The post Hunting the Snark originally appeared on the HLFF SciLogs blog.

]]>Imagine an infinite grid of squares – initially all empty (white). Some of these squares could be coloured in black, which we would call ‘alive’ cells, and the white cells can be thought of as ‘dead’. The way the game works is that at each step in time, we consider all the cells in their current state, and based on some rules we decide whether that cell will be alive, or dead, in the next time step.

The cells in the Game of Life have preferences fairly similar to real living organisms – they do not like to be lonely, so any cell which is only next to one other living cell, or no living cells, will die of loneliness. (‘Next to’ in this case includes horizontally, vertically or diagonally adjacent). They also prefer not to be overcrowded, so any cell which has four or more living neighbours will die out in the next iteration. And anything with exactly two or three neighbours stays alive to fight another day, or comes alive if it was not already.

Obviously, we could have picked any set of rules here – a slightly different definition of ‘neighbour’, or different thresholds for loneliness and overcrowding – but if your settings are not right, you might find it is very difficult to keep anything alive and that things all die out quickly, or that your board gets infested with a growing mass of black cells you cannot stop.

But with these particular rules, there is just enough balance for it to be interesting – some configurations die out, others stay alive forever (some examples above), and some behave in fun ways. For example, it is possible to make a glider, which iterates through a series of steps to create a version of itself one cell across from where it was before. Watching this loop repeat, the shape ‘glides’ across the grid and will keep going forever unless it is interrupted.

Mathematicians seek out interesting structures that you can build in this universe of rules. One configuration of interest is the idea of an oscillator – something which loops through a set of states and returns to its exact original configuration. A simple example might be a ‘blinker’ – something which flips between two possible states, the simplest of which is a straight line of three cells. This will oscillate between a vertical line and horizontal line, unless disturbed by any other living cells nearby.

It is also possible to construct oscillators of other periods – ones which take three, four and five steps to return to their original state are seen below.

**Left-right:** Caterer, Mazing and the Pseudo-Barberpole

This obviously leads to a further question: Which periods of oscillator can be created in this game? The people who study this question are probably aware that such discoveries are not going to shatter the foundations of mathematics, or necessarily lead to much else beyond the satisfaction of finding the answer – but whole websites exist of enthusiasts seeking out and excitedly studying and cataloguing examples of life they have found, like Charles Darwin on a voyage of discovery.

The story of periodic oscillators has an interesting history: Oscillators of small periods (up to about 15) were found within a few years of the invention of the game in 1970, as well as some interesting examples of higher periods – the twin bees shuttle, a period 46 oscillator, was found by Life expert Bill Gosper in 1971.

But the rest of the list took a bit longer to fill in. People continued to discover new oscillators over the years, but ticking off the list one at a time was unsatisfying. Surely we can find something that knocks out a bunch of cases in one go, like mathematical induction? Well, yes, we can. The existence of gliders in the game (structures which move in a straight line) means we can build more complex structures using a glider, and tweak them to get different periods.

A reflector is a structure in Life which is stable, but when a glider approaches it from the right direction, it will absorb it and create a new glider going off in a different direction. The first such structure was discovered by Paul Callahan in 1996, and over the next couple of decades many people discovered others, including ones which took fewer and fewer steps to complete the reflection, and using fewer initial live cells. The smallest known reflector is called the Snark, and was discovered in 2013 by Mike Playle – it reflects a glider through 90 degrees, and uses 52 cells (in a box of 23×17 cells).

This kind of structure can be used to build oscillators: Imagine pointing four of these at each other, so they pass a glider around between them. By tweaking the distance between the reflectors, and the types of reflectors used, we can build oscillators of any large period – and in fact this discovery meant that it is possible to construct an oscillator of any period greater than 59. The discovery of the Snark made this even smaller, showing that a non-trivial oscillator of any period 43 or above can be constructed.

This means, excitingly (for people who find this kind of thing sufficiently exciting), that we might be close to solving the problem in general, and finding an oscillator of every possible period. All we would need is to find oscillators of every period up to 43, and then we are covered. In July this year, one of the missing cases was fixed – a period 19 oscillator named Cribbage was found by Mitchell Riley.

This really put the pressure on Life-ologists to fill the one missing hole: The only period for which we did not know of a non-trivial oscillator was 41. But luckily, mathematicians do some of their best work under pressure and it took less than a week for Nico Brown to find one (casually posting it on the ConwayLife forums with the understated caption “Interesting loop”!) Consisting of 204 cells, the so-far-boringly named 204P41 oscillator (below – click for animation) completes a search which started over 50 years ago when this game was first invented.

This confirms that the Game of Life cellular automaton is **omniperiodic** – it has oscillators of every possible period. And yes, this research is not going to win any Nobel Prizes or spin off into whole new fields of research – but it has closed a gap in human knowledge and given us a new fun thing to play with – and is that not what maths is all about, really?

The post Hunting the Snark originally appeared on the HLFF SciLogs blog.

]]>The post Die seltsamen Fehlleistungen neuronaler Netze originally appeared on the HLFF SciLogs blog.

]]>Aber Adi Shamir hat sich auf seinen frühen Meisterleistungen nicht ausgeruht. Auf dem diesjährigen Heidelberg Laureate Forum überrascht er sein Publikum mit Neuigkeiten aus einem Gebiet, das mit Kryptografie – bis auf die Zugehörigkeit zum großen Fach Informatik – nichts zu tun hat: neuronale Netze.

Das sind diese Schichten aus lauter einfachen Elementen, die Neuronen nachempfunden sind. Information fließt von den Neuronen einer Schicht zu denen der nächsthöheren Schicht, wird von diesen in relativ einfacher Weise verarbeitet und an die übernächste Schicht weitergereicht. Die Einzelheiten dieser Verarbeitung legt sich das Netz auf eine sehr spezielle Weise zu: durch Lernen an Beispielen. Neuronale Netze haben in den letzten Jahren sensationelle Erfolge erzielt, zum Beispiel den Weltmeister im Go-Spiel entthront. Sie stecken auch hinter den „large language models“ wie ChatGPT, die zumindest den Anschein eines Denk- oder gar Einfühlungsvermögens erwecken.

Angesichts dieser Erfolge ist es umso beunruhigender, dass neuronale Netze auf ihrem ursprünglichen Spielfeld, dem Erkennen von Bildern, unerklärliche Schwächen aufweisen. Ein Netz hat sehr viele Bilder von Katzen präsentiert bekommen und dabei gelernt, eine Katze zuverlässig von jedem anderen abgebildeten Gegenstand zu unterscheiden. Dann stört man ein Bild, das eindeutig eine Katze zeigt, mit einer minimalen Menge zufälliger Abweichungen, so klein, dass ein menschlicher Betrachter überhaupt keinen Unterschied sieht – und schon glaubt das Netz, das Bild zeige einen Weißstorch, eine Schüssel mit Guacamole oder was weiß ich.

Na ja – wenn ein solches neuronales Netz in einem autonomen Auto steckt und die Bilder seiner Kamera analysiert, legt man irgendwie schon Wert darauf, dass es nicht eine freie Strecke mit einer roten Ampel verwechselt oder umgekehrt. Entsprechend eifrig haben sich die Fachleute um eine Erklärung des Phänomens bemüht, bisher ohne nennenswerten Erfolg. An dieser Stelle bietet Adi Shamir gemeinsam mit Odelia Melamed und Oriel BenShmuel vom Weizmann Institute of Science in Rehovot (Israel) eine neue Idee an. Die entscheidenden Gedanken kommen dabei bemerkenswerterweise aus der Geometrie.

In ihrer Arbeit nehmen Shamir, Melamed und BenShmuel der Klarheit der Darstellung zuliebe einige heftige Vereinfachungen vor. So unterstellen sie, ihr neuronales Netz habe nur gelernt, zwischen zwei Sorten von Bildern zu unterscheiden: „Katze“ und „Pampe“ (im Original Guacamole). Und während echte neuronale Netze in ihrer untersten Schicht die Farbwerte der Pixel entgegennehmen, aus denen das Bild besteht, ist es in der Vereinfachung nur eine einzige reelle Zahl pro Pixel.

Ein ausgelerntes neuronales Netz tut dasselbe wie ein Computerprogramm, das zu einem *x* das *f*(*x*) berechnet: eine Funktion auswerten. Das *x* ist ein Bild und das *f*(*x*) eine reelle Zahl, die angibt, wie stark dieses Bild das Merkmal „Katze“ aufweist. Wie die Funktion *f* definiert ist und wie das Netz sie berechnet: Das wissen wir nicht so genau. Die Einzelheiten der Berechnung hat es ja nicht einprogrammiert bekommen, sondern gelernt: indem ein „Lehrer“ ihm viele Beispielbilder vorlegte und dazusagte, ob es sich um Katze oder Pampe handelt. Immerhin wissen wir, dass unsere Funktion bei jedem der gelernten Katzenbilder einen hohen positiven Wert annimmt und bei jedem Pampenbild einen sehr negativen.

