Mathematicians From England

"The worst... medical care would be just say, "Oh well, it's completely up to you. I'm not going to give you any advice." Well... you're the guy who knows... [the] supposed... expert. ...I want your advice. ...[T]his is an invaluable process which I hope all medical students have been taught. How to engage... It's not . ...Paternalism is ...immediately saying, "...I think you should do this," without really allowing that person a full autonomy to choose."

"[I]n medicine... ... not manipulation or coercion, is when... as a doctor or... authority, you genuinely believe that this action is in the person's best interests, but they don't... want to do it. ...How do you make that an ethical persuasion? It's based on... two things, first... respecting the autonomy of the individuals, that they can refuse... no matter what, respect their ability to choose... the other thing is your authenticity, your integrity, that... you are doing this on behalf of that individual, for their best interest; not... to keep your clinic numbers up or to stop this person being a nuisance...[etc.]"

"What we use in the book... are ways which we think might... provide gripping narratives, and yet provide a realistic way of communicating small risks..."

"This current COVID-19 virus... is a classic situation... of... an uncertainty problem, rather than a risk problem because we... don't know the parameters. We don't know... how it might spread in Great Britain. We don't know the effectiveness of the interventions that are going to be made. So... when you're making projections... over the next 6 months, there's a massive range of possibilities, up to 1/2 million deaths... from about 5... the most optimistic... [A]ny quantification, giving any probabilities would be... very ambitious..."

"There'a a nice thing about 1/2 hour because an adult life expectancy is 55 to 60 years... is actually a million 1/2 hours. A million 1/2 hours is 57 years. So you have got, not all of you... some of you have got a million 1/2 hours to fritter away."

"I learned mathematics on my own from textbooks which is perhaps strange given that both my parents were involved in the subject. At the same time, I spent a good deal of time studying art and wanted to follow a career in that direction until I was eventually convinced by my family that I should first work for a mathematics degree to ensure that I could earn a living."

"My parents separated after my father resigned his university position to focus on his inventions, and my mother then finished her education and became a school mathematics teacher. We moved to a very cosmopolitan area of London, which was like a new birth for me; it was there that my interest in mathematics really began."

"I would like to thank John Garnett for a lot of very helpful advice."

"My father ... went on to a position in the mathematics department of the University of East Anglia. While I was growing up, the elementary school I attended was extremely ethnically homogeneous. I was unable to escape from heavy issues concerning race, which my mother always explained in a political context."

"I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. What is Spectral geometry? Spectral geometry most usually means the study of how the geometry of an object is related to the natural frequencies of the object. These are the frequencies at which the object can vibrate. A vibrating object often produces a sound, and the frequencies can be heard as the dominant tone and the overtones of the sound. The well known question highlighting what spectral geometry is all about is the question "Can one hear the shape of a drum?" I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. In mathematical terms, the natural frequencies of an object (or rather their squares) are the eigenvalues of a partial differential operator called the Laplacian. This Laplacian takes each function defined on the object and differentiates it twice to give a new function. The eigenvalues of the Laplacian form an infinite sequence of numbers tending to infinity. In spectral geometry we study how these numbers depends on the shape of the object. For people who like to know the full story, I should mention that many spectral geometers (including me) who work on the Laplacian on smooth manifolds study the whole sequence of eigenvalues of the Laplacian. Now the low eigenvalues can give accurate values for the frequencies at which a real life object vibrates, but the very high eigenvalues do not correspond to genuine physical vibrations of the object because of molecular forces and damping. These effects are not included in the model where the vibration is driven by the Laplacian alone. This means that my research is rather different from that of an engineer who wishes to model precisely the vibrations of a real life object. In actual fact the questions I work on are more closely related to mathematics arising in quantum physics and string theory. In addition, I don't always study the Laplacian, but also the eigenvalues of other operators, which might represent other physical quantities than the frequencies of vibration. I mostly study spectral geometry for nice smooth objects such as spheres and tori, but some people work on rough objects and even discrete objects like graphs. In the last eight years, I have worked mostly on the spectral zeta function, which is an infinite sum of powers of the eigenvalues. In particular, I have worked on the zeta-regularised determinant, which is used in topology, quantum field theory, and string theory. Recently, I have been very interested in the sum of squares of the wavelength of a surface, which is related to all kinds of different things including vortex theory."

