## Archive for the ‘life’ Category

### My COVID-19 vaccination, part 2

August 5, 2021

Since my last post on this subject a few things have changed. In Germany 53% of people are fully vaccinated against COVID-19, which is good news. We are now in a situation where in this country any adult who wants the vaccination can get it. Of course this percentage is still a lot lower than what is desirable and the number of people being vaccinated per day has dropped to less than half what it was in mid June. I find it sad, and at the same time difficult to understand, that there are so many people who are not motivated enough to go out and get the vaccination.

Yesterday my wife and I got our second vaccination. In the meantime the relevant authority (STIKO) has recommended that those vaccinated once with the product of AstraZeneca should get an mRNA vaccine the second time. The fact that we waited the rather long time suggested to get our second injection meant that the new recommendation had already come out and we were able to get the vaccine of Biontech the second time around. There have not been many studies of the combination vaccination but as far as I have seen those that there are gave very positive results. So we are happy that it turned out this way. This time the arm where I got the injection was sensitive to pressure during the night but this effect was almost gone by this morning. The only other side effect I noticed was an increased production of endorphins. In other words, I was very happy to have reached this point although I know that it takes a couple of weeks before the maximal protection is there.

Every second year there is an event in Mainz devoted to the popularization of science called the Wissenschaftsmarkt. It has been taking place for the last twenty years. Normally it is in the centre of town but due to the pandemic it will be largely digital this year. This year it is on 11th and 12th September and has the title ‘Mensch und Gesundheit’ [rough translation: human beings and their health]. I will contribute a video with the title ‘Gegen COVID-19 mit Mathematik’ [against COVID-19 with mathematics]. The aim of this video is to explain to non-scientists the importance of mathematics in fighting infectious diseases. I talk about what mathematical models can contribute in this domain but also, which is just as important, about what they cannot do. If the public is to trust statements by scientists it is important to take measures against creating false expectations. I do not go into too much detail about COVID-19 itself since at the moment there is too little information available and too much public controversy. Instead I concentrate on an example from long ago where it is easier to see clearly. It also happens to be the example where the basic reproductive number was discovered. This is the work of Ronald Ross on the control of malaria. Ross was the one who demonstrated that malaria is transmitted by mosquito bites and he was rewarded for that discovery with a Nobel Prize in 1902. After that he studied ways of controlling the disease. This was for instance important in the context of the construction of the Panama Canal. There the first attempt failed because so many workers died of infectious diseases, mainly malaria and yellow fever, both transmitted by mosquitos. The question came up, whether killing a certain percentage of mosquitos could lead to a long-term elimination of malaria or whether the disease would simply come back. Ross, a man of many talents, set up a simple mathematical model and used it to show that elimination is possible and was even able to estimate the percentage necessary. This provided him with a powerful argument which he could use against the many people who were sceptical about the idea.

### Lisa Eckhart and her novel Omama

October 2, 2020

Let me finally come to the novel itself. It struck me as a curate’s egg. Parts of it are very good. There are passages where I appreciate the humour and I find the author’s use of language impressive. On a more global level I do not find the text attractive. It is the story of the narrator’s grandmother. (Here is a marginal note for the mathematical reader. Walter Rudin, known for his analysis textbooks, was born in Austria. In  a biographical text about him I read that one of his grandmothers was referred to as ‘Omama’.) The expressions are often very crude, with a large dose of excrement and other unpleasant aspects of the human body, and many elements of the story seem to me pointless. There is no single character in the novel who I find attractive. This is in contrast to the novel of Banine which I previously wrote about, where I find the narrator attractive. That novel also contains plenty of crude expressions but there are more than enough positive things to make up for it. I would like to emphasize that just because I find a novel unpleasant to read it does not mean I judge it negatively. A book which I found very unpleasant was ‘Alexis ou le traite du vain combat’ by Marguerite  Yourcenar but in that case my conclusion was that it could only be so unpleasant because it was so well written. I do not have the same feeling about Omama. As to the insight which I hoped I might get for Eckhart’s stage performances I have not seen it yet, but maybe I will notice a benefit the next time I experience a stage performance by her.