In dem abstrakten Raum aller denkbaren Bilder – ja, jedes Bild ist genau ein Punkt in diesem Raum! – wandern wir jetzt in Gedanken von einem Punkt mit sehr positivem *f*-Wert (sprich Katzenbild) zu einem mit sehr negativem *f*-Wert (Pampenbild). Die Punkte unterwegs sind dann Bilder, die einen sehr allmählichen Übergang vom einen zum anderen Bild darstellen. Dann ist irgendwo auf dem Weg *f*(*x*) = 0. (Ja, *f* ist stetig, und es gilt der Zwischenwertsatz.) Das gilt für alle Wege von der einen zur anderen Bildersorte.

Wenn der Raum aller Bilder jetzt zweidimensional wäre, so dass wir ihn uns richtig leicht vorstellen könnten, dann gäbe es in diesem Raum Katzenbezirke und Pampenbezirke. Möglicherweise gibt es von jeder Sorte mehrere Bezirke, die nicht miteinander zusammenhängen. Auf jeden Fall sind die Bezirke säuberlich getrennt, und zwar durch eine Kurve: die Menge aller Punkte, auf denen *f*(*x*) = 0 ist.

Leider hat der Raum aller Bilder nicht nur zwei Dimensionen, sondern so viele, wie ein Bild Pixel hat: Größenordnung eine Million. Die Funktion* f* gibt es immer noch, und deren Nullstellenmenge ebenfalls. In drei Dimensionen wäre es eine Fläche, und standardmäßig hat sie eine Dimension weniger als der Raum, in dem sie lebt. Die Fachleute sagen an dieser Stelle „Mannigfaltigkeit“ statt „Fläche“; denn zweidimensonal ist sie beim besten Willen nicht. Eine 999999-dimensionale Teilmenge eines millionendimensionalen Raums ist eine erhebliche Herausforderung für das Vorstellungsvermögen; aber man kann abstrakte Aussagen über sie machen.

Zum Beispiel kann die Trennfläche nicht beliebig verknittert sein. Denn sie ist das Ergebnis einer Berechnung durch – zugegeben: zahlreiche – Neuronen, die jedes für sich sehr einfach gebaut sind. Da können so exotische Dinge wie fraktale Verknitterungen gar nicht vorkommen. Und das oben genannte Verwechslungsproblem dürfte eigentlich auch nicht auftreten.

Unsere Funktion *f* ist stetig: erstens, weil die Neuronen konstruktionsbedingt nicht anders können, zweitens weil sich das beim Lernprozess ohnehin ergeben sollte. Das Netz soll ja ein Katzenbild mit geringen Abweichungen, also einen Punkt in unmittelbarer Nähe des Katzenbilds, noch als Katzenbild erkennen. Anders ausgedrückt: Die Funktion *f* soll von einem positiven Wert nicht plötzlich steil auf null und gar darunter abfallen. Genau das passiert aber.

Shamir und Kollegen bieten dafür folgende Erklärung an: Alle von einer Digitalkamera aufgenommenen und nicht raffiniert manipulierten Bilder sind in einem speziellen Sinne „ordentlich“. Sie enthalten eben nicht die zufälligen kleinen Abweichungen von der „richtigen“ Bildgestalt, die ein neuronales Netz so spektakulär in die Irre führen können. Insbesondere sind alle Bilder, Katze wie Pampe, an denen das Netz trainiert wird, ordentlich. In dem abstrakten Raum aller Bilder sind die ordentlichen eine sehr kleine und vor allem dünne Teilmenge. Je nachdem, wie man den Begriff „ordentlich“ definiert, was nicht einfach ist, liegt ihre Dimension um das Zehn- bis Hundertfache unter der des ganzen Raums.

Wie dem auch sei: Im Verlauf des Lernprozesses merkt das Netz gewissermaßen sehr schnell, dass es überhaupt nur um ordentliche Bilder geht, und legt seine Trennfläche zwischen Katze und Pampe in einer ersten Phase so, dass sie im Wesentlichen der Teilmenge der ordentlichen Bilder folgt. Erst in der zweiten Phase kommt die Feinabstimmung: Wenn sich herausstellt, dass ein Katzenbild noch auf der falschen Seite der Trennfläche liegt, kommen die Zwerge mit kleinen Hämmerchen und schlagen eine Delle in die Trennfläche, bis die auf der richtigen Seite am Katzenbild vorbei verläuft. Dasselbe geschieht in umgekehrter Richtung, falls ein Pampenbild sich als fehlplatziert herausstellen sollte.

Was wollen uns die Autoren mit der seltsamen Metapher von den Zwergen und den Hämmerchen sagen? Mehrere Dinge. Erstens ist es eine gute Idee, sich die Trennfläche aus dünnem Blech vorzustellen. Klopft man es an einer Stelle zurecht, dann geht die unmittelbare Umgebung mit. Das ist eine andere Ausdrucksweise dafür, dass die Funktion *f* stetig und die Trennfläche nicht zu heftig gekrümmt sein sollte.

Zweitens: Wenn man auf das Blech klopft, beult es sich in einer Richtung aus, die senkrecht zur Ausbreitungsrichtung des Blechs ist. Jawohl, auch in hochdimensionalen Räumen kann man sinnvoll von rechten Winkeln reden. Allerdings gibt es hier sehr viele verschiedene Richtungen, die alle senkrecht auf dem Blech stehen. Da nun die Trennfläche im Wesentlichen der Menge der ordentlichen Bilder folgt, treibt jeder Hammerschlag sie von dieser Menge weg, in die wüsten Gefilde der unordentlichen Bilder. Einmal dort hingeraten, wird sie in der Tendenz dort bleiben. Denn da das Netz nie ein unordentliches Bild zu sehen bekommt, hat es auch keine Gelegenheit, die Position der Trennfläche im Reich des Unordentlichen zu korrigieren.

Drittens: Es sind sehr viele sehr kleine Hammerschläge. So ist das übliche Fehlerkorrekturverfahren („backpropagation“) gebaut. Das heißt, die Dellen im Blech sind ziemlich flach, gerade so tief, wie es sein muss, damit das Netz ein ordentliches Bild richtig erkennt. Das wiederum hat zur Folge, dass man von einem ordentlichen Bild nur ein kurzes Stück in die falsche Richtung wandern muss, um auf die Trennfläche zu treffen, und noch ein kleines Stück darüber hinaus, um auf die falsche Seite zu geraten.

So weit die stark vereinfacht dargestellte Idee. Wenn man die oben genannten Einschränkungen aufhebt – Farb- statt Schwarzweißwerte, mehr als zwei Klassen von Bildern –, wird die Sache unübersichtlicher, aber nicht prinzipiell schwieriger. Andere Einzelheiten, über die ich hinweggegangen bin, wollen ausgearbeitet werden, was die Autoren erhebliche Mühe gekostet hat.

Und nachdem das Problem erkannt ist, liegt eine Abhilfe nicht unmittelbar auf der Hand. Natürlich kann man eine große Menge unordentlicher Bilder erzeugen und dem Netz als Lernstoff vorlegen. Aber dabei steigt der Trainingsaufwand leicht auf das Tausendfache oder mehr, was die Sache unpraktikabel macht. Für eine wirksame Abhilfe braucht es wohl noch neue Ideen.

The post Die seltsamen Fehlleistungen neuronaler Netze originally appeared on the HLFF SciLogs blog.

]]>summarizes the week of the 10th Heidelberg Laureate Forum, which once a year brings together bright young mathematicians and computer scientists with the most prominent scientists in the field, in sketch notes! Read more

The post The 10th Heidelberg Laureate Forum in Sketchnotes originally appeared on the HLFF SciLogs blog.

]]>The week of the 10th Heidelberg Laureate Forum, which once a year brings together bright young mathematicians and computer scientists with the most prominent scientists in the field, started with fresh new lectures by old and new friends of the HLF, Laureates Raj Reddy, Yael Tauman Kalai, and Robert Metcalfe, all very popular among the audience!

This year’s HLF has a brand new format that turned out to be a revelation: the Lightning Talks, moderated by mathematician and communicator Dr. Tom (Rocks Maths) Crawford.

But the surprise didn’t stop there! The Spark Session’s purpose was to inspire!

This week welcomed new Laureates who shared with the audience their passion for mathematics, phase transitions and the connections between maths and computer science

The week couldn’t be complete without a Panel discussion on what is probably ”the” hot topic of the year: Generative AI, moderated by science journalist Anil Ananthaswamy.

Throughout the week, the participants of the HLF had the opportunity to visit an exhibition with a selection from the last 10 years of outreach at HLF. The HLF Foundation’s outreach activities are situated in the MAINS, the Mathematics and Information Station, opened by the HLFF in 2017. The Head of Outreach and Exhibitions, Dr. Volker Gaibler, gave us a tour!

The post The 10th Heidelberg Laureate Forum in Sketchnotes originally appeared on the HLFF SciLogs blog.

]]>The post Can You Tell Cats Apart from Guacamole? originally appeared on the HLFF SciLogs blog.

]]>Shamir is mostly celebrated for his work in cryptography but has made contributions to computer science outside of cryptography. Recently, he has increasingly been focusing on adversarial examples in machine learning.

Adversarial attacks are techniques used to manipulate the output of machine learning models, particularly neural networks, by introducing carefully crafted input data. In the case of an image, the end result may look very similar to the original data to an observer, but they feature very slight changes that can cause the model to make incorrect predictions or classifications. Essentially, by changing just a few key pixels in a specific way, the classifier can be completely deceived.