"Although I cannot claim to find it easy to balance my ambitions in mathematical research with the desire to be a good parent, to be an inspiring teacher, or to effect positive social change in the world, I do feel very fortunate to be able to spend my life tackling these challenges, which are extremely interesting and important to me."

"I went to Cambridge, which represented a second major change in my life. As I learned more mathematics, I saw that it is an entire world of its own which many people choose to live in, a world in many ways more real than the real world; it feels permanent, eternal, and offers a deep sense of security because nearly everyone who understands it agrees on what is truth. By the time I had finished at Cambridge, I was very involved with mathematics and did not consider other careers."

"I do research on a variety of problems in condensed matter physics. My primary interests are in the general field of statistical mechanics."

"My research is in the field of spectral geometry, the study of how the shape of an object affects the modes in which it can resonate. A famous question in the field is, can one hear the shape of a drum? Spectral geometry bridges different branches of science, including engineering and physics, as well as a number of different fields of mathematics. However, quite different sorts of questions are studied within each discipline. I am a mathematical analyst, which gives me an appreciation for the infinite and the infinitesimal. At the moment, one of the things I am working on understanding is the total wavelength of a surface like a sphere or something of greater complexity, such as the surface of a bagel or a pretzel. What is this total wavelength? If you strike a surface it can resonate at any one of a list of frequencies, and the wavelength of the sound produced by the vibration is inversely proportional to the frequency. In the mathematically idealized model there are infinitely many possible wavelengths. The total wavelength should be the sum of all of these individual wavelengths except that this infinite sum equals infinity. Fortunately, a finite number can be assigned to it by a slightly elusive process called regularization. (This process is also used in mathematical physics to mysteriously obtain true answers from formulas which do not really make sense!) I first became interested in the total wavelength as a model related to a question which can be roughly stated as, can one hear the shape of the universe? However, the total wavelength shows up in many quite different areas of mathematics and I am finding these connections intriguing."

"I cannot claim to find it easy to balance my ambitions in mathematical research with the desire to be a good parent, to be an inspiring teacher, or to effect positive social change in the world, I do feel very fortunate to be able to spend my life tackling these challenges, which are extremely interesting and important to me."

"I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. What is Spectral geometry? Spectral geometry most usually means the study of how the geometry of an object is related to the natural frequencies of the object. These are the frequencies at which the object can vibrate. A vibrating object often produces a sound, and the frequencies can be heard as the dominant tone and the overtones of the sound. The well-known question highlighting what spectral geometry is all about is the question "Can one hear the shape of a drum?" I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. In mathematical terms, the natural frequencies of an object (or rather their squares) are the eigenvalues of a partial differential operator called the Laplacian. This Laplacian takes each function defined on the object and differentiates it twice to give a new function. The eigenvalues of the Laplacian form an infinite sequence of numbers tending to infinity. In spectral geometry we study how these numbers depends on the shape of the object. For people who like to know the full story, I should mention that many spectral geometers (including me) who work on the Laplacian on smooth manifolds study the whole sequence of eigenvalues of the Laplacian. Now the low eigenvalues can give accurate values for the frequencies at which a real-life object vibrates, but the very high eigenvalues do not correspond to genuine physical vibrations of the object because of molecular forces and damping. These effects are not included in the model where the vibration is driven by the Laplacian alone. This means that my research is rather different from that of an engineer who wishes to model precisely the vibrations of a real-life object. In actual fact the questions I work on are more closely related to mathematics arising in quantum physics and string theory. In addition, I don't always study the Laplacian, but also the eigenvalues of other operators, which might represent other physical quantities than the frequencies of vibration. I mostly study spectral geometry for nice smooth objects such as spheres and tori, but some people work on rough objects and even discrete objects like graphs. In the last eight years, I have worked mostly on the spectral zeta function, which is an infinite sum of powers of the eigenvalues. In particular, I have worked on the zeta-regularised determinant, which is used in topology, quantum field theory, and string theory. Recently, I have been very interested in the sum of squares of the wavelength of a surface, which is related to all kinds of different things including vortex theory."

"I went to Cambridge, which represented a second major change in my life. As I learned more mathematics, I saw that it is an entire world of its own which many people choose to live in, a world in many ways more real than the real world: it feels permanent, eternal, and offers a deep sense of security because nearly everyone who understands it agrees on what is truth. By the time I had finished at Cambridge, I was very involved with mathematics and did not consider other careers."