### The plague priest of Annaberg

March 26, 2020

I find accounts of epidemics, whether documentary or fictional, fascinating. I appreciated texts of this kind by Camus (La Peste), Defoe (Journal of the Plague Year) and Giono (Le Hussard sur le Toit). This interest is reflected in a number of posts in this blog, for instance this one on the influenza pandemic of 1918. At the moment we all have the opportunity to experience what a pandemic is like, some of us more than others. In such a situation there are two basic points of view, depending on whether you see the events as concerning other people or whether you feel that you are yourself one of the potential victims. The choice of one of these points of view probably does not depend mainly on the external circumstances, except in extreme cases, and is more dependent on individual psychology. I do feel that the present COVID-19 pandemic concerns me personally. This is because Germany, where I live, is one of the countries with the most total cases at the moment, after China, Italy, USA and Spain. Every evening I study the new data in the Situation Reports of the WHO. The numbers to be found in the Internet are sometimes quite inconsistent. This can be explained by the time delays in reporting, the differences in the definitions of classes of infected individuals used by different people or organizations and unfortunately in some cases by poltically motivated lies. My strategy for extracting real information from this data is to stick to one source I believe to be competent and trustworthy (the WHO) and to concentrate on the relative differences between one day and the next and one country and another in order to be able to see trends. I find interesting the extent to which diagrams coming from mathematical models have found their way into the media reporting of this subject. Prediction is a high priority for many people at the moment.

Motivated by this background I started to read a historical novel by Gertrud Busch called ‘Der Pestpfarrer von Annaberg’ [the plague priest of Annaberg] which I got from my wife. The main character in the book is a person who really existed but many of the events reported there are fictional. Annaberg is a town in Germany, in the area called ‘Erzgebirge’, the literal English translation of whose name is ‘Ore mountains’. This mountain range lies on the border between Germany and the Czech republic. People were attracted there by the discovery of valuable mineral deposits. In particular, starting in the late fifteenth century, there was a kind of gold rush there (Berggeschrey), with the difference that the metal which caused it was silver rather than gold. My wife was born and grew up in that area and for this reason I have spent some time in Annaberg and other places close to there. The narrator of the book is Wolfgang Uhle, a priest in the Erzgebirge in the sixteenth century active in Annaberg during the outbreak of plague there. In fact in the end only a small part of the book concerns the plague itself but I am glad I read it. The author has created a striking picture of the point of view of the narrator, at a great distance from the modern world.

During Uhle’s first period as a priest there was a fire in a neighbouring village which destroyed many houses. He saved the life of a young girl, in fact a small child, who was playing in a burning house. Much to the amusement of the adults the girl said she would marry him when she was old enough. In fact she meant it very seriously and when she was old enough it did happen that after some difficulties she got engaged to him. The tragedy of Wolfgang Uhle is that he had a temper which was sometimes uncontrollable. Before the marriage took place he once got into a rage due to the disgraceful behaviour of the judge in his village. Unfortunately at that moment he was holding a large hammer in his hand. A young girl had asked him if a stone she had brought him was valuable. He had some knowledge of geology and he intended to use the hammer to break open the stone and find out more about its composition. In his sudden rage he hit the judge on the head with the hammer and killed him. He went home in a state of shock without any plan but his housekeeper brought him to flee over the border into Bohemia. He was sentenced to death in absentia and hid in the woods for five years. The girl who he was engaged to repudiated him, stamped on his engagement ring and quickly married another man. He partly lived from what he could find in nature, living at first in a cave. Later he started working together with a charcoal burner. I learned something about what that industry was like when I visited those woods myself a few years ago. Eventually he revealed his identity and had to leave.

In the woods he met a man who had got lost and asked him the way. The man wanted to go to Bärenstein, which is the town where my wife spent her childhood. He agreed to show him the way. The man told him that the plague had broken out in Annaberg and that the town was desperately searching for a priest to tend to the spiritual needs of the sick. Uhle decided that he should volunteer, despite the danger. He saw this as God giving him a chance to make amends for his crime. He wrote letters to the local prince and the authorities of the town. The prince agreed to grant him a pardon in return for his service as priest for the people infected with the plague. He then went to Annaberg and tended to the sick, without regard to the danger he was putting himself in. There is not much description of the plague itself in the book. There is a key scene where he meets his former love on her deathbed and it turns out that she had continued to love him and felt guilty for having abandoned him. Uhle survives the plague, gets a new position as a priest, marries and has children. This book was different from what I expected when I started reading it. Actually the fact that it was so different from things I otherwise encounter made it worthwhile for me to read it.