Researchers started figuring out that tiny perturbations can cause severe effects in algorithms in 2013, Shamir notes in a recent paper. This became even more apparent when image neural networks became really powerful.

This was brilliantly illustrated in 2017 by a student-run group at MIT who picked the ever-popular cat-guacamole example to show how neural networks can be fooled. This has since become the go-to example, the “hello world” of adversarial attacks.

Adversarial attacks on image neural networks involve introducing small, often imperceptible changes to an image to deceive a neural network into misclassifying it. While the altered image still looks the same to humans, the neural network might see it differently due to the subtle manipulations. These attacks exploit the complex, high-dimensional space in which neural networks operate, finding weaknesses that cause the network to make errors. The goal can be simply to make the network misclassify the image or to misclassify it in a specific, targeted way

Imagine you have a well-trained neural network that can correctly identify a picture of a cat 99% of the time. An adversarial attack aims to subtly and maliciously modify this picture so that the neural network misclassifies it, perhaps thinking it’s a dog, while to the human eye, the image still clearly looks like a cat.

Initially, such attacks needed to know how the neural network functioned, but that is no longer the case. Attacks can work on multiple types of networks trained in different way, and there’s no clear way to prevent such attacks.

Such adversarial attacks are dangerous, but they are themselves vulnerable. Here is the same image as above, only rotated slightly: it is now correctly identified as a cat (and a tabby at that).

In his HLF lecture, Shamir proposed a framework to understand why this happens and how you can fool networks. Of course, the laureate does not want to actually attack neural networks. He wants to make them safer – and there is a very good reason for that.

We have been hearing some version of “AI is coming” for a decade or so. But now AI is here, in particular generative AI. The number of AI science papers has skyrocketed; the ways in which companies are using AI is surging; and we are already seeing it in our day-to-day life. You only need to look at ChatGPT to see how big of an impact a single example can have.

Nowadays, neural networks can also create strikingly complex images with ease. Research in this field has matured to the point where several companies offer such services to end users at a relatively low cost – a sign of a matured technology.

However, these networks are also fragile, Shamir notes.

It is not hard to understand why adversarial attacks can cause major problems. These attacks could be used maliciously to deceive AI systems in critical applications like autonomous vehicles, facial recognition, and medical imaging. Imagine a self-driving car that uses cameras to recognize traffic signs. An adversarial attack could subtly alter the appearance of a stop sign so that the car’s AI interprets it as a yield sign. This could lead to dangerous situations on the road. Adversarial attacks could be used to fool security systems: For example, by wearing a specially crafted pair of glasses or using a modified photo, an intruder might be incorrectly identified as an authorized user. It does not have to be a visual attack, either. With an adversarial voice attack, you could fool personal assistants like Siri or Alexa into executing commands without the user’s knowledge.

But perhaps nowhere is this as dangerous as in facial recognition.

By the end of 2021, an estimated one billion surveillance cameras were in operation globally, over half of them in China. Governments and companies are increasingly looking at deploying facial recognition using these cameras. This is in itself a pretty questionable decision, but let us focus on the technical aspects.

Over the past decade, there has been an “amazing improvement in the accuracy of facial recognition systems,” says Shamir. This accuracy is now at over 99%, and given the large number of cameras it is “the only way to deal with this massive data stream,” the laureate adds.

As with the cat-guacamole example, Shamir showed that you can fool algorithms into believing one person is another. He mentions the example of Morgan Freeman and Scarlett Johansson. Algorithms believe one is the other. There was no difference for anyone else in the world, but if you want to make an attack on two people in particular, it can be done.

Shamir even detailed another type of attack he has developed. Instead of changing pixels, he mathematically modified a few weights in the neural network. This is a new attack type that he called “weight surgery.” It is extremely difficult to predict what will be the effect of such changes on the performance of the neural network, but it shows once again how vulnerable such systems can be.

“Since we don’t understand the role of the weights in Deep Neural Networks [multi-layered models often used for image generation], hackers can easily embed hard to detect trapdoors into open source models,” Shamir explained in the lecture.

So then, where does this leave us?

Being aware of the problem is an important first step. It is easy to get swept away in the hype and disregard some of the issues that can stem from it. Understanding exactly what the problem is would be the next step – this is what Shamir (and many other researchers) are working on. There is important progress being made on this front, the laureate mentions.

However, overcoming the problem is a different challenge. Simply put, we are not able to do this at the moment. There are currently no effective ways to recognize adversarial examples and their effect, the laureate notes.

So for now, the best thing to do is simply tread carefully when deploying such networks.

“Deep Neural Networks are extremely vulnerable to many types of attacks, and we should take great care in deploying them in a safe way,” Shamir concludes.

This was not lost on the young researchers in Heidelberg. During the boat trip, someone took the time to reclassify what was served as a dip. So, would you like some cat to go with the nachos?

The post Can You Tell Cats Apart from Guacamole? originally appeared on the HLFF SciLogs blog.

]]>The post A Conversation with Karen Uhlenbeck originally appeared on the HLFF SciLogs blog.

]]>I took a deep breath. Then I joined the Zoom call.

It very soon became clear that I need not have worried. Karen Uhlenbeck is a brilliant woman but what’s more, she is delightfully normal. The next day, at the panel discussion, Ingrid Daubechies would confess to Uhlenbeck how meeting her was a high point in her life. “Not only did you really exist and were real but you were wonderful,” Daubechies said. I could not agree more.

Karen Uhlenbeck’s career is full of prestige and acclaim. She was awarded the National Medal of Science in 2000 and received the Leroy P. Steele Prize for Seminal Contribution to Research of the American Mathematical Society in 2007. She became an honorary member of the London Mathematical Society in 2008 and a Fellow of the American Mathematical Society in 2012 – and this is only a small selection of her accolades.

You would be forgiven for thinking, given her success, that Uhlenbeck was following a plan. This could not be further from the truth. “I just wander around and when something catches my eye, I work on it,” Uhlenbeck told young researchers during her panel discussion. She confessed that she does not always choose wisely, but stressed the importance of making sure that the problems you choose to tackle are ones that genuinely interest you, not just your supervisor: “Getting problems that are the right fit for one’s imagination is the important step.”

Finding what fitted her imagination was not a simple process for Uhlenbeck. She intended to major in physics when she enrolled at the University of Michigan. It was during her second semester, when attendance started to be taken in lectures, that she started to consider a change. She had also been finding the lab work – and her lab partners –challenging.

Around that time, Uhlenbeck met Daniel Hughes, a finite geometer and group theorist. Hughes (or “Dan” as he was known to Karen), took her under his wing. By the end of her second year, Uhlenbeck was grading linear algebra papers. It was immediately obvious that mathematics was something she was very good at.

Following her time in Michigan, Uhlenbeck attended graduate school, taught at MIT for a year, and then taught at Berkley for another two. She held faculty positions at various universities in the state of Illinois from 1981 to 1988, before making her final move to the University of Texas in Austin. Following her retirement, she moved to Princeton where, in 2019, she became a Distinguished Visiting Professor at the Institute for Advanced Study.

I asked Karen if she had any regrets regarding her mathematical career. “I wish I had recruited more students and post docs to work with me at the University of Texas,” she replied. “In a similar vein, I wish I had advertised my students’ work more aggressively. I tended to take things as they came and left my students to fend for themselves once they completed theses.”

Any researcher aiming for a long-term career in mathematics has hoops to jump through. Uhlenbeck stresses the importance of holding on to the enjoyment of maths. “Don’t let it get to you or embitter you,” she told the young researchers during her panel at the HLF. “It’s very important to keep your soul.” In fact, one of Uhlenbeck’s greatest assets is her ability to not let the external world get to her. “I judge myself,” she replied when a young researcher at the HLF asked her how she deals with imposter syndrome. Her validation comes first and foremost from within.

Karen Uhlenbeck was awarded the Abel Prize in 2019 in recognition of her “pioneering achievements in geometric partial differential equations, gauge theory and integrable systems, and for the fundamental impact of her work on analysis, geometry and mathematical physics.” The citation is not exaggerating when describing her contributions as fundamental – Uhlenbeck is a founder of modern geometric analysis, and made great strides in Yang-Mills theory as well.

Geometric analysis describes both the application of geometric methods to differential equations, and the use of tools from differential equations to solve problems in differential geometry and differential topology. Geometric analysis’ breadth means it can lead to a wide variety of results in both mathematics and physics.

Gauge theory is a subfield of geometric analysis, studying fields in which a certain function known as the Lagrangian does not change following the application of a particular type of smooth operator (known as a Lie Group). If this seems difficult, it’s because it is! Uhlenbeck studied gauge theory extensively, in particular working with Yang-Mills theory, which focusses on a specific type of Lie Group. Yang-Mills theory is a key piece of the jigsaw of unifying the electromagnetic force and the weak force. It searches for a theory of quantum gravity and plays a vital role in our understanding of the Standard Model of particle physics. Karen Uhlenbeck’s work analysing the Yang-Mills equations in four dimensions inspired many breakthroughs from other mathematicians and eventually led to the work for which Simon Donaldson won the Fields Medal in 1986.

