## Biography of John D. Rockefeller by Ron Chernow

February 20, 2023

I have just read the biography ‘Titan’ of John D. Rockefeller by Ron Chernow. Rockefeller was a contemporary of Andrew Carnegie who I wrote about in a previous post, being just four years younger. Rockefeller became the richest man in the world after Carnegie had occupied that position. This book is a biography and not an autobiography and that is not an accident. While Carnegie had a talent and an inclination for writing and freely revealed many things about himself Rockefeller was extremely secretive. Given this it is surprising that so much is known about his life. At one time he gave extensive interviews to a journalist and Chernow was able to access the transcripts of these. He also engaged in a form of reverse engineering. Rockefeller wrote many letters but in doing so he tended to conceal the most important things. By contrast the writers of the letters he received were often less discrete and so Chernow could use those as a source of information about their recipient. Like Carnegie Rockefeller grew up in modest circumstances. However in another way his family background was very different. While Carnegie grew up in an atmosphere of honesty and hard work Rockefeller’s father ‘Big Bill’ was a swindler, quack doctor and bigamist. He used to abandon his wife and children for months at a time, although he did pay their bills when he returned after an unspecified length of time. The family frequently moved house due to the schemes of the father. It was very important for Rockefeller to achieve financial independence from his father. It was also important for him to fulfill moral standards which his father had violated. He was a dutiful father.

Rockefeller was very religious and his wife even more so. He belonged to the Baptist church and starting from a young age supported the church he went to with work, money and fundraising. He was strictly against drinking, smoking and even less obviously sinful things such as theatre and opera. The children were mostly confined at home, being taught by private tutors. The regime was very strict so that, for instance, a child who ate two pieces of cheese on one day received extensive reproaches. The mother stated that no woman needs more than two dresses. The children were encouraged to earn their own money. At a time when the family was already rich the parents concealed this fact from the children.

A well-known characteristic of Rockefeller is his philanthropy. He admired the corresponding activities of Carnegie but privately said that Carnegie was vain. He, Rockefeller, went to great lengths to stay in the background in the context of his philanthropic gifts. He generally did not want things named after him just because he had paid for them. He invested tremendous effort in trying to decide what were the most valuable causes he should contribute to. In this sense it seems that for him giving away money was more strenuous than earning it. Eventually, after the strain was damaging his health he delegated a lot of his philanthropic activity. An important principle of his was that he would only give money if it was matched by a certain sum raised for the same project from other sources. In the end he was often not too successful in implementing this policy – he was not able to stand up to the pressure from the beneficiaries. A good example of this were his huge contributions to the beginnings of University of Chicago. Rockefeller had a key influence on establishing medical research in the United States. He founded what is now called the Rockefeller University. He made a very important contribution to fighting hookworm in the southern US and later in other parts of the world. He himself believed in homeopathy but due to the fact that the people who managed his philanthropy were more enlightened that he was the money donated contributed essentially to establishing evidence-based medicine in the US and weakening the influence of homeopathy. Thus in a way he came to doing exactly the opposite of what his father had done.

This is a very high quality biography and contains a host of interesting things which I did not even mention.

## Grammatical gender and politics in Germany

February 17, 2023

I find it a little difficult to write about this subject in English and in such a way that it is comprehensible for an international audience. Nevertheless I think it is worthwhile to do so. The content of this post is related to a political theme which is specific to Germany, or at least German-speaking countries, but it is related to political issues of much wider significance. In German a noun belongs to one of three genders, masculine, feminine or neuter. These names have special connotations but fundamentally they are just names for three different classes of nouns, which behave in different ways within the language, such as influencing the form of adjectives which describe them. The English Wikipedia article Grammatical gender mentions that the term ‘noun class’ is sometimes used as a synonym for this. I think that mathematicians in particular will easily understand the logical status of a concept like this which is defined by its logical relations. I am reminded of the famous (and apocryphal?) story about Hilbert saying that in Euclidean geometry instead of points, lines and planes we could say tables, chairs and beer-mugs and it would make no difference. What is important is the relations between the concepts, not their names.