### Cedric Villani’s autobiography

November 1, 2019

I have just read Cédric Villani’s autobiographical book ‘Théorème Vivant’. I gave the German translation of the book to Eva as a present. I thought it might give her some more insight into what it is like to be a mathematician and give her some fortitude in putting up with a mathematician as a husband. Since I had not read the book before I decided to read it in parallel. I preferred to read the original and so got myself that. With hindsight I do not think it made so much difference that I read it in French instead of German. I think that the book is useful for giving non-experts a picture of the life of a mathematician (and not just that of a mathematician who is as famous as Villani has become). For this I believe that it is useful that the book contains some pieces of mathematical text which are incomprehensible for the lay person and some raw TeX source-code. I think that they convey information even in the absence of an understanding of the content. On the other hand, this does require a high level of tolerance on the part of the reader. Fortunately Eva was able to show this tolerance and I think she did enjoy the book and learn something more about mathematics and mathematicians.

For me the experience was of course different. The central theme of the book is a proof of Villani and Clement Mouhot of the existence of Landau damping, a phenomenon in plasma physics. I have not tried to enter into the details of that proof but it is a subject which is relatively close to things I used to work on in the past and I was familiar with the concept of Landau damping a long time ago. I even invested quite a lot of time into the related phenomenon of the Jeans instability in astrophysics, unfortunately without significant results. Thus I had some relation to the mathematics. It is also the case that I know many of the people mentioned in the book personally. Sometimes when Villani mentions a person without revealing their name I know who is meant. As far as I remember the first time I met Villani was at a conference in the village of Anogia in Crete in the summer of 2001. At that time he struck me as the number one climber of peaks of technical difficulty in the study of the Boltzmann equation. I do not know if at that time he already dressed in the eccentric way he does today. I do not remember anything like that.

For me the book was pleasant to read and entertaining and I can recommend it to mathematicians and non-mathematicians. If I ask myself what I really learned from the book in the end then I am not sure. One thing it has made me think of is how far I have got away from mainstream mathematics. A key element of the book is that the work described there got Villani a Fields medal, the most prestigious of mathematical prizes. These days the work of most Fields medallists is on things to which I do not have the slightest relation. Villani was the last exception to that rule. Of course this is a result of the general fact that communication between different mathematical specialities is so hard. The Fields medal is awarded at the International Congress of Mathematicians which takes place every four years. That conference used to be very attractive for me but now I have not been to one since that in 2006 in Madrid and I imagine that I will not go to another. That one was marked by the special excitement surrounding Perelman’s refusal of the Fields medal which he was offered for his work on the Poincaré conjecture. Another sign of the change in my orientation is that I am no longer even a member of the American Mathematical Society, probably the most important such society in the world. I will continue to follow my dreams, whatever they may be. Villani is also following his dreams. I knew that he had gone into politics, becoming a member of parliament. I was surprised to learn that he has recently become a candidate for the next election to become mayor of Paris.

### Science as a literary pursuit

August 24, 2019

I found something in a footnote in the book of Oliver Sacks I mentioned in the previous post which attracted my attention. There is a citation from a letter of Jonathan Miller to Sacks with the idea of a love of science which is purely literary. Sacks suggests that his own love of science was of this type and that that is the reason that he had no success as a laboratory scientist. I feel that my own love for science has a strong literary component, or at least a strong component which is under the control of language. In molecular biology there are many things which have to be named and people have demonstrated a lot of originality in inventing those names. I find the language of molecular biology very attractive in a way which has a considerable independence from the actual meaning of the words. I expect that there are other people for whom this jungle of terminology acts as a barrier to entering a certain subject. In my case it draws me in. In my basic field, mathematics, the terminology and language is also a source of pleasure for me. I find it stimulating that everyday words are often used with a quite different meaning in mathematics. This bane of many starting students is a charm of the subject for me. Personal taste plays a strong role in these things. String theory is another area where there is a considerable need for inventing names. There too a lot of originality has been invested but in that case the result is not at at all to my taste. I emphasize that when I say that I am not talking about the content, but about the form.