A major unsolved problem of Yang-Mills theory is the Yang-Mills existence and mass gap problem. This is one of the seven problems chosen by the Clay Mathematics Institute in 2000 to form their “Millennium Problems”, with a $1 million reward promised to the solvers of each problem.

The Yang-Mills existence and mass gap problem is not the only one of the Millennium problems that appears to be amenable to geometric analytic techniques. Grigori Perelman’s work in geometric analysis, specifically on Ricci Flow, resulted in a solution to the Poincaré Conjecture. At the time of writing, in 2023, the Poincaré Conjecture is the only one of the problems to have been solved. Grigori Perelman refused the money.

Despite this incredible work, Uhlenbeck’s favourite result is her work on regularity of elliptic systems, published in 1976. Uhlenbeck explained to me that “It might be my most technical and difficult paper. It proved to me that I was a good mathematician.” She worked on the paper immediately after graduate school, spending five or six years on it. At the time, relations between the USA and Soviet Russia were poor, impacting the flow of ideas, so Uhlenbeck was unaware until many years later that the same problem had been solved by Nina Uraltseva in 1972. Upon hearing the news though, Karen Uhlenbeck was pleased: “She was one of my heroes as a graduate student.”

Something Karen Uhlenbeck and I have in common is that we are both huge proponents of women in maths. Uhlenbeck feels a huge responsibility towards other women in mathematics and has spent large parts of her career encouraging others. In her current position, she has organised monthly lunches for female mathematicians to introduce them to ideas from different fields. She has also long been engaged with organisations such as the Association for Women in Mathematics. Uhlenbeck even donated half of her Abel Prize money to the promotion of minorities in mathematics, splitting it between the EDGE Foundation, and the Institute for Advanced Study.

Uhlenbeck describes her early career as being like the film Oppenheimer – “all men, cigarettes and alcohol.” When asked why she succeeded in such a male-dominated world, while many other women did not, she is stumped. “I don’t really have any idea how I did it.” Uhlenbeck states that she does not believe she was smarter or “more savvy” than other women.

She is similarly uncertain of how to encourage women in mathematics in the future, especially in a genuine and helpful way. She describes how upon meeting young female mathematicians, she immediately thinks she should help them. Even as a woman, she confessed that she immediately assumes that men are brilliant and women are struggling. “This is actually a form of prejudice,” Uhlenbeck acknowledges. “These are brilliant mathematicians I should be excited to meet.” She says she has spent the last 30 years of her career trying to encourage women. Now, Uhlenbeck realises, she needs to sit back and admire them.

Uhlenbeck is optimistic about the future, too. She has noticed the attitudes towards women in maths, and that the environments of maths departments have slowly been changing. She does point out that, as women had not been part of the intellectual world for 4000 years, it will take more than a couple of decades to change that.

Whilst at HLF, I found myself in several conversations about the lack of female Fields Medallists. One repeated discussion was how capping the age of those eligible to receive the medal at 40 precludes almost all women who become mothers. Uhlenbeck herself does not have children, but she tells me that this was not her choice “I was sad that I was not able to get pregnant, at a time when there were not easy medical procedures to help solve the problem.”

I am touched and moved by her honesty and openness. I had not directly asked about children, but Uhlenbeck gave a strong impression of wanting to talk about it. “It is understandable that no one asks me why I did not have children. It must be a common unspoken question. I am not the only woman in history with this story.” In many ways, maths became a solace to Uhlenbeck, going some way to fill the gap that she had hoped children would fill: “[It] would have left my life empty if I hadn’t had mathematics to fill it.”

Uhlenbeck and I spoke in depth about her life and career, but there was still time to chat about her hobbies. It didn’t take long for us to discover our shared love for the New York Times Spelling Bee puzzle – a puzzle where you must make as many words as you can with the seven letters you are given. Letters can be used more than once, but each word must contain the central, yellow letter. Uhlenbeck enjoys the challenge and explained that feeling of satisfaction after finding a word containing all seven letters is much like how she feels after solving a particularly difficult mathematics question.

Over the course of her career, Uhlenbeck has come across many problems that have taken her years to solve. Rather than this being discouraging, she relishes the challenges. She described to me the comfort in returning to the same problem again, revisiting previous chains of thinking and prying out new ideas. In fact, she said she often felt disappointed when she finally did crack the puzzle. “I like to work on problems that are hard and take a long time and there’s always a bit of sadness when I solve them.”

Unfortunately for me, it was now my turn to feel a bit of sadness as my chat with Karen Uhlenbeck was nearly over. I wish I had had longer to speak with her, and I wish the Zoom call had not cut out. But despite the brevity of our conversation, it was clear what a brilliant person she is. Yes, brilliant in the sense of her mathematical achievements, but also brilliant in her generosity and in her humility. They say to never meet your heroes. Thankfully, Karen Uhlenbeck proves that adage wrong.

The post A Conversation with Karen Uhlenbeck originally appeared on the HLFF SciLogs blog.

]]>The post Kann ChatGPT sich in die Gedanken anderer einfühlen? originally appeared on the HLFF SciLogs blog.

]]>Für ein gute Druckqualität empfahl es sich, auf der mechanischen Schreibmaschine – richtige Typenhebel, nix Kugelkopf, nix Typenrad – kraftvoll in die Tasten zu hauen, damit auch reichlich Kohle aufs Papier kam. Da konnte im Eifer des Gefechts der Typenhebel fürs kleine „o“ schon mal ein Loch ins Papier stanzen, wodurch am Ende von dem Buchstaben nur ein schmaler äußerer Rand im Druck sichtbar war.

In einer Diplomprüfung in angewandter Mathematik stellt der Prüfer die Frage, wie das Skalarprodukt im Raum *L*^{2} definiert sei, und erhält die überraschende Antwort „Integral über *f*(*x*) komponiert mit *g*(*x*) d*x*“. Nach einer Pause der Ratlosigkeit versucht der Prüfer, mit gezielten Fragen dem Kandidaten die richtige Antwort „Integral über *f*(*x*) mal *g*(*x*) d*x*“ zu entlocken – vergebens. Der Kandidat beharrt auf seiner Version, fürchtet sogar, er würde aufs Glatteis geführt, und kramt schließlich zum Beweis das Skript heraus, nach dem er gelernt hat. Da steht \( \int f(x) \circ g(x) dx \), und der Kringel in der Mitte ist in der Tat das Zeichen für die Komposition (Hintereinanderausführung) von Funktionen. Zu dumm, dass der Mensch, der die Vorlage für das – im Umdruckverfahren hergestellte – Skript tippte, bei dem Malpunkt zwischen *f*(*x*) und *g*(*x*) so heftig zugeschlagen hatte, dass nur noch der Kringel stehen blieb.

Der Kandidat hat also zweifelsfrei sehr exakt gelernt; aber das hilft nichts. Denn hätte er die Formel fürs Skalarprodukt auch nur an den einfachsten Beispielen anzuwenden versucht, wäre ihm klargeworden, dass die Version mit dem Kringel nicht stimmen kann.

Warum erzähle ich diese – ziemlich alte – Geschichte? Weil in einem kürzlich durchgeführten Experiment die Software ChatGPT noch wesentlich schlechter aussieht als damals der Prüfling. Joachim Escher, Professor für Mathematik an der Universität Hannover, stellt ChatGPT ein paar einfache Fragen zur elementaren Zahlentheorie ^{[1]}. Nichts Tiefsinniges; es geht um Teilbarkeit und Primzahlen, was man so in der 7. Klasse lernt. ChatGPT gibt haarsträubende Antworten. Es behauptet, 2023 sei eine Primzahl (nein, 2023 = 7 ^{.} 17^{2}), macht dann auf die Bitte, 2023 durch 119 (= 7 ^{.} 17) zu teilen, elementare Rechenfehler mit dem Ergebnis, dass die Division nicht ohne Rest aufgeht, und beharrt auf diesem falschen Ergebnis. Wenig später behauptet es, 2023 sei gleich 43 ^{.} 47 (nein, es kommt 2021 heraus). „Merkwürdig, wenn die eine Zahl auf 3 und die andere auf 7 endet, mit welcher Ziffer endet dann das Produkt?“ fragt Escher, ganz der wohlwollende Prüfer. Und selbst dann kommt ChatGPT nicht auf die Idee, es könnte ein Rechenfehler vorliegen. Ein echter Mensch wäre bei dieser Prüfung glatt durchgefallen.

Sowohl ChatGPT als auch der unglückliche Prüfling hatten sich fleißig eine große Menge an Material reingezogen (ersteres nutzte beim Training reichlich 175 Milliarden Dokumente), aber eben nicht ernsthaft verarbeitet. Deswegen mussten sie beide scheitern. Der Mensch, weil er ohne diese Verarbeitung einen – sagen wir – Schreibfehler in der Quelle nicht erkennen und entsprechend nicht ausbessern konnte. Die künstliche Intelligenz, weil… das wissen wir nicht so genau. Vielleicht hat eine der Quellen tatsächlich einen falschen Rechenweg enthalten. Wahrscheinlicher ist es, dass ChatGPT, wie das so seine Art ist, an eine vorliegende Sequenz von Worten das statistisch plausibelste Wort angehängt hat, und das immer wieder. So konstruiert das Programm diese erstaunlich eloquenten Sätze. Aber diese statistische Methode hilft natürlich nicht gegen Rechenfehler.