Problems arise from the fact that the names for the genders have strong connotations. Words for male human beings usually belong to the gender ‘masculine’, words for female human beings usually belong to the gender ‘feminine’. In other words, there is a correlation between grammatical and biological gender. This has led some people to give grammatical gender a significance beyond the formal one I have introduced previously. This happens although there are obvious examples where the two notions associated to the word ‘gender’ do not agree. For instance the German word ‘Person’ meaning person is feminine while the word ‘Mensch’ meaning human being is masculine. This exaggeration of the meaning of grammatical gender leads to problems. Let us for instance take the German word ‘Professor’ meaning professor. It can mean a professor of either sex. However if we want to emphasize that a particular professor is female we can use the word ‘Professorin’. The ending ‘in’ indicates a female person. The argument of some people is that if we use the word ‘Professor’ in the inclusive sense, denoting a person who may be female then this can lead to the implicit assumption that the person concerned is a man, due to the fact that the grammatical gender of the word is masculine. Thus they say that this practise leads to women becoming ‘invisible’ and thus discriminates against them. Other people have other opinions on this subject. It is something which could in principle be discussed in a civilised way. In practise this is often not possible. An elaborate system of feminine forms of various words has been developed (sometimes called ‘Gendern’) and anyone who publicly critises the use of this system risks being branded a right-wing extremist. The system is widespread in various areas: politics, state media, universities, public administration, private companies etc. In many cases the system is not just used but imposed, either through official regulations or through peer pressure.

I find ‘Gendern’ unpleasant and avoid it whenever possible. What are my motives for that? The first is an aesthetic one – I love language and ‘Gendern’ makes a mess of the language. German is not my native language but I still do not like seeing it being mistreated. Various surveys show that a majority of Germans are against ‘Gendern’ but it is nevertheless widespread. Its proponents are loud and sometimes also aggressive. Another argument againt ‘Gendern’ is that it not only makes language ugly but it makes it so clumsy as to often make a text difficult to understand and tiring to read. Moving in a language like English where gender in this sense does not exist seems like a luxury in comparison. As I said at the beginning this seems like a very special topic. However I also claimed that it is of wider significance. Why do I say that? I see it as an example of the way in which politics has come to suppress free rational discussion, a phenomenon which occurs in many areas of life and in many countries. Moral convictions are regarded as more important than facts and logic. As yet this has not had much effect on me personally. A mathematician is likely to be less affected than a social scientist. I enjoy using language in different ways, private and professional, and I see ‘Gendern’ as a threat to being able to feel comfortable and at home in the German language. I also see it as an example of the way in which political ideology can have a negative impact on the daily lives of normal people. It is important that the ‘silent majority’ speaks out and does not leave public discourse to vocal minorities.

## Visit to Switzerland

January 6, 2023

Over New Year Eva and I spent some time in Switzerland. It was the first time I had made a private trip there. I have been in Zurich several times for my work and I was once at a conference in Ascona but that was all my experience of the country. We spent three nights in Chur which we found a very pleasant town. The following three nights we were in Lucerne which I liked even better. In some ways visiting Switzerland was like returning to a better past. In what we saw there were no signs of hardship or other problems due to the pandemic or the Ukraine war. The restaurants were full. In the hotels the personnel was plentiful. The complete absence of requirements to wear a mask was refreshing. It was strange at the end of the trip when we reentered Germany on the train and had to put on our masks again. In Switzerland it was easy to forget many of the negative aspects of everyday life at present.

From Chur we took a day trip by train to Arosa, a place I had not heard of before. The town itself is not specially attractive but the scenery on the way there is spectacular. I discovered that the sanatorium in Thomas Mann’s novel The Magic Mountain was based on an institution in Arosa. In the novel the sanatorium is situated in Davos but in fact Mann’s wife stayed for extended periods in both these places. Mann visited her in Arosa. Robert Louis Stevenson also spent time there. In fact I do not like Mann very much. The first book of his I read was Doctor Faustus, in English. While I admired some of the descriptions, particularly that of the music shop, the book as a whole did not appeal to me. Next I read Death in Venice which I did not like better. Finally, after having learned German I read Death in Venice in the original, hoping that this would be an improvement. Unfortunately it was not the case. This is in contrast to my experience with Kafka. When I originally read Kafka’s work in English I liked it a lot and when I later read it in the original I found it even better. In Arosa I saw a lot of Alpine Choughs, the first of many I encountered on the trip.