The idea of using the same words with different meanings has a systematic development in mathematics in context of topos theory. I learned about this through a lecture of Ioan James which I heard many years ago with the title ‘topology over a base’. What is the idea? For topological spaces $X$ there are many definitions and many statements which can be formulated using them, true or false. Suppose now we have two topological spaces $X$ and $B$ and a suitable continuous mapping from $X$ to $B$. Given a definition for a topological space $X$ (a topological space is called (A) if it has the property (1)) we may think of a corresponding property for topological spaces over a base. A topological space $X$ over a base $B$ is called (A) if it has property (2). Suppose now that I formulate a true sentence for topological spaces and suppose that each property which is used in the sentence has an analogue for topological spaces over a base. If I now interpret the sentence as relating to topological spaces over a base under what circumstances is it still true? If we have a large supply of statements where the truth of the statement is preserved then this provides a powerful machine for proving new theorems with no extra effort. A similar example which is better known and where it is easier (at least for me) to guess good definitions is where each property is replaced by one including equivariance under the action of a certain group.

Different mathematicians have different channels by which they make contact with their subject. There is an algebraic channel which means starting to calculate, to manipulate symbols, as a route to understanding. There is a geometric channel which means using schematic pictures to aid understanding. There is a combinatoric channel which means arranging the mathematical objects to be studied in a certain way. There is a linguistic channel, where the names of the objects play an important role. There is a logical channel, where formal implications are the centre of the process. There may be many more possibilities. For me the linguistic channel is very important. The intriguing name of a mathematical object can be enough to provide me with a strong motivation to understand what it means. The geometric channel is also very important. In my work schematic pictures which may be purely mental are of key importance for formulating conjectures or carrying out proofs. By contrast the other channels are less accessible to me. The algebraic channel is problematic because I tend to make many mistakes when calculating. I find it difficult enough just to transfer a formula correctly from one piece of paper to another. As a child I was good in mental arithmetic but somehow that and related abilities got lost quite early. The combinatoric channel is one where I have a psychological problem. Sometimes I see myself surrounded by a large number of mathematical objects which should be arranged in a clever way and this leads to a feeling of helplessness. Of course I use the logical channel but that is usually on a relatively concrete level and not the level of building abstract constructs.

Does all this lead to any conclusion? It would make sense for me to think more about my motivations in doing (and teaching) mathematics in one way or another. This might allow me to do better mathematics on the one hand and to have more pleasure in doing so on the other hand.

### Encounter with an aardvark

August 21, 2019

When I was a schoolboy we did not have many books at home. As a result I spent a lot of time reading those which were available to me. One of them was a middle-sized dictionary. It is perhaps not surprising that I attached a special significance to the first word which was defined in that dictionary. At that time it was usual, and I see it as reasonable, that articles did not belong to the list of words which the dictionary was responsible for defining. For this reason ‘a’ was not the first word on the list and instead it was ‘aardvark’. From the dictionary I learned that an aardvark is an animal and roughly what kind of animal it is. I also learned something about its etymology (it was an etymological dictionary) and that it originates from Dutch words meaning ‘earth’ and ‘pig’. Later in life I saw pictures of aardvarks in books and saw them in TV programmes, but without paying special attention to them. The aardvark remained more of an intriguing abstraction for me than an animal.

Yesterday, in Saarbrücken zoo, I walked into a room and saw an aardvark in front of me. Suddenly the abstraction turned into a very concrete animal pacing methodically around its enclosure. I had a certain feeling of unreality. I do not know if aardvarks always walk like that or whether it was just a habit which this individual had acquired by being confined to a limited space. Each time it returned (reappearing after having disappeared into a region not visible to me) the impression of unreality was heightened. I was reminded of the films of dinosaurs which sometimes come on TV, where the computer-reconstructed movements of the animals look very unrealistic to me. Seeing the aardvark I asked myself, ‘if mankind only knew this animal from fossil remains would it ever have been possible to reconstruct the gait I now see before me?’

Another animal I encountered in the Saarbrücken zoo is a species whose existence I did not know of before. This is Pallas’s cat. This is a wild cat with a very unusual and engaging look. The name Pallas has a special meaning for me for the following reason. When I was young and a keen birdwatcher some of the birds which were most exciting for me were rare vagrants from Siberia which had been brought to Europe by unusual weather conditions. A number of these are named after Pallas. I knew almost nothing about the man Pallas. Now I have filled in some background. In particular I learned that he was a German born in Berlin who was sent on expeditions to Siberia by Catherine the Great.