Ich komme zu dem Schluss, dass diese künstliche Intelligenz eben doch ziemlich blöde ist, und lehne mich beruhigt zurück, da ich mich meiner überlegenen Intelligenz vergewissert habe.

Nur drei Monate später wird diese Gewissheit empfindlich gestört. In derselben Zeitschrift wie Joachim Escher, nur eine Ausgabe später, berichtet Christian Spannagel, Professor an der PH Heidelberg, dass ChatGPT auf klassische Aufgaben korrekte Antworten gibt, sogar auf mehrfaches Befragen hin verschiedene Wege zur Lösung derselben Aufgabe beschreitet und insgesamt die Leistungen eines guten Studierenden erbringt ^{[2]}. Von der Vorstellung, 2023 sei eine Primzahl, lässt es zwar nicht ab, und auch sonst gibt es immer wieder mal falsche Antworten. Aber auf eine Textaufgabe, die erst in eine mathematische Formulierung umzusetzen ist, liefert es, zwanzigmal gefragt, zwanzigmal die richtige Lösung, mit verschiedenen Lösungswegen und jedesmal etwas anders formuliert. Die neue Version GPT-4, wohlgemerkt; dem Vorgänger GPT-3.5 gelangen nur 13 von 20 Versuchen.

So wie es aussieht, hat diese künstliche Intelligenz in wenigen Monaten mächtig zugelernt. Zu allem Überfluss kann sie mittlerweile auf ein sehr mächtiges Rechengerät zurückgreifen: Die Firma Wolfram, welche die Universal-Mathematik-Software Mathematica entwickelt hat und vertreibt, hat einen „Adapterstecker“ zu ChatGPT programmiert, ein Programm, das eine Frage in natürlicher Sprache – vom menschlichen Nutzer oder von ChatGPT selbst formuliert – entgegennimmt, in den Formalismus von Mathematica umsetzt, an dieses weiterreicht und das Ergebnis der Berechnung, in natürliche Sprache umformuliert, zurückgibt. Um genau zu sein, es ist nicht Mathematica allein, sondern dessen Erweiterung Wolfram alpha, die außer den Rechenkapazitäten noch allerlei geografische und andere Daten bereithält.

Seit seiner Erstveröffentlichung im November 2022 hat ChatGPT nicht nur gewaltiges Aufsehen erregt, sondern auch an Fähigkeiten erheblich zugelegt. Wenn dieser steile Aufstieg so weitergeht: Ist die Software auf dem Weg zu einer echten Intelligenz?

Der renommierte und fachkundige Journalist Anil Ananthaswamy hat ChatGPT auf eine spezielle Intelligenzleistung getestet, die unter den Fachleuten als *theory of mind* diskutiert wird: Ist der Kandidat in der Lage, sich in die Gedankenwelt eines anderen hineinzuversetzen? Für kleine Kinder pflegt man dafür den „Sally-Anne-Test“ heranzuziehen. Sally legt eine Murmel in den linken von zwei Körben und verlässt dann den Raum. Währenddessen legt Anne, die im Raum geblieben ist, die Murmel vom linken in den rechten Korb. Sally kommt zurück. An dieser Stelle wird der Kandidat, der das alles beobachtet hat, gefragt: „In welchem Korb wird Sally die Murmel suchen?“

Sehr kleine Kinder pflegen zu antworten: „Im rechten“, da ist die Murmel ja schließlich. Erst wenn sie ungefähr drei bis vier Jahre alt sind, können sie erkennen, dass Sally den linken Korb wählen wird, weil sie Annes Verlegeaktion nicht mitbekommen hat. Erst dann haben sie sich eine *theory of mind* zugelegt – in diesem Fall von Sallys *mind*.

Der Sally-Anne-Test ist in der Literatur ausgiebig beschrieben worden. Zweifellos waren diese Beschreibungen auch unter den Milliarden Texten, die ChatGPT während seiner Trainingsphase verarbeitet hat. Auf eine Frage, die auch nur ungefähr die Geschichte vom Sally-Anne-Test erzählt, würde es ohne weiteres eine korrekte Antwort „aus dem Gedächtnis rekonstruieren“ können, wie der Kandidat, der zwar die Formel fürs Skalarprodukt richtig rezitieren kann, aber im Übrigen keine Ahnung davon hat. Das wäre also noch kein Beweis dafür, dass ChatGPT eine *theory of mind* hat.

Also legt Ananthaswamy seine Frage raffinierter an. Er erzählt nach wie vor die Sally-Anne-Geschichte, aber die Beteilgten heißen nicht Sally und Anne, sondern Alice und Bob, der Kontext ist völlig anders, und damit es noch ein bisschen schwerer ist, kommt die Aufgabe hinzu, aus der gewonnenen Erkenntnis eine Schlussfolgerung zu ziehen. In einem Vortrag, den Ananthaswamy am 5. Juli 2023 in der Mathematik-Informatik-Station (MAINS) in Heidelberg gehalten hat, führt er seine Frage an ChatGPT und dessen Antwort vor (ab Minute 7).

ChatGPT besteht den Test mit einer glatten Eins. Es zieht nicht nur die richtige Schlussfolgerung, sondern begründet auch, warum Alice (statt Sally) annimmt, ihre richtige Brille liege in der linken statt der rechten Schublade, und deswegen am nächsten Tag heftige Kopfschmerzen hat.

Hat ChatGPT also eine *theory of mind*? Schwer zu sagen, vor allem weil OpenAI, die Firma, die es entwickelt hat, wesentliche Einzelheiten für sich behält. Stephen Wolfram („Mathematica“) hat in einem sehr ausführlichen Überblick zusammengetragen, was man trotzdem weiß. Und eine *theory of mind *oder Ähnliches ist ChatGPT definitiv nicht explizit einprogrammiert worden. Noch ist sein Verhalten auch sehr instabil. In einem Artikel in „Science“ weiß Melanie Mitchell zu berichten, dass es gewisse amerikanische Standard-Prüfungsfragen korrekt beantwortet, aber an denselben Fragen, geringfügig anders formuliert, kläglich scheitert.

Aber nehmen wir an, dass die Software mit der nächsten Version noch erheblich an Stabilität zulegt und dann auch noch härtere Tests besteht. Wird dann die Grenze zwischen deren Fähigkeiten und echtem Verständnis verschwimmen? Und was ist eigentlich echtes Verständnis?

Für einen Mathematiker wie mich wird diese Frage besonders pikant, denn bei den Gegenständen der Mathematik weiß man typischerweise nicht – kann es nicht wissen –, was sie „eigentlich“ sind. Was man wissen – und in der Prüfung abfragen – kann, ist nur, wie man mit ihnen umgeht. Und das wiederum ist in Definitionen festgelegt, die man auswendig lernen kann.

Was also – sagen wir – ein Skalarprodukt ist, erschließt sich erst durch den Umgang mit diesem Begriff, das heißt durch den Sprachgebrauch. Und hier drehen die drei Philosophen Christoph Durt, Tom Froese und Thomas Fuchs den Spieß um: Allein im alltäglichen Gebrauch der Sprache ist so viel außersprachliche Bedeutung (*meaning*) enthalten, dass ein *large language model *(LLM) wie ChatGPT allein durch den Konsum großer Mengen geschriebener Sprache während der Trainingsphase genug *meaning* aufsaugt, um damit zumindest diverse Tests zu bestehen.

Das ganze Gebiet ist in heftiger Bewegung. Mit weiteren Überraschungen ist zu rechnen.

[1] Joachim Escher & ChatGPT: Mündliche Prüfung mit ChatGPT. Oder warum die Primzahl 2023 = 43 x 47 ist. Mitteilungen der Deutschen Mathematiker-Vereinigung 31 (2), S. 102–103, 2023

[2] Christian Spannagel: Hat ChatGPT eine Zukunft in der Mathematik? Mitteilungen der Deutschen Mathematiker-Vereinigung 31 (3), S. 168–172, 2023

The post Kann ChatGPT sich in die Gedanken anderer einfühlen? originally appeared on the HLFF SciLogs blog.

]]>The post How to Guarantee You Win the Lottery originally appeared on the HLFF SciLogs blog.

]]>Lotteries take place in many places around the world, and are an interesting case study in human behaviour and our relationship with mathematics. It is fascinating how something which in most cases has an incredibly small chance of happening is such an attraction for players – the probability of winning the lottery jackpot is conventionally one in tens of millions. People continue to buy tickets, drawn by the tiny chance of a life-changing amount of money.

Since lotteries are usually drawn using a completely random method in the interests of fairness, it is definitely not possible to increase your chances of winning by choosing particular numbers. However, it is possible to increase the size of your potential prize through your choice of numbers.

In lotteries where the top prize is split between everyone who matches all the numbers, you are playing against everyone else, as well as the random nature of the draw. In these situations, knowing which numbers are more likely to be popular is useful – generally, this tends to include numbers locally considered “lucky” – as is avoiding unlucky numbers like 13 (in Western cultures), 4 (China, Japan and Korea), 9 and 43 (Japan), 17 (Italy) and 39 (Afghanistan).