One thing I liked about Lucerne was its situation on the lake, surrounded by mountains. On one evening the lake provided an impressive variety of colour combinations as the light changed. One striking aspect of the lake is the powerful stream caused by the flow of water being largely held back at one place and confined to a narrow region. (The river Reuss flows into the lake at one end and out again at the other, close to Lucerne.) We took a boat trip on the lake. From there is it possible to see a house where Richard Wagner lived for several years and wrote some of his most famous works. He had a number of famous visitors there, including Nietzsche and Liszt. In Germany it is typical for people to set off fireworks on New Year’s Eve. I do not like that much. Firstly it makes me feel that it is unsafe to go onto the streets at that time. This year it was particularly extreme. On the other hand I do not like being kept awake for much of the night by the noise. In Lucerne they had a variant which I found preferable. Fireworks were forbidden on New Year’s Eve. People did occasionally violate this rule but it was a minor annoyance. Then on the first of January there was a professional firework display over the lake. It was spectacular, it was safe and it did not cost anyone sleep. In Lucerne we visited the Rosengart art collection which only served to show me once again how little I am able to appreciate modern art. The painters represented most there are Picasso and Klee. From what we heard on the guided tour I had the impression that Picasso was a very unpleasant person, in love with himself. I find his paintings ugly and I do not appreciate the paintings of Klee we saw either. Eva has more understanding for such things. The few paintings we saw that I liked were by impressionists, on the side of the border to modern art where I still feel comfortable. I am thinking in particular of paintings of Monet, Renoir and Seurat.

## Matched asymptotic expansions and the Kaplun-Lagerstrom model

December 13, 2022

In a post I wrote a long time ago I described matched asymptotic expansions as being like black magic. Now I have understood some more about how to get from there to rigorous mathematics. My main guide in doing so has been Chapter 7 of the book ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. There they give an extensive treatment of a model problem by Kaplun and Lagerstrom. The ultimate source of this is work of Stokes on hydrodynamics around 1850. In his calculations he found some paradoxical phenomena. Roughly speaking, attempting to obtain an asymptotic expansion for a solution led to inconsistencies. These things remained a mystery for many years. A big step forward came in the work of Kaplun and Lagerstrom in 1957. There they introduced an ODE model which, while having no direct physical interpretation, provides a relatively simple mathematical context in which to understand these phenomena. It is this model problem which is treated in detail by Hastings and McLeod. The model is a boundary value problem for the equation $y''+\frac{n-1}{r}y'+\epsilon yy'=0$. We look for a solution with $y(1)=0$ and $\lim_{r\to\infty}y(r)=1$. The first two terms look like the expression for the Laplacian of a spherically symmetric function in $n$ dimensions and for this reason the motivation is strong to look at the cases $n=2$ and $n=3$ which are vaguely related to fluid flow around a cylinder and flow around a sphere, respectively. It turns out that the case $n=2$ is a lot harder to analyse than the case $n=3$. When $n=3$ the problem has a unique solution for $\epsilon>0$. We would like to understand what happens to this solution as $\epsilon\to 0$. It is possible to find an asymptotic expansion in $\epsilon$ but it is not enough to use powers of $\epsilon$ when building the expansion. There occurs a so-called switchback term containing $\log\epsilon$. This is a singular limit although the parameter in the equation only occurs in a lower order term. This happens because the equation is defined on a non-compact region.

Consider the case $n=3$. In applying matched asymptotic expansions to this problem the first step is to do a straightforward (formal) expansion of the equation in powers of $\epsilon$. This gives differential equations for the expansion coefficients. At order zero there is no problem solving the equation with the desired boundary conditions. At order one this changes and it is not possible to implement the desired boundary condition at infinity. This has to do with the fact that in the correct asymptotic expansion the second term is not of order $\epsilon$ but of order $\epsilon\log\epsilon$. This extra term is the switchback term. Up to this point all this is formal. One method of obtaining rigorous proofs for the asymptotics is to use GSPT, as done in two papers of Popovic and Szmolyan (J. Diff. Eq. 199, 290 and Nonlin. Anal. 59, 531). There is an introduction to this work in the book but I felt the need to go deeper and I looked at the original papers as well. To fit the notation of those papers I replace $y$ by $u$. Reducing the equation to first order by introducing $v=u'$ as a new variable leads to a non-autonomous system of two equations. Introducing $\eta=1/r$ as a new dependent variable and using it to eliminate $r$ from the right hand side of the equations in favour of $\eta$ leads to an autonomous system of three equations. This allows the original problem to be reformulated in the following geometric way. The $u$-axis consists of steady states. The point $(1,0,0)$ is denoted by $Q$. The aim is to find a solution which starts at a point of the form $(0,v,1)$ and tends to $Q$ as $r\to\infty$. A solution of this form for $\epsilon$ small and positive is to be found by perturbation of a corresponding solution in the case $\epsilon=0$. For $\epsilon>0$ the centre manifold of $Q$ is two-dimensional and given explicitly by $v=0$. In the case $\epsilon=0$ it is more degenerate and has an additional zero eigenvalue. To prove the existence of the desired connecting orbit we may note that for $\epsilon>0$ this is equivalent to showing that the manifold consisting of solutions starting at points of the form $(0,v,1)$ and the manifold consisting of solutions converging to $Q$ intersect. The first of these manifolds is obviously a deformation of a manifold for $\epsilon=0$. We would like the corresponding statement for the second manifold. This is difficult to get because of the singularity of the limit. To overcome this $\epsilon$ is introduced as a new dynamical variable and a suitable blow-up is carried out near the $u$-axis. In this way it is possible to get to a situation where there are two manifolds which exist for all $\epsilon\ge 0$ and depend smoothly on $\epsilon$. They intersect for $\epsilon=0$ and in fact do so transversely. It follows that they also intersect for $\epsilon$ small and positive. What I have said here only scratches the surface of this subject but it indicates the direction in which progress could be made and this is a fundamental insight.