### Light and lighthouses

June 3, 2019

I recently had the idea that I should improve my university web pages. The most important thing was to give a new presentation of my research. At the same time I had the idea that the picture of me on the main page was not very appropriate for attracting people’s attention and I decided to replace it with a different one. Now I have a picture of me in front of the lighthouse ‘Les Éclaireurs’ in the Beagle Channel, taken by my wife. I always felt a special attachment to lighthouses. This was related to the fact that as a child I very much liked the adventure of visiting uninhabited or sparsely inhabited small islands and these islands usually had lighthouses on them. This was in particular true in the case of Auskerry, an island which I visited during several summers to ring birds, especially storm petrels. I wrote some more about this in my very first post on this blog. For me the lighthouse is a symbol of adventure and of things which are far away and not so easy to reach. In this sense it is an appropriate symbol for how I feel about research. There too the goals are far away and hard to reach. In this context I am reminded of a text of Marcel Proust which is quoted by Mikhail Gromov in the preface to his book ‘Metric structures for Riemannian and non-Riemannian spaces’:

‘Même ceux qui furent favorables à ma perception des vérités que je voulais ensuite graver dans le temple, me félicitèrent de les avoir découvertes au microscope, quand je m’étais au contraire servi d’un télescope pour apercevoir des choses, très petites en effet, mais parce qu’elles étaient situées à une grande distance, et qui étaient chacune un monde’

[Even those who were favourable to my perception of the truths which I wanted to engrave in the temple, congratulated me on having discovered them with a microscope, when on the contrary I used a telescope to perceive things, in fact very small, but because they were situated at a great distance, and each of which was a world in itself.]

I feel absolutely in harmony with that text. Returning to lighthouses, I think they are also embedded in my unconscious. Years ago, I was fascinated by lucid dreams. A lucid dream usually includes a key moment, where lucidity begins, i.e. where the dreamer becomes conscious of being in a dream. In one example I experienced this moment was brought about by the fact of simultaneously seeing three lighthouses, those of Copinsay, Auskerry and the Brough of Birsay. Since I knew that in reality it is impossible to see all three at the same time this made it clear to me that I must be dreaming.

The function of a lighthouse is to use light to convey information and to allow people (seafarers) to recognise things which are important for them. Thus a lighthouse is a natural symbol for such concepts as truth, reason, reliability, learning and science. These concepts are of course also associated with the idea of light itself, that which allows us to see things. These are the elements which characterize the phase of history called the enlightenment. Sometimes I fear that we are now entering a phase which is just the opposite of that. Perhaps it could be called the age of obscurity. It is characterized by an increasing amount of lies, deceit, ignorance and superstition. Science continues its progress but sometimes it seems to me like a thin ray among gathering darkness. A future historian might describe the arch leading from the eighteenth to the twenty-first century. I recently watched a video of the Commencement speech of Angela Merkel in Harvard. In a way many of the things she said were commonplaces, nothing new, but listening to her speech and seeing the reactions of the audience it became clear to me that it is important these days to repeat these simple truths. Those of us who have not forgotten them should propagate them. And with some luck, the age of obscurity may yet be averted.

### Banine’s ‘Jours caucasiens’

April 11, 2019

I have just read the novel ‘Jours caucasiens’ by Banine. This is an autobiographical account of the author’s childhood in Baku. I find it difficult to judge how much of what she writes there is true and how much is a product of her vivid imagination. I do not find that so important. In any case I found it very interesting to read. It is not for readers who are easily shocked. Banine is the pen name of Umm-El-Banine Assadoulaeff. She was born in Baku into a family of oil magnates and multimillionaires. In fact she herself was in principle a multimillionaire for a few days after the death of her grandfather, until her fortune was destroyed when Azerbaijan was invaded by the Soviet Union. In later years she lived in Paris and wrote in French. To my taste she writes very beautifully in French. I first heard of her through the diaries of Ernst Jünger. While he was an officer in the German army occupying Paris during the Second World War he got to know Banine and visited her regularly. It was not entirely unproblematic for her during the occupation when she was visited at her appartment by a German army officer in uniform. She seemed to regard this with humour. The two had a close but platonic relationship.

