugely important if we’re a bit wrong – this just gives us a place to start.
- A probability density function (PDF) for the locations of the crime sites. This is a function which describes how the criminal chooses the crime site, and therefore how the criminal is influenced by z. If we have a number of crimes committed at known locations, then the PDF describes the probability that a criminal with anchor point z commits crimes at these locations. Working out what we should choose for this is a little trickier…
We’ll see why we need these in a minute, but first, how do we choose our PDF? The answer is that it depends on the type of criminal, because different criminals behave in different ways. There are two main categories of offenders – resident offenders and non-resident offenders.
Resident offenders are those who commit crimes near to their anchor point, so their criminal region (the zone in which they commit crimes) and anchor region (a zone around their anchor point where they are often likely to be) largely overlap, as shown in the diagram:
If we think that we may have this type of criminal, then we can use the famous normal distribution for our density function. Because we’re working in two dimensions, it looks like a little hill, with the peak at the anchor point:
Alternatively, if we think the criminal has a buffer zone, meaning that they only commit crimes at least a certain distance from home, then we can adjust our distribution slightly to reflect this. In this case, we use something that looks like a hollowed-out hill, where the most likely region is in a ring around the centre as shown below:
The second type of offenders are non-resident offenders. They commit crimes relatively far from their anchor point, so that their criminal region and anchor region do not overlap, as shown in the diagram:
If we think that we have this type of criminal, then for our PDF we can pick something that looks a little like the normal distribution used above, but shifted away from the centre:
Now, the million-dollar question is which model should we pick? Determining between resident and non-resident offenders in advance is often difficult. Some information can be made deduced from the geography of the region, but often assumptions are made based on the crime itself – for example more complex/clever crimes have a higher likelihood of being committed by non-residents.
Once we’ve decided on our type of offender, selected the prior distribution (1) and the PDF (2), how do we actually use the model to help us to find our criminal? This is where the mathematical magic happens in the form of Bayesian statistics (named after statistician and philosopher Thomas Bayes).
Bayes’ theorem tells us that if we multiply together our prior distribution and our PDF, then we’ll end up with a new probability distribution for the anchor point z, which now takes into account the locations of the crime scenes! We call this the posterior distribution, and it tells us the most likely locations for the criminal’s anchor point given the locations of the crime scenes, and therefore the best places to begin our search.
This fascinating technique is actually used today by police detectives when trying to locate serial offenders. They implement the same steps described above using an extremely sophisticated computer algorithm called Rigel, which has a very high accuracy of correctly locating criminals.
So, what about Jack?
If we apply this geospatial profiling technique to the locations of the crimes attributed to Jack the Ripper, then we can predict that it is most likely that his base location was in a road called Flower and Deane Street. This is marked on the map below, along with the five crime locations used to work it out.
Unfortunately, we’re a little too late to know whether this prediction is accurate, because Flower and Deane street no longer exists, so any evidence is certainly long gone! However, if the detectives in Victorian London had known about geospatial profiling and the mathematics behind catching criminals, then it’s possible that the most infamous serial killer in British history might never have become quite so famous…
Growing a human heart from a single cell may seem like science fiction, but scientists at the Gladstone Institute at the University of California San Francisco, have taken a huge step forward, by producing the first three-dimensional, beating, human heart chamber. Previously, it had been possible to produce a two dimensional sheet of beating heart cells, but to really gain an understanding of heart formation in a developing foetus and perhaps more importantly, how drugs given to women during pregnancy may affect this development, a three dimensional structure was needed. By treating stem cells with drugs and then confining them to a very small spherical geometry, Bruce Conklin and his team have managed to grow their very own three dimensional model of a human heart, as he explains…
Bruce – the cells around the edge became fibroblasts – a particular type of cell that you use to heal wounds and then only in the very centre were cardiac cells that beat. What this is forming is more of a little organoid is what we call it, where there’s beating cells but there’s also multiple other cell types and that’s what makes it so interesting is that these cell types are somehow talking to each other and somehow collaborating in some way so that they can actually make this structure that we didn’t expect.
Tom – It’s almost like they’re trying to form a heart…
Bruce – That certainly is the impression. They’re heart cells, they’re forming cavities so it could be a model of how parts of human development occurs, but it certainly is not a real human heart in the sense that there’s probably many things that we’re missing. We just have a simplified version with just one chamber, but having it in a controlled way where it happens the same way over and over again we can start asking questions about ‘how do these cells talk to each other?’ So once you have a system which is reproducible you can do experiments to break it in some way or to enhance it in another way.
Tom – What are the applications of this work then?
Bruce – The most obvious application of the work is to study human development. How do cells actually form a heart is something of basic interest. And also, the most common form of birth defects is actually cardiac defects. But the other application is that we can expose these developing human micro-chambers to drugs which are thought to cause developmental defects, specifically of the heart, and in fact one of the key experiments in this study was to use the drug thalidomide which is notorious for causing birth defects. When we expose these cells to the thalidomide they had a dramatic change in the morphology so that you could see that it was altering the developmental process in this micro-chamber. Thalidomide was tested in rodents before it was tried in people and there were no cardiac defects in the rodents. I think that more and more we’re thinking how do we get tests which use real human cells so that we can actually make safer drugs. And in this case say you turn back the hands of time and you had this sort of test perhaps you would have discovered that thalidomide was dangerous before it had gone on to be given to people.
You can listen to the full interview with the Naked Scientists here.
War has always happened throughout human history and, chances are, it will continue to do so in the future. With this in mind, it’s important to ensure that if it does occur it’s carried out as humanely as possible, which is why treaties such as the Geneva Convention exists. Violating certain parts of this treaty, such as the use of chemical and biological weapons, for example, constitutes a war crime. With recent developments in artificial intelligence, a new version of the convention may be required. There have been two major revolutions in warfare so far: gunpowder and nuclear weapons, and the use of artificial intelligence is seen by many as the third such revolution. In an open letter to the United Nations, more than 100 leading robotics experts, including Elon Musk, Stephen Hawking, and the founder of Google’s Deepmind have called for a ban on the use of AI in managing weapons systems. I spoke to Peter Clark, founder of Resurgo Genetics and an expert in machine learning…
- The letter aims to trigger a debate about having international legislation for AI weapons systems, much in the same way that we have for nuclear or chemical weapons.
- Current drones require a pilot (even if thousands of miles away) and therefore still maintain an element of human morals and ethics, which means they are very different to a fully autonomous weapons system.
- One possible example of this technology could be a swarm of mini drones carrying small packets of explosives that could target individuals in a population.
- Techniques that are currently used to profile people’s online behaviour could be easily applied to such weapons systems to identify and eliminate people that opposed a particular ideology.
- The technologies being discussed are all available, and could be put together now into a system that could be catastrophic for the globe, which is why this letter is so important.
You can listen to the full interview for the Naked Scientists here.