December 10, 2022

## The shooting method for ordinary differential equations

November 25, 2022

A book which has been sitting on a shelf in my office for some years without getting much attention is ‘Classical Methods in Ordinary Differential Equations’ by Hastings and McLeod. Recently I read the first two chapters of the book and I found the material both very pleasant to read and enlightening. A central theme of those chapters is the shooting method. I have previously used the method more than once in my own research. I used it together with Juan Velázquez to investigate the existence of self-similar solutions of the Einstein-Vlasov system as discussed here. I used it together with Pia Brechmann to prove the existence of unbounded oscillatory solutions of the Selkov system for glycolysis as discussed here. This method is associated with the idea of a certain type of numerical experiment. Suppose we consider a first order ODE with initial datum $x(0)=\alpha$. Suppose that for some value $\alpha_1$ the solution tends to $+\infty$ for $t\to\infty$ and for another value $\alpha_2>\alpha_1$ the solution tends to $-\infty$. Then we might expect that in between there is a value $\alpha^*$ for which the solution is bounded. We could try to home in on the value of $\alpha^*$ by a bisection method. What I am interested in here is a corresponding analytical procedure which sometimes provides existence theorems.

In the book the procedure is explained in topological terms. We consider a connected parameter space and a property $P$. Let $A$ be the subset where $P$ does not hold. If we can show that $A=A_1\cup A_2$ where $A_1$ and $A_2$ are non-empty and open and $A=A_1\cap A_2=\emptyset$ then $A$ is disconnected and so cannot be the whole of the parameter space. Hence there is at least one point in the complement of $A$ and there property $P$ holds. The most common case is where the parameter space is an interval in the real numbers. For some authors this is the only case where the term ‘shooting method’ is used. In the book it is used in a more general sense, which might be called multi-parameter shooting. The book discusses a number of cases where this type of method can be used to get an existence theorem. The first example is to show that $x'=-x^3+\sin t$ has a periodic solution. In fact this is related to the Brouwer fixed point theorem specialised to dimension one (which of course is elementary to prove). The next example is to show that $x'=x^3+\sin t$ has a periodic solution. After that this is generalised to the case where $\sin t$ is replaced by an arbitrary bounded continuous function on $[0,\infty)$ and we look for a bounded solution. The next example is a kind of forced pendulum equation $x''+\sin x=f(t)$ and the aim is to find a solution which is at the origin at two given times. In the second chapter a wide variety of examples is presented, including those just mentioned, and used to illustrate a number of general points. The key point in a given application is to find a good choice for the subsets. There is also a discussion of two-parameter shooting and its relation to the topology of the plane. This has a very different flavour from the arguments I am familiar with. It is related to Wazewski’s theorem (which I never looked at before) and the Conley index. The latter is a subject which has crossed my path a few times in various guises but where I never really developed a warm relationship. I did spend some time looking at Conley’s book. I found it nicely written but so intense as to require more commitment than I was prepared to make at that time. Perhaps the book of Hastings and McLeod can provide me with an easier way to move in that direction.