The society in which Banine grew up was the result of the discovery of oil. Her ancestors had been poor farmers who suddenly became very rich because oil-wells were built on their land. She presents her family as being very uncivilised. They were muslims but had already been strongly affected by western culture. I found an article in the magazine ‘Der Spiegel’ from 1947 where ‘Jours caucasiens’ is described by the words ‘gehören zu den skandalösesten Neuerscheinungen in Paris’ [is one of the most scandalous new publications in Paris]. It also says that her family was very unhappy about the way they were presented in the book and I can well understand that. It seems that she had a low opinion of her family and their friends and the culture they belonged to, although she herself did not seem to mind being part of it. She was attracted by Western culture and Paris was the place of her dreams. As a child she had a German governess. Her mother died when she was very young and after her father had remarried she had a French and an English teacher for those languages. She quickly fell in love with French. On the other hand, she saw having to learn English as a bit of a nuisance. Her impression was that the English had just taken the words from German and French and changed them in a strange way.

After the Russian invasion Banine’s father, who had been a government minister in the short-lived Azerbaijan Republic, was imprisoned. He was released due to the efforts of a man whose motivation for doing so was the desire to marry Banine. She was very much against this. Perhaps the strongest reason was that he had red hair. There was a superstition that red-haired people, who were not very common in that region, had evil supernatural powers. Banine’s grandmother told her a story about an alchemist who discovered the secret of red-haired people. According to him they should be treated in the following way. He cut off their head, boiled it in a pot and put the head on a pedestal. If this was done correctly then the heads would start to speak and make prophecies which were always true. Banine could not help associating her potential husband with this horrible myth. Unfortunately she was under a lot of social pressure and after hesitating a bit agreed to the marriage. Apart from being a sign of gratitude for her father’s release this was also a way of persuading her suitor to use his influence to get a visa for her father to allow him to leave Russia. In the end she accepted this arrangement instead of running away with the man she loved. At this time she was fifteen years old. Her father got the visa and left the country. Later she also got a visa and was able to leave. The last stage of her journey was with the Orient Express from Constantinople to Paris. The book ends as the train is approaching Paris and a new life is starting for her.

### The probability space as a fiction

February 12, 2019

Looking at this book also led to progress of a different kind. I started thinking about the question of why I found probability theory so difficult. One superficial view of the subject is that it is just measure theory except that the known objects are called by different names. Since I do understand measure theory and I have a strong affinity for language if that was the only problem I should have been able to overcome it. Then I noticed a more serious difficulty, which had previously only been hovering on the edge of my consciousness. In elementary probability the concept of a probability space is clear – it is a measure space with total measure one. In more sophisticated probability theory it seems to vanish almost completely from the discussion. My impression in reading texts or listening to talks on the subject is that there is a probability space around in the background but that you never get your hands on it. You begin to wonder if it exists at all and this is the reason for the title of this post. I began to wonder if it is like the embedding into Euclidean space which any manifold in principle has but which plays no role in large parts of differential geometry. An internet search starting from this suspicion let me to an enlightening blog post of Terry Tao called ‘Notes 0: A review of probability theory‘. There he reviews ‘foundational aspects of probability theory’. Fairly early in this text he compares the situation with that in differential geometry. He compares the role of the probability space to that of a coordinate system in differential geometry, a probably better variant of my thought with the embeddings. He talks about a ‘probabilistic way of thinking’ as an analogue of the ‘geometric way of thinking’. So I believe that I have now discovered the basic thing I did not understand in this context – I have not yet understood the probabilistic way of thinking. When I consider the importance when doing differential geometry of (not) understanding the geometric way of thinking I see what an enormous problem this is. It is the key to understanding the questions of ‘what things are’ and ‘where things live’. For instance, to take an example from Tao’s notes, Poisson distributions are probability measures (‘distribution’ is the probabilistic translation of the word ‘measure’) on the natural numbers, the latter being thought of as a potential codomain of a random variable. Tao writes ‘With this probabilistic viewpoint, we shall soon see the sample space essentially disappear from view altogether …’ Why I am thinking about the Cheshire cat?