"My mother is British, from a family with a trade union background and a central interest in class struggle; she met my father, who is Nigerian, while both were students of mathematics in London. My father was a very talented mathematician, and after my parents married, he went on to a position in the mathematics department of the University of East Anglia."

"While I was growing up, the elementary school I attended was extremely ethnically homogeneous. I was unable to escape from heavy issues concerning race, which my mother always explained in a political context."

"My research is in the field of spectral geometry, the study of how the shape of an object affects the modes in which it can resonate. A famous question in the field is, Can one hear the shape of a drum? Spectral geometry bridges different branches of science, including engineering and physics, as well as a number of different fields of mathematics. However, quite different sorts of questions are studied within each discipline. I am a mathematical analyst, which gives me an appreciation for the infinite and the infinitesimal. At the moment, one of the things I am working on understanding is the total wavelength of a surface like a sphere or something of greater complexity, such as the surface of a bagel or a pretzel. What is this total wavelength? If you strike a surface it can resonate at any one of a list of frequencies, and the wavelength of the sound produced by the vibration is inversely proportional to the frequency. In the mathematically idealized model, there are infinitely many possible wavelengths. The total wavelength should be the sum of all of these individual wavelengths except that this infinite sum equals infinity. Fortunately, a finite number can be assigned to it by a slightly elusive process called regularization. (This process is also used in mathematical physics to mysteriously obtain true answers from formulas which do not really make sense!) I first became interested in the total wavelength as a model related to a question which can be roughly stated as, can one hear the shape of the universe? However, the total wavelength shows up in many quite different areas of mathematics and I am finding these connections intriguing."

"I learned mathematics on my own from textbooks, which is perhaps strange given that both my parents were involved in the subject. At the same time, I spent a good deal of time studying art and wanted to follow a career in that direction until I was eventually convinced by my family that I should first work for a mathematics degree to ensure that I could earn a living."

"My parents separated after my father resigned his university position to focus on his inventions, and my mother then finished her education and became a school mathematics teacher. We moved to a very cosmopolitan area of London, which was like a new birth to me; it was there that my interest in mathematics really began."

"I also attended his eightieth birthday celebration in , in 2003. Peter gave a wonderful polished talk about his experiences at in World War II, which was informative and moving and made a political point. I noticed that he frequently paused to refer to a very small sheaf of notes in his hand. He left the papers on the rostrum after the talk, and out of curiosity I took a look. They were blank! It was a ."

"I would have wished that I could write in some detail of the nature of our work in those wonderfully exciting days. For we were regularly reading the highest grade cipher messages passing between the German High Command and a the senior echelons of the German army, the German navy (including the U-boat fleet) and the Luftwaffe; moreover, we were reading those messages within a few hours of their original transmission. We were thus able to provide as perfect and complete picture of the enemy's plans and dispositions as any nation at war has ever had at its disposal — not lightly did Churchill described our work as his "secret weapon," far more potent than anything Werner von Braun could deploy against us. Unfortunately, the British government currently is behaving in a remarkably paranoid fashion with respect to the revelations of "secrets" by those who at some time (as, of course, I had to do) taken an oath of confidentiality."

"These notions Anaximenes received from Anaximander, Anaximander from Thales himself, who was the Head and Founder of the Ionic Philosophy; and spread this opinion of the Gravity of the Fix'd Stars among his Sect."

"[A]s we are told by Democritus these notions about the Sun and Moon are not to be ascrib'd to Anaxagoras as their original... He had them from his Master Anaximenes whose Opinion... was, that the Stars were of a fiery nature and substance, that there were also mingled with them certain Earthly Bodies... [H]e plainly means, Planets of a terrestrial nature, performing their revolutions in the System of every Fix'd Star."

"[A]fterwards it diffused it self thro' the Italic Philosophy, the followers of which taught, that each Star was a World in the infinite Æthereal Space, containing Earth, Air and Æther; and that the Moon, not only was like our Earth, but inhabited by Animals of a larger size, and furnish'd with Plants of a more beautiful appearance."

"[I]f we look back to the first Rise of Astronomy... we shall find nothing better approv'd of, nothing more universally entertained among the several Sects of Philosophers, than this notion of the Gravity of the Celestial Bodies."

"[W]e do still tread in the steps of the Ancients in this Physical Astronomy; inasmuch as they knew that the Celestial Bodies gravitated towards each other, and were retain'd in their Orbits by the force of Gravity; and were also apprized of the Law of this Gravity."