## The autobiography of Andrew Carnegie

November 4, 2022

Carnegie later got involved in the production of iron and things constructed out of it, such as rails and bridges. He got an advantage over his competitors by employing a chemist. The iron ore from some mines was unpopular and correspondingly relatively cheap. There had been problems with smelting it. A chemical analysis revealed the source of the problem – that ore contained too much iron for the smelting process to work well. The solution was to modify the process (with the help of scientific considerations) and then it was possible to buy the high quality ore at a cheap price while others continued to buy low quality ore at a high price. Previously nobody really knew what they were buying. Carnegie believed in the value of real knowledge. Carnegie did not like the stock exchange and emphasized that except for once at the beginning of his career he never speculated. It was always his policy to buy and sell things on the basis of their real value. Carnegie was no friend of unions and often fought them hard. On the other hand he was, or claimed to be, a friend of the working man. His idea was not to give people money just like that but to give them the opportunity to improve their own situation. In later years he gave a huge amount of money, about 300 million dollars in total for various causes. He gave money for libraries (more than two hundred), for scientific research, for church organs and of course for the Carnegie Hall. These were all things which he believed would do people good.

Carnegie was committed to the goal of world peace. He had a lot of influence with powerful politicians and it seems that in at least one case he used it to prevent the US becoming involved in a war. He got into contact with Kaiser Wilhelm II. It turned out that both of them were admirers of Robert the Bruce. He had great hopes for the Kaiser as someone who could help to bring peace and he must have been bitterly disappointed in 1914 when things went a very different way. Then he transferred his hopes to President Wilson. At that point his autobiography breaks off. Here I have only been able to present a few selected things from a fascinating book which I thoroughly recommend. I find Carnegie an admirable character.

## Andrew Carnegie and me

October 10, 2022

At the moment I am rereading the autobiography of Andrew Carnegie. For me he is a leading example of a ‘good capitalist’. I see some parallels between my early life and that of Carnegie and I have been thinking about similarities and differences. We were both born in Scotland to parents who were not rich. His father was a weaver. At some stage advances in technology meant that his form of industry was no longer viable. The family had difficulty earning their living and decided to emigrate to America. While still in Scotland Carnegie’s mother started a business to supplement the income of her husband.

My father was a farmer. When he began the size of the farm (70 acres) was sufficient. (I recently noticed the coincidence that the father of Robert Burns, Scotland’s national poet, also had a farm of 70 acres.) Later there was a trend where the size of farms increased and the machinery used to work them became more advanced. In order to do this the farmers, who were mostly not rich, had to borrow money to buy more land and better machines. The banks were eager to lend them that money. Of course this meant a certain risk but many of the people concerned were prepared to take that risk. My father, on the other hand, never borrowed any money in his life and so he missed taking part in this development. This meant that under the new conditions the farm was too small (and in fact some of it was not very good land – it was too wet) to support our family (my parents, my grandmother and myself) very easily. My mother did everything she could to supplement the income of my father. In particular she took in bed and breakfast guests during the summer. I should point out that we were not poor. We did not lack anything essential, living in part from our own produce such as milk, butter, cheese, eggs, potatoes and meat. My grandmother kept a pig and hens. The school I attended, Kirkwall Grammar School, was the school for all children in the area – there was no alternative. The parents of many of the other children I went to school with were better off financially than my parents. As a sign of this, I mention an exchange between our school and one in Canada. Many of the other pupils took part in that. My parents could not afford to finance it for me. At a time when many people were getting their first colour TV we still had a very old black and white device where with time ‘black’ and ‘white’ were becoming ever more similar. I did not feel disadvantaged but I just mention these things to avoid anyone claiming that I grew up in particularly fortunate economic circumstances.

Both Carnegie and I benefitted from the good educational system in Scotland. School was already free and compulsory in his time. My university education was mostly financed by the state, although I did win a couple of bursaries in competitions which helped to make my life more comfortable. In my time parents had to pay a part of the expenses for their childrens’ university education, depending on their incomes. My parents did not have to pay anything. Some of the people I studied with should have got a contribution from their parents but did not get as much as they should have. Thus I actually had an advantage compared to them. Carnegie’s father was involved in politics and had quite a few connections. My parents had nothing like that. It might be thought that since my parents did not have very much money or connections and since there were very few books in our house I started life with some major disadvantages. I would never make this complaint since I know that my parents gave me some things which were much more important than that and which helped me to build a good life. I grew up in a family where I felt secure. My parents taught me to behave in certain ways, not by command but by their example. They taught me the qualities of honesty, reliability, hard work and humility. Carnegie received the same gifts from his parents.