A survey by Alex Bellos back in 2014 found that 7 is considered to be a lot of people’s favourite number, so that is likely to be popular; and many people choose their numbers based on their own and family birthdays, so numbers between 1 and 31 tend to be more frequently chosen than numbers above 31. So potentially, picking numbers some consider unlucky or are otherwise less likely to choose means that if you do win big, you are less likely to share the jackpot with as many others.

Of course, all of this is subject to the same annoying probabilities – your chances of winning are still vanishingly small. Is there anything we can do to make it more likely – or even guarantee – that we will be a lottery winner?

Theoretically, it should be possible to buy as many lottery tickets as you want – with a sufficiently coordinated network of buyers and probably quite a big spreadsheet, every possible combination of numbers could be calculated, allocated and purchased. The cost of doing so would be – with exact values subject to the specifics of the lottery you are playing – a significant investment, but has been theorised as a way of guaranteeing yourself a win. You would definitely have a ticket that matches all the numbers!

But the numbers do not necessarily stack up if you calculate the expected return on this technique. For example, the UK lottery, when it started in 1994, used six numbers drawn from 49 and had tickets costing £1. From this we can work out the expected return of this strategy, by looking at how much it would cost and how much you would win.

The number of possible sets of numbers you could choose can be calculated as 49 × 48 × 47 × 46 × 45 × 44 = 10,068,347,520, since each successive number is drawn from a slightly smaller set of options. We then need to divide by 6 × 5 × 4 × 3 × 2 × 1 = 720, as we do not care what order the balls are drawn in, and this is the number of ways to rearrange them (my previous post on combinatorics covers this in more detail).

This gives us 13,983,816 possible combinations (and also tells us that the probability of winning the jackpot by matching all six numbers is 1 in 13,983,816, or 0.000007% – less likely than being struck by an asteroid). The cost of buying one ticket for each would then be just under £14 million; but how much would you be likely to win?

Things get slightly complicated here, since the UK lottery, like many others, has prizes for matching fewer than six numbers, and within your collection of tickets you would also have all of these possible outcomes too (although you may wish to outsource taking the 246,820 tickets that each match three numbers down to the shops to claim £10 for each of them). You would also win prizes for matching 4 and 5 numbers; and the UK lottery also has a “bonus ball” drawn separately as a seventh number, which comes into play when you have also matched 5 of the main numbers, and gives you an extra prize.

The size of the jackpot itself depends on the number of tickets sold – and admittedly, a shadowy group of mathematicians purchasing a huge chunk of tickets for one particular draw would increase this amount fairly significantly. However, if you run the numbers on an average weekly draw under the original prize system, from the jackpot and all the smaller prizes together you would net a total of around £6,292,717 – less than half what you spent on tickets (not to mention logistics). Of the 14 million tickets you bought, only 260,624 would win any kind of prize – of which the bulk would be your £10 tickets – and the rest would go straight in the recycling.

The UK lottery also operates a “rollover” on occasions when nobody wins the jackpot – a portion of the prize fund that was set aside for the jackpot gets added to the jackpot for the following week. But even when there is a rollover, a double rollover (when this happens twice in a row) or a triple rollover, you are still expected to lose a good chunk of your £14m, with the amount you can expect to win being still less than £10m.

More recently, this has gotten even worse: the UK lottery has increased its number set to picking 1 out of 59 – now meaning there are 45,057,474 combinations – and tickets now cost £2 per draw, so you would have to put in nearly £100m to play all the tickets, and would definitely make an even bigger loss – the average maximum jackpot is only £13m (with a cap on how many times it can roll over), and smaller prizes would not add much to that total.

It makes sense that it is not possible to make a profit doing this – on the UK lottery, for every £1 ticket purchased approximately 45p went into the prize fund, whilst the other 55p goes into charitable donations – as well as covering operational costs and creating a profit for the company running the lottery. So there is no way you could ever get out more than you put in – like any gambling-based activity, if someone is making money out of running it, you can always expect to lose out overall.

This does not mean there is no hope at all, though: people do still win the lottery, and net huge amounts of money. With any gambling game, it is always a possibility that you will walk away with more money than you spent – if you are lucky. While you cannot control fortune, it is possible to change the way these probabilities interact, mathematically, by making careful choices.

For example, when playing roulette, your options include betting on a particular colour (around half the slots on the wheel) or betting on a number (one of the slots); you win a lot more money for correctly matching a single number, but you can control through this choice based on whether you would prefer to have a higher chance of winning a bit of money, or a lower chance to win a lot – you can think of this as ‘concentrating’ the probabilities in a way that suits you.

The good news is, if all you care about is the buzz of winning, some mathematicians have now found a way to guarantee this for you. Some researchers at Manchester University recently released a paper entitled “You need 27 tickets to guarantee a win on the UK National Lottery”, which uses clever combinatorial arguments to suggest a small set of tickets which between them will definitely match at least two numbers, winning you a small prize under the current system (a free random play on the following lottery).

The system involves using a clever piece of mathematics called **finite geometry** – I have discussed it here previously when talking about **block designs**. These are structures that include all the possible combinations of objects from a given set of sets – in my previous post, this was about arranging objects with given properties into groups. The diagram below shows these structures represented as diagrams – in the bottom three, which are a structure called the **Fano Plane**, each line contains three points, and each point lies on three of the lines (one of which is not a straight line, but we will forgive it).

Having assigned two numbers to each of the points, buying a ticket with the six numbers attached to each line, and creating planes that cover all the numbers, guarantees that for the six numbers in the lottery draw, one of your tickets definitely will contain at least a pair. The other two diagrams are included to make sure all 59 of the UK lottery’s numbers are covered – 3 lots of 14 numbers on Fano planes makes 42, and we could just use a fourth to include the rest. In this case, the remaining numbers are instead arranged in simpler structures to let you buy the minimal number of tickets – proven in the paper to be 27.

You can of course choose to assign the numbers differently to the way it has been done in the diagram – and it is probably a good idea to do this, since the paper has been published and I imagine there are a non-trivial number of people using the numbers suggested in the paper, so – as we have seen – avoiding those specific ones will decrease your chances of sharing a prize. (For a more in-depth explanation, Matt Parker has made a video, in which he goes and buys the 27 tickets for one week’s draw – you will have to watch to see what happens.)

The problem here is again that you should not expect massive returns: your “win” might just be a free extra play (which is notionally worth another £2, but as we have discussed, is unlikely to be worth much more than that) – and even matching three numbers, which is not guaranteed here, only nets you £30. Since this is an example of a way to manipulate and combine probabilities, guaranteeing yourself a minimum of a two-number match will slightly reduce your chances of making a three-number match if you play these tickets instead of 27 random draws (around 23.5% instead of the usual 24.4%).

Peter Rowlett at The Aperiodical did some quick simulations, creating all 45 million possible lottery draws and counting which ones would win you enough to cover your initial investment (£54, for 27 tickets at £2 each). The probability of making a profit, based on the proportion of all possible draws where this happened, is less than 5% – and in around 75% of draws the only win was a match of two numbers. So maybe this is not a guaranteed path to fortune – but it is an interesting bit of maths!

The post How to Guarantee You Win the Lottery originally appeared on the HLFF SciLogs blog.

]]>The post Posters of Progress: All Eyes on the Young Researchers Taking the Stage originally appeared on the HLFF SciLogs blog.

]]>While the laureates are the main attraction of the Heidelberg Laureate Forum (HLF), the young researchers are the heart and soul of the event. It is only fitting that during the first day of this year’s 10th HLF, young researchers took to the stage to present some of their own work.

From Ishika Ghosh, who described her use of complex topology to unravel the secrets of tree leaf shapes, to Yasra Chandio who delved into the rapidly evolving field of mixed reality (merging the tactile, physical world with intricate computer-generated elements), young researchers got a few minutes to explain their specialized work and try to convey its broader relevance. The session was followed by a poster session in which the audience was invited to visit the posters and discuss them in more detail.

Presentations ranged from pure mathematics to applied computer science and everything in between and there was no shortage of intriguing topics. Using mathematics and computer science was a common thread between presentations, but pure science was also given center stage.

What do snakes, jellyfish, and astronauts have in common? You could say that they are all fascinating creatures and all interesting in their own way, but Oliver Karl Klaus Gross sees something else. They all use specific sequences of poses in order to generate motion, Gross mentions. He models motion with the help of the fiber bundle of positioned shapes. There is a geometric formulation of this phenomenon, which he used to show that in many scenarios, physical motion is actually a horizontal lift. Modeling motion is an important emerging area of research with applications in different fields. In robotics, for instance, motion modeling helps in planning and controlling movements. Robots rely on accurate motion models to perform tasks like picking up objects, moving around, and even performing surgeries. Plus, you get to model an armadillo springboard diving, which is an excellent perk.