In a sequel to the blog post just mentioned Tao continues to discuss free probability. This is a kind of non-commutative extension of ordinary probability. It is a subject I do not feel I have to learn at this moment but I do think that it would be useful to have an idea how it reduces to ordinary probability in the commutative case. There is an analogy between this and non-commutative geometry. The latter subject is one which fascinated me sufficiently at the time I was at IHES to motivate me to attend a lecture course of Alain Connes at the College de France. The common idea is to first replace a space (in some sense) by the algebra of (suitably regular) functions on that space with pointwise operations. In practise this is usually done in the context of complex functions so that we have a * operation defined by complex conjugation. This then means that continuous functions on a compact topological space define a commutative $C^*$-algebra. The space can be reconstructed from the algebra. This leads to the idea that a $C^*$-algebra can be thought of as a non-commutative topological space. I came into contact with these things as an undergraduate through my honours project, supervised by Ian Craw. Non-commutative geometry has to do with extending this to replace the topological space by a manifold. Coming back to the original subject, this procedure has an analogue for probability theory. Here we replace the continuous functions by $L^\infty$ functions, which also form an algebra under pointwise operations. In fact, as discussed in Tao’s notes, it may be necessary to replace this by a restricted class of $L^\infty$ functions which are in particular in $L^1$. The reason for this is that a key structure on the algebra of functions (random variables) is the expectation. In this case the * operation is also important. The non-commutative analogue of a probability space is then a $W^*$-algebra (von Neumann algebra). Comparing with the start of this discussion, the connection here is that while the probability space fades into the background the random variables (elements of the algebra) become central.

### Broken foot

November 17, 2018

Last Saturday morning, while getting up from my desk at home, I tripped over something. I fell on the ground but my foot was stuck under the desk and could not accompany me properly in my fall. After that my foot was rather painful. Nevertheless the pain was not extreme and I did not take the matter very seriously. I limped around and even gave my lectures at the blackboard as normal on Tuesday and Thursday morning. On Wednesday I went to my GP to get his opinion on a non-acute matter. Actually I did not even really know him since I have not been to the doctor for five years apart from getting a tetanus booster. The pain in my foot had been getting less from day to day and I imagined I was on the way to recovery. At the same time the foot looked a bit funny, with some strange bruises. For this reason I showed the doctor my foot. He said that these bruises could be a sign of an injury deeper in the foot and suspected it was broken. He sent me to get it X-rayed to make sure. This is the first time in my life I have broken a bone and so I might be excused for not thinking of the possibility.

When I got to the place where the X-ray was to be done it turned out that their machine was broken and would only be working again the next day. Having already wasted so much time in attending to the foot I decided to look for another radiologist. By this time it was lunchtime and the first radiologist I tried was closed for lunch. The next one I tried did not accept me as a patient, for reasons I did not quite understand. The next one was closed the whole afternoon for a training course. In the late afternoon the fifth attempt was finally successful. The X-ray revealed that my foot was broken. Technically it is what is called a Jones fracture, which is a certain kind of break of the fifth metatarsal bone, a bone of whose existence I had known nothing up to that point in my life. It has the reputation of healing rather badly due to the poor blood supply in that area. I was told that I should see a surgeon as soon as possible. Since it was already very late in the afternoon that had to wait for the next day.

On Thursday I went back to my GP to discuss the further strategy. His practise was closed in the morning for a training course (where did I hear that phrase recently?). It opened again at three in the afternoon. Once I managed to see the doctor he called up a practise of a surgeon to check someone could see me that day. He got a positive answer and so I went there as quickly as possible. When I arrived the ‘friendly’ lady at the desk greeted me with the sceptical phrase, ‘what, you expect the surgeon to see you today?’. I just replied ‘That was the idea.’ Fortunately he did accept to see me. He explained the mechanism of the fracture. The bone is attached to a tendon and when that tendon is pulled with a large enough force the bone breaks apart. Fortunately it was a clean break and the two pieces had moved very little from the place they would normally be. I left the practise with crutches and wearing a surgical boot which looks very futuristic, like the first installment towards a space suit. I took a taxi home.

Now I learned something about how it is to have limited mobility. It just happens that Eva is away for a week so that she was not there to help me. (On Monday she was still there and took me to the university and back by car.) It is an interesting mental training. You realize how often you needlessly go from one place to another under normal circumstances, even within the home. Now it is necessary to plan an optimal route and minimise the number of times I go up and down the stairs. Yesterday I went to the university with the tram as a method to reduce the necessary walking distance to one I could manage. I also had the interesting experience of giving myself an anti-thrombosis injection. Actually it was not so bad as I expected and I suppose it will soon become routine. The professionals gave me the first one on Thursday to explain how it works. This morning I did some necessary shopping at the supermarket and took the tram one stop to get there. A neighbour saw me leaving home and was kind enough to transport me in the one direction in her car. Now I am looking forward to a weekend at home where the I will have no bigger physical obstacle to overcome than occasionally climbing the stairs.