Let me now come back to the question of books. As a child I was hungry for them. We had a good school library which included some unusual things which I suppose not all parents would have been happy about if they had known the library as well as I did. For instance there was a copy of the ‘Malleus Maleficarum’. What was important for my future was that there were current and back issues of Scientific American and New Scientist. There was also a public library from which I benefitted a lot. Apparently this was the first public library in Scotland, founded in 1683. In the beginning it was by subscription. It became free due to a gift of money from Andrew Carnegie in 1889. He spent a huge amount of time and effort in supporting public libraries in many places. In 1903 Carnegie also gave the money to construct a building for the library and that is the building it was still in when I was using it. He visited Kirkwall to open the library in 1909. Thus it can be said that I personally received a gift of huge value from Carnegie, the privilege of using that library. His activity in this area was his way of returning the gift which he received as a young boy when someone in Pittsburgh opened his private library to working boys.

## Is mathematics being driven out by computers?

September 28, 2022

In the past two weeks I attended two conferences. The first was the annual meeting of the Deutsche Mathematikervereinigung (DMV, the German mathematical society) in Berlin. The second was the joint annual meeting of the ESMTB (European Society for Mathematical and Theoretical Biology) and the SMB (Society for Mathematical Biology) in Heidelberg. I had the impression that the participation of the SMB was relatively small compared to previous years. (Was this mainly due to the pandemic or due to other problems in international travel?) There were about 500 participants in total who were present in person and about another 100 online. I was disappointed with the plenary talks at both conferences. The only one which I found reasonably good was that of Benoit Perthame. One reason I did not like them was the dominance of topics like machine learning and artificial intelligence. This brings me to the title of this post. I have the impression that mathematics (at least in applied areas) is becoming ever weaker and being replaced by the procedure of developing computer programmes which could be applied (and sometimes are) to the masses of data which our society produces these days. This was very noticeable in these two conferences. I would prefer if we human beings would continue to learn something and not just leave it to the machines. The idea that some day the work of mathematicians might be replaced by computers is an old one. Perhaps it is now happening, but in a different way from that which I would have expected. Computers are replacing humans but not because they are doing everything better. There is no doubt there are some things they can do better but I think there are many things which they cannot. The plenary talks at the DMV conference on topics of this kind were partly critical. There occurred examples of a type I had not encountered before. A computer is presented with a picture of a pig and recognizes it as a pig. Then the picture is changed in a very specific way. The change is quantitatively small and is hardly noticeable to the human eye. The computer identifies the modified picture as an aeroplane. In another similar example the starting picture is easily recognizable as a somewhat irregular seven and is recognized by the computer as such. After modification the computer recognizes it as an eight. This seems to provide a huge potential for mistakes and wonderful opportunities for criminals. I feel that the trend to machine learning and related topics in mathematics is driven by fashion. It reminds me a little of the ‘successes’ of string theory in physics some years ago. Another aspect of the plenary talks at these conferences I did not like was that the speakers seemed to be showing off with how much they had done instead of presenting something simple and fascinating. At the conference in Heidelberg there were three talks by young prizewinners which were shorter than the plenaries. I found that they were on average of better quality and I know that I was not the only one who was of that opinion.

In the end there were not many talks at these conferences I liked much but let me now mention some that I did. Amber Smith gave a talk on the behaviour of the immune system in situations where bacterial infections of the lung arise during influenza. In that talk I really enjoyed how connections were made all the way from simple mathematical models to insights for clinical practise. This is mathematical biology of the kind I love. In a similar vein Stanca Ciupe gave a talk about aspects of COVID-19 beyond those which are common knowledge. In particular she discussed experiments on hamsters which can be used to study the infectiousness of droplets in the air. A talk of Harsh Chhajer gave me a new perspective on the intracellular machinery for virus production used by hepatitis C, which is of relevance to my research. I saw this as something which is special for HCV and what I learned is that it is a feature of many positive strand RNA viruses. I obtained another useful insight on in-host models for virus dynamics from a talk of James Watmough.