As if drawing inspiration from Gross’ mathematical models of motion, Sascha Robert Gaudlitz dove into the equally fascinating world of complex mathematical equations. Gaudlitz gave a mini-masterclass on communicating mathematics. He works on what is arguably one of the most challenging aspects in mathematics: modeling complex, stochastic, spatiotemporal phenomena with partial differential equations. How does one present that in a couple minutes? Well, “think of them as being a mathematical recipe,” the young researcher explained. With any kind of recipe, you have the ingredients you work with and the cooking techniques that you need to follow. His work is somewhat akin to that, except you start from the meal and see whether you can infer the ingredients and cooking procedures just by tasting it. After all, who said mathematics is not delicious?

Science is not necessarily always practical. Nevertheless, it is just fantastic when it improves something about our society. Several young researchers focused exactly on that.

For instance, Mohak Chada works on sustainable computing. If you think this is not a problem, you should probably know that data centers already consume over 1% of global energy and are projected to consume 8-13% of global energy by 2030. That means that if data centers were a country, they would rank the top 10 energy consumers. So making this type of computing more sustainable is essential for our transition to a sustainable future. This is one of the reasons Chada is working on something called GreenCourier.

GreenCourier takes a new cloud computing approach called serverless computing. In serverless computing, users are only responsible for writing small pieces of code called functions while all infrastructure management is handled by the cloud service provider. This reduces carbon emissions through an intelligent scheduling policy that schedules serverless functions depending on their carbon-efficiency. Chada found that this approach can significantly reduce carbon emissions for data centers.

Life science and healthcare also took the stage. Catherine Chen works on how the human brain represents semantic information from different languages. She found that the cortical representation of lexical semantics is shared between English and Chinese. This is somewhat surprising, because if you look at clinical or behavioral data, you get contrasting evidence regarding how semantics is processed in different languages. Chen and her colleagues gave participants some stories to listen to and then carried out fMRI scans of their brains. From that, they developed computational models to compare brain representations of lexical semantics, finding a common cortical representation across these different languages.

Laura Margaret Stegner worked on something else: robots; specifically, care robots. Many countries are facing an aging population and there is a growing need for assisted care. At the same time, the work force in this field is steadily decreasing in many countries. Can robots be of assistance? If so, how do we integrate robots into these complex and delicate ecosystems? These are the questions Stegner is working on. She conducted design studies to understand each stakeholders’ need and envision what systems are necessary to successfully integrate robots into existing workflows and habits. She found that the system needs to support quick, on-the-fly inputs because caregivers often have hectic and dynamic schedules. At the same time, the robots should support simple, natural inputs to encourage the assisted adults to use it. Because the care environment and the necessity can change rapidly, the robot must also exhibit context awareness and potentially even modify the originally specified task. These are not simple tasks, but Stegner believes that end-user development (EUD) could be a tool to aid these tasks, personalizing care robots’ behaviors and actions. These tools use a combination of AI and formal methods and could help alleviate a very real problem.

As the session moved downstairs for the poster discussion, one thing was abundantly clear: The young researchers present are not just the future – they are the ‘now’. They are tackling complex questions that span across disciplines, from pure mathematics and environmental sustainability to healthcare and human cognition. They are pushing the boundaries of what is known and, perhaps more importantly, challenging the ways we approach these problems.

Their projects have demonstrated how impactful interdisciplinary research can be, but also how important it is to have a solid theoretical foundation.

As they take their research from the halls of the Heidelberg Laureate Forum into the wider world, we can only wait in eager anticipation for the breakthroughs that are sure to come.

The post Posters of Progress: All Eyes on the Young Researchers Taking the Stage originally appeared on the HLFF SciLogs blog.

]]>The post Molecule of the Year 1992 originally appeared on the HLFF SciLogs blog.

]]>At the 10th HLF in September 2023, Nobel Prize recipient and pharmacologist Louis J. Ignarro held the Lindau Lecture. This year marked 25 years since he received the Nobel Prize in Physiology or Medicine for discovering the role of nitric oxide as a cardiovascular signalling module. So what is nitric oxide? And what does it do?

Nitric oxide is a molecule with the chemical formula NO. It’s formed by a nitrogen and an oxygen atom, joined by a double covalent bond, with one unpaired electron.

In chemistry, molecules with an unpaired electron are called “free radicals”. Radicals are highly unstable, and do not naturally exist for long. Indeed, it has been shown that the half life of NO in an aqueous solution due to the reaction with oxygen is less than 10 seconds, i.e. after 10 seconds you can expect at least half of a sample of NO to have reacted with oxygen to become NO_2.

The fact that NO so readily oxidises to NO_2 is a problem. When released into the atmosphere, NO_2 reacts with OH radicals already there to form HNO_3, also known as nitric acid. This is one form of acid rain, hence NO_2 is a major air pollutant.

It may therefore also not be surprising to learn that NO_2 can be lethal to humans, but we will discuss this more later. At this point, it should be noted that nitric oxide (NO) is not to be confused with nitrous oxide (N_2O), an anaesthetic gas.

Though nitric oxide has some uses outside of biology (for example, it is sometimes used in catalytic converters), it’s safe to say that it’s most celebrated for its use is medicine and physiology.

What’s largely considered to be the most important function of NO is vasodilation i.e. the widening of blood vessels. Ignarro was put onto this idea when he spotted how samples in his lab reacted in the presence of his heavily smoking colleagues. He correctly surmised that something in the cigarettes was causing arterial tissue to relax and set about finding out which compound was responsible.

The process behind how NO causes vasodilation is fairly complicated but essentially it causes a reaction in which a phosphoryl (PO_3) group is added to several proteins in a process known as phosphorylation. This causes smooth muscle to relax, resulting in vasodilation.

Vasodilation causes increased blood flow and a drop in blood pressure, and as such, NO has many uses in a medical setting. One condition that it is used to treat is pulmonary hypertension, i.e. increased blood pressure in the arteries of the lungs. It can also improve hypoxemia (low oxygen levels) in some types of respiratory failure including acute respiratory distress syndrome. There are even trials underway investigating if NO can be used to treat COVID-19.

Another condition that can be treated with NO is angina – chest pain caused by reduced blood flow to the heart. Vasodilation lowers the pressure in arteries and also lowers left ventricular filling pressure, which has the effect of reducing cardiac workload.

A corollary of NO being an effective treatment of heart disease is that some heart diseases are caused by issues in synthesising NO. It was this that first drew Louis Ignarro’s attention to the possibility of such a molecule existing. As a child in high school, he observed how much more some people seemed to suffer from cardio-vascular disease, compared to others. He wondered whether there was a molecule that could protect from the diseases, but wasn’t until he was older that he proved this hypothesis.

Increased oxygen flow has obvious benefits in sport. which raises questions about whether or not nitric oxide could be used for doping. At the time of writing, nitric oxide is not on any WADA (World Anti-Doping Agency) prohibited list. A possible explanation for this is that it is actually very difficult to administer large quantities of NO, and the current methods are not suitable for sport.

You may recall that NO is present in cigarettes, and in the very short term there is anecdotal evidence that asthma patients find that cigarettes relieve their symptoms. However, there are many compounds aside from nitric oxide, some of which make asthma worse over time, and some of which are carcinogens. So cigarettes are not a good idea for asthma patients, athletes, or anyone else.

Nitric oxide can be inhaled but pure NO would be too concentrated by orders of magnitude. NO readily oxidises to NO2 which is lethal to humans so it needs to be administered in a hospital setting. A solution to this is to use very low concentrations of NO, which react very slowly with O2 and form negligible quantities of NO2, but caution is still advised here.

NO can also be administered via nitroglycerine, but these methods both have their issues too. Nitroglycerine is an explosive and hence must be handled with care. Even without this though, this is still a risky method. If vasodilation occurs too quickly (which can happen when nitroglycerine is given) then the patient’s blood pressure might drop too fast. This could cause the patient to pass out, and reduce blood flow to the heart and brain which has the potential to cause serious damage.

Fear not though! There are safe ways to increase nitric oxide in the human body. Nitric oxide is naturally produced in mammals by cells so rather than artificially introducing NO to the body, it’s possible to aid the body in producing more NO.

One really simple way of increasing your NO production is breathing through your nose. This is because some of the cells producing nitric oxide are located in the nose. This is one reason why athletes are encouraged to breathe through their nose as much as possible.

Nitric oxide is produced from nitrates, which can be found in some vegetables. By eating more leafy greens, such as spinach, lettuce, cress and rocket (or arugula if you prefer), one can increase their dietary intake of nitrates. Some studies have indicated that eating vegetables rich in nitrates can be as effective at lowering blood pressure as some medications.

Another dietary change that can increase NO production is eating more foods rich in antioxidants. Anti-oxidants do what they say on the tin – they reduce oxidation. This increases the half-life of nitric oxide in the body, and so increases its concentration. Examples of anti-oxidants include vitamin C and vitamin E, hence a good source of antioxidants are plant-based foods, such as carrots, citrus fruits and blueberries. Dark chocolate also contains a wide variety of anti-oxidants, and if you’re anything like me, it also makes you very happy.

Louis Ignarro was awarded the Nobel Prize in 1998, along with Robert Furchgott and Ferid Murad. Their breakthrough papers about how NO caused vasodilation were all published in the 1970s and early 1980s. So what caused the Nobel committee to award them their prizes so much later? Ignarro has a theory.