"That saying is well known, so often used by Anaxagoras, and his Scholars, Achelaus and Euripides, Namely, "That the Sun and Stars were fiery or red-hot Stones and Golden Clods." Of the same mind also were Democritus, Metrodorus, and Diogenes..."

"Nor were they so absurd in their conceptions about Gravity, as to think that it was done by the virtue of any point within the Earth, or of a Center, to which all heavy Bodies placed any where tended; but they thought it was done by the power of the whole Matter in the Terrestrial Globe attracting all things to it self: And as the power of the is composed of the powers of the several parts combin'd together, so they believed that the Gravity towards the whole Earth, resulted from the Gravity towards each single part of it. ...[T]hey believ'd there was a Gravity towards the Moon and Sun, acting in the same manner as it does towards the Earth; and that each Planet, like a Stone, whirl'd in a sling, was kept in its Orbit by the same principle, and for the same reason revolving always about us."

"Upon this account it is, that every Problem in the Terrestrial Physics is very operose and perplex'd, on the contrary, in the Celestial Physics, much more easy and simple; tho' even the latter has its difficulties, arising from the different distances and magnitudes of the Celestial Bodies, For the Fix'd Stars are so vastly distant asunder, that they have no mutual action upon each other, observable by us..."

"For the Sun and Planets are separated from one another by so immense a distance, as renders them incapable of exerting most of those forces whereby all Bodies act upon one another; so that they have no other force left them whereby they can affect one another, but the single force of universal Gravity: Whereas in the production of several Phænomena, that are observ'd upon our Earth, innumerable other forces are exerted, such as are very hard to be distinguish'd from one another; which notwithstanding, if not accurately done, in vain do we attempt Nature, and make any inquiry into it."

"[T]hose persons seem to apply their thoughts but to a very indifferent purpose in the study of Nature, that overlook this part of Astronomy, from whence the principal and most simple Laws of Nature are to be learn'd."

"[T]hose who are less vers'd in the more abstruse parts of Geometry, or less concerned about the Physical parts, may pass over, and only read the Astronomy separately and distinct..."

"[T]he Physics, it is all taken out of the above mention'd Authors; but is here intermix'd with Astronomy, in such places as seem'd proper and convenient; the Geometry to be met with in it, I have either borrowed elsewhere, and quoted... or delivered it Lemmatrically."

"The Celestial Physics, or Physical Astronomy, is not only the first in dignity of all inquiries into Nature... but the first in order, because it is the easiest."

"[T]he Physics delivered in the following Work... was both known and diligently cultivated by the most ancient Philosophers. ...[T]he true System of the World, approv'd of Pythagoras, and others among the Ancients..."

"From some things mention'd by Diogenes Laertius concerning Plato, which also are obscurely hinted at in his Timæus I am apt to believe with Galileo that the divine Philosopher suppos'd the Mundane Bodies, when they were first formed, were moved with a Rectilinear motion (by the means of Gravity,) but after that they had arrived to some determined places, they began to revolve by degrees in a Curve, the Rectilinear Motion being chang'd into a Curvilinear one."

"The phenomena prove that the tail of a comet is vapour from its head which, when the comet is greatly heated in the neighbourhood of the sun in perihelion, continually rises up and moves away into the regions opposite the sun."

"[T]he motion of comets is found to be not at all dissimilar to the motion of the planets, depending on the same principles and undergoing continual revolutions. But, since these planets move in ellipses in which the distance of the foci has a great ratio to the transverse axis, and that major axis is immense, their periodic times greatly exceed the periodic times of the common planets, for they are in the sesquialteral ratio [3:2] of the transverse axes [their squares are as the cubes of the axes]. Thus Descartes’s rectilinear cometary trajectory, which he filched from Kepler, collapses, and many things about comets in his fable of the world may be added, which almost two thousand years ago had been shown to be impossible by Lucretius. Such is the free motion in full spaces, to say nothing of the fact that in his system the motion is from time to time against the motion of the vortex. But, in place of so eccentric ellipses whose more remote parts cannot be observed because the comet is not visible there, parabolae may be assumed in calculation, and many things useful for improving astronomy and physics and advancing them further may thus be deduced with the help of the more intricate geometry."