Returning to the issue of mathematics and computers another aspect I want to mention is arXiv. For many years I have put copies of all my papers in preprint form on that online archive and I have monitored the parts of it which are relevant for my research interests for papers by other people. When I was working on gravitational physics it was gr-qc and since I have been working on mathematical biology it has been q-bio (quantitative biology) which I saw as the natural place for papers in that area. q-bio stands for ‘quantitative biology’ and I interpreted the word ‘quantitative’ as relating to mathematics. Now the nature of the papers on that archive has changed and it is also dominated by topics strongly related to computers such as machine learning. I no longer feel at home there. (To be fair I should say there are still quite a lot of papers there which are on stochastic topics which are mathematics in the classical sense, just in a part of mathematics which is not my speciality.) In the past I often cross-listed my papers to dynamical systems and maybe I should exchange the roles of these two in future – post to dynamical systems and cross-list to q-bio. If I succeed in moving further towards biology in my research, which I would like to I might consider sending things to bioRxiv instead of arXiv.

In this post I have written a lot which is negative. I feel the danger of falling into the role of a ‘grumpy old man’. Nevertheless I think it is good that I have done so. Talking openly about what you are unsatisfied with is a good starting point for going out and starting in new positive directions.

## The centre manifold theorem and its proof

September 2, 2022

In my research I have often used centre manifolds but I have not thoroughly studied the proof of their existence. The standard reference I have quoted for this topic is the book of Carr. The basic statement is the following. Let $\dot x=f(x)$ be a dynamical system on $R^n$ and $x_0$ a point with $f(x_0)=0$. Let $A=Df(x_0)$. Then $R^n$ can be written as a direct sum of invariant subspaces of $A$, $V_-\oplus V_c\oplus V_+$, such that the real parts of the eigenvalues of the restrictions of $A$ to these subspaces are negative, zero and positive, respectively. $V_c$ is the centre subspace. The centre manifold theorem states that there exists an invariant manifold of the system passing through $x_0$ whose tangent space at $x_0$ is equal to $V_c$. This manifold, which is in general not unique, is called a centre manifold for the system at $x_0$. Theorem 1 on p. 16 of the book of Carr is a statement of this type. I want to make two comments on this theorem. The first is that Carr states and proves the theorem only in the case that the subspace $V_+$ is trivial although he states vaguely that this restriction is not necessary. The other concerns the issue of regularity. Carr assumes that the system is $C^2$ and states that the centre manifold obtained is also $C^2$. In the book of Perko on dynamical systems the result is stated in the general case with the regularity $C^r$ for any $r\ge 1$. No proof is given there. Perko cites a book of Guckenheimer and Holmes and one of Ruelle for this but as far as I can see neither of them contains a proof of this statement. Looking through the literature the situation of what order of differentiability is required to get a result and whether the regularity which comes out is equal to that which goes in or whether it is a bit less seems quite chaotic. Having been frustrated by this situation a trip to the library finally allowed me to find what I now see as the best source. This is a book called ‘Normal forms and bifurcations of planar vector fields’ by Chow, Li and Wang. Despite the title it offers an extensive treatment of the existence theory in any (finite) dimension and proves, among other things, the result stated by Perko. I feel grateful to those authors for their effort.

A general approach to proving the existence of a local centre manifold, which is what I am interested in here, is first to do a cut-off of the system and prove the existence of a global centre manifold for the cut-off system. It is unique and can be obtained as a fixed point of a contraction mapping. A suitable restriction of it is then the desired local centre manifold for the original system. Due to the arbitrariness involved in the cut-off the uniqueness gets lost in this process. A mapping whose fixed points correspond to (global) centre manifolds is described by Carr and is defined as follows. We look for the centre manifold as the graph of a function $y=h(x)$. The cut-off is done only in the $x$ variables. If a suitable function $h$ is chosen then setting $y=h(x)$ gives a system of ODE for $x$ which we can solve with a prescribed initial value $x_0$ at $t=0$. Substituting the solution into the nonlinearity in the evolution equation for $y$ defines a function of time. If this function were given we could solve the equation for $y$ by variation of constants. A special solution is singled out by requiring that it vanishes sufficiently fast as $t\to -\infty$. This leads to an integral equation of the general form $y=I(h)$. If $y=h$, i.e. $h$ is a fixed point of the integral operator then the graph of $h$ is a centre manifold. It is shown that when certain parameters in the problem are chosen correctly (small enough) this mapping is a contraction in a suitable space of Lipschitz functions. Proving higher regularity of the manifold defined by the fixed point requires more work and this is not presented by Carr. As far as I can see the arguments he does present in the existence proof nowhere use that the system is $C^2$ and it would be enough to assume $C^1$ for them to work. It is only necessary to replace $O(|z|^2)$ by $o(|z|)$ in some places.