In the 1990s, Ignarro showed how nitric oxide acts as a neurotransmitter mediating the erection of the penis. It does this by causing the accumulation of a molecule known as cyclic GMP (or cGMP), which in turn causes vasodilation and increased blood flow to the penis.

This discovery got a lot of attention. It should also be noted that NO plays a similar role in the erection of the clitoris, though it’s perhaps a measure of the world that we live in that this received significantly less attention.

On 9 January 1992, the front page of the New York Times included a short article with the heading “Chemical A Factor in Male Impotence”. Newspapers all around the world ran similar features, with one newspaper in Italy even including a cartoon!

By 1998, this discovery had led to the production of a drug to treat erectile dysfunction. This drug is called sildenafil, but you’re more likely to recognise the brand name: Viagra. Viagra doesn’t itself produce NO but instead enhances its effect as a neurotransmitter. Viagra inhibits a molecule known as PDE-5, which normally breaks down cGMP. As a result, there is a noticeable increase in cGMP accumulation in response to even a small amount of NO.

Interestingly, Viagra’s ability to cause vasodilation means that it can sometimes be taken to treat pulmonary arterial hypertension. It’s even been suggested that it could be used to treat some symptoms of Reynaud’s phenomenon. You probably know someone who suffers from Reynaud’s phenomenon – sufferers experience reduced blood flow to the end of arterioles, in particular extremities such as fingers and toes, which can cause the skin to turn white and then blue.

Because Viagra works by promoting vasodilation though, care must also be taken. Those taking nitroglycerine, as mentioned before, or indeed any other medication to increase nitric oxide, risk experiencing a rapid drop in blood tension, with serious consequences.

You’d be right if you’ve guessed – perhaps from the sheer number of applications – that Ignarro’s work on NO wasn’t plain sailing. Ignarro faced a lot of failure throughout the course of his research. It was through his conviction that he was onto something that he was able to push through and keep his enthusiasm up.

It will be clear by now how important nitric oxide is. It may therefore to surprise you that Louis Ignarro faced serious pushback. Before the work of Ignarro, Furchgott and Murad, nitric oxide had a reputation as a pollutant. Other scientists discouraged Ignarro from pursuing his work, not believing it would come to anything. Once again, it was the strength of Ignarro’s belief that his work would yield results that allowed him to not be deterred.

The Lindau lecture is unique in the Heidelberg Laureate Forum academic program in that it is not directly related to mathematics or computer science. Despite this, it still proves to be a great inspiration to young researchers as the experience of research has similarities across all disciplines. Ignarro’s talk demonstrated this excellently.

Louis Ignarro first began thinking about the work that led to his Nobel Prize as a child, something that many mathematicians and computer scientists will relate to. One need look no further than Andrew Wiles’ work on Fermat’s Last Theorem to see a famous example of a mathematician discovering a problem at a young age and chipping away until they’ve solved it.

Medicine, like mathematics and computer science, can also be very collaborative. Louis Ignarro won his Nobel Prize alongside two other scientists (Robert Furchgott and Ferid Murad). Similarly, a large number of maths and computer science papers have several authors. In fact, 2022 Fields medallist Hugo Duminil-Copin spoke in his HLF press conference about how every single paper that earned him his Fields Medal had multiple authors.

One final similarity that Ignarro’s work has to maths and computer science is how something designed for one application can have far-reaching consequences. Just as number theorists had no idea that their work on prime numbers would form the basis of modern cryptography, Louis Ignarro could not have predicted that his research would eventually be used to help combat a global pandemic.

These are just some of the insights that young researchers will have gained from listening to Louis Ignarro. What makes Ignarro’s work so inspiring, though, is the roadblocks he overcame along the way. Just like Ignarro, the young researchers may well have to keep going through failure and persevere in the face of others not seeing the value of their work. Here, they could learn a lot from Ignarro. In his lecture, Louis Ignarro shared a quote by Ralph Waldo Emerson: “Do not go where the path may lead, go instead where there is no path and leave a trail”. This sums up his philosophy very well, and to a room of young scientists about to start their careers, Louis Ignarro’s advice couldn’t have been better.

The post Molecule of the Year 1992 originally appeared on the HLFF SciLogs blog.

]]>The post The Man Who Wired the World: Robert Metcalfe and Connectivity originally appeared on the HLFF SciLogs blog.

]]>Global connectivity has come a long way in the past couple of decades. It can be hard to imagine as you read this on your smartphone or laptop, connected to a 4G or 5G network, but there was a time, not that long ago, when the internet and high speed data transfer seemed like a pipe dream. In our modern times, we are just a few clicks away from most of human knowledge, and we can reach our friends and family in text, audio, or video. But we should not take the openness of the internet for granted.

In fact, Metcalfe says, the story of global connectivity begins with monopolies. To get an idea of how connectivity used to work in his youth, Metcalfe humorously recalls what his mother told him when he left for college. “When you get to Boston, Bobby, call us to let us know you’re okay. We’ll let it ring three times to know you’re okay.” Beyond the humorous intent lies an important point: In that time, even the cost of a phone call was significant. Metcalfe is one of the people that changed that and brought us to where we are today.

However, his path was rarely straightforward.

His first advisor, Marvin Minsky (also a laureate in computer science), did not like his initial work. Metcalfe had to change paths, and eventually started work on computer connectivity, where his work would go on to change the world.

Metcalfe recalled how in the early stages of his work, he was confronted with a problem. Inventors wanted to put “a computer on every desk.” It seemed outrageous at the time. But for Metcalfe, the problem was how to get these computers to “talk” to each other and exchange information in a practical way.

He would go on to invent and develop Ethernet technology. Yet, the journey to Ethernet’s success also had its share of complexities.

Ethernet is a foundational technology for local area networks (LANs) that enables devices within the same physical location to communicate with each other. Ethernet provides a set of protocols and standards for how data packets should be placed on the network and transported between computers, employing a combination of hardware and software to facilitate fast, reliable data transfers. Ethernet laid the groundwork for modern networking and has had a transformative impact on everything from education and technology to healthcare. It enabled the advent of the World Wide Web, dial-up internet, and eventually, broadband.

Before all that, though, there was ALOHAnet.

ALOHAnet, short for “Additive Links On-line Hawaii Area,” was an experimental wireless network created at the University of Hawaii. It became operational in 1971. In its purest form, ALOHAnet used remote units communicated with a base station over two separate radio frequencies (for inbound and outbound transmission respectively). It was the first public demonstration of a wireless packet data network.

Metcalfe was looking for a way to have PCs take turns communicating on a shared cable in Ethernet. He learned that radio (which ALOHA used) is not a suitable technology for a local area network, but he did use the ALOHA random-access techniques and he incorporated them in Ethernet.

Other technical innovations, like replacing the original coaxial cable with a flexible cable gradually made Ethernet more appealing. In 1976, when Metcalfe published a seminal paper discussing the advantages and potential of Ethernet, the protocol was already functional.

But for a time, it was not clear whether Ethernet would become the dominant protocol.

The creation of Ethernet was an innovative leap in packet-mode communication. But it was not the only competing protocol.

Other tech companies developed their own protocols as well. “But we won,” recalls Metcalfe.

Ethernet became established for several reasons: “For starters, it worked,” says Metcalfe – which is a pretty important part of why things become popular. Specifically, it was fast. Ethernet offered speeds that were approximately 10,000 times faster than its predecessors, dramatically outperforming competing protocols in both speed and reliability.

It was also open. Ethernet is an open standard, which spurred innovation and collaboration. It is only fitting that the most common standard for computer connectivity also favors human connectivity.

As computers and the internet have changed, Ethernet has managed to keep up the pace. For instance, video accounts for over 80% of the internet traffic and the protocol was never designed to work with this type of load. Nevertheless, subsequent updates and improvements have helped Ethernet keep up with the changes.

An interesting segment of Metcalfe’s talk was devoted to Metcalfe’s Law, which states that the value of a network increases proportionally with the square of the number of connected users. In simple terms, the more people and devices that are connected, the more valuable the network becomes. This insight has implications far beyond computing, affecting social networks, online marketplaces, and even political movements.

Though optimistic about the potential of universal connectivity, Metcalfe did not shy away from addressing its challenges. These include issues related to disinformation, security, and even hacking. The internet has faced various “pathologies,” as Metcalfe calls them, but the laureate is confident that while technology can create these problems, it can also be instrumental in solving them. Metcalfe seems optimistic about the future of connectivity, even mentioning the Voyager spacecraft as an extreme example of how far connectivity has come.

It was a fitting start to the HLF. Metcalfe is best known as the inventor of the Ethernet technology, but his own path is a testament to connectivity between different worlds. Throughout his career, Metcalfe has been an academic, inventor, entrepreneur, and pundit. His path has shown how these different worlds can be connected through our actions. Of course, connectivity also lies at the core of the HLF. This event is all about the connections we make here.

As the audience left the auditorium, it was clear that Metcalfe’s words had struck a chord. The room was buzzing with talk of past innovations and future possibilities. It was a powerful reminder that connectivity is not just about hardware and software; it is about human relationships and how they shape our world.

The post The Man Who Wired the World: Robert Metcalfe and Connectivity originally appeared on the HLFF SciLogs blog.

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