"[T]he increases of optics, geography and other sciences... are also due to the application of the more intricate geometry to philosophical matters. Hence has been made clear the curvature of the rays of light in the same medium; hence the causes of extraordinary s have been laid bare; hence, given one surface of a lens, another may be determined by means of which a ray entering the lens with given position will have a given position in emerging from it; hence in geography the excess of the normal diameters of the axis over the axis is found, and also the al figure of any planet; hence the varying gravity of the same body in different parts of the Earth, and the varying length of an isochronous pendulum according to the latitude of its place, and then indeed, after the due correction, the construction of a universal measure and of a perfect ."

"Although the celestial spaces in which the planets move around are... unresisting, yet media are considered in which the moving body is resisted, and this resistance is considered in conjunction with gravitation or centripetal force. Among others, this problem now presents itself for solution: Given the direction, the law of centripetal force, and the law of resistance, to construct the path of the projectile. In particular, if the law of centripetal force is posited as reciprocally duplicate to the distances and the resistance is in the duplicate ratio of the speed, then indeed the problem of Galileo will be solved, as is fitting."

"But, since the law of centripetal force employed by nature is to be discovered from its symptoms, the indisputably elliptical orbit and the sesquialteral ratio of the periodic times and the distances from the centre of forces, the same great Newton solved not only the universal problem of determining the trajectory and the motion in it for any given centripetal force, but also its converse. After this universal problem had been solved the sequel was to find other [quantities] in the geometric figure that are measures of physical qualities; for example, that the periodic times in ellipses are in the sesquiplicate ratio of the transverse axes [the squares of the times are as the cubes of the axes], and as many other things similar to these as possible. Also, for instance, to compare this force, which we experience in the planets, with another given force near to us, namely gravity. But also the new philosophy was to concern itself with movable elliptical orbits, in which the line of apsides either advances or retires. Also, for instance, a more exact [theory] of rectilinear descent and of the motion of pendulous bodies than the Huygenian one, since that supposes the centre to be infinitely removed. Therefore also, other s different from the common one and variously devised according as the pendulum oscillates inside or outside the surface of the Earth. And let that suffice for this problem. But also on account of the mutual actions of bodies moving around a centre the orbits usually turn out to be deformed, and also an investigation of these actions and of the deformity arising from them, whence arise many minor inequalities of the planets, such as the motion of the nodes, the variation of maximum latitude, and other things in the moon."

"Descartes's cosmic system, which he jokingly called his fable of the world, is shown to be a fable indeed."

"[F]or the further improvement of natural philosophy a more advanced geometry must be found. ...[T]he reason why physical science has here been brought to a level that is the envy of foreigners is the knowledge... of some more universal geometry. Of what part of this the learned owe to this renowned university and in it to the prince of geometers I shall not speak lest I appear to be fawning, which in a mathematician would be unseemly."

"Mr Issac Newton in addition to the geometric figure in any orbit of a projectile sought also to find the measure of the (tending to a given centre) of the body borne in that orbit, from whatever cause that force may arise, be it from a deeper mechanical one or from a law imposed by the supreme creator of all things. He inquires geometrically into the law of centripetal force of a body moved in the circumference of a circle with the force tending to a given point either on the circumference or anywhere outside it or inside it, or even infinitely removed. By the same method he seeks the law of centripetal force tending to the centre of a plane nautical spiral (that is one that the radii cut in a given angle) which will drive a body in that spiral. Also the law of centripetal force that would make a body rotate in an ellipse when the centre of the ellipse coincides with the centre of forces. If the ellipse is changed into a hyperbola and the centripetal force into a centrifugal one the same things apply to the hyperbola. Also the resolution of the same problem when the centre of forces coincides with either focus of the ellipse shows that the law of centripetal force is reciprocally in the duplicate ratio of the distance [as the inverse square of the distance]; others had long before shown that this was the one and only law that would satisfy the other phenomenon observed by Kepler in the motion of the planets. These results also apply to the hyperbola and the parabola when the centre of forces is situated in a focus of the conic section."

"[W]e have come into the age where questions that were once cosmographical are being transformed into geometrical problems. For it has now been shown not only that the areas which bodies driven in a circuit describe by the radii drawn to the centre of forces are in immobile planes and proportional to the times, but also conversely that every body moved in this way is impelled by a centripetal force that tends towards the aforesaid point. By this proposition alone the Ptolemaic system is destroyed, for the primary planets by radii drawn to the Earth describe areas in no way proportional to the times, while with the radii drawn to the sun it is established that they run over areas sufficiently proportional to the times."

"Since two or more mutually gravitating bodies describe orbits around a common immobile centre of gravity, and since by common consent there is an immense difference between the quantity of matter in the sun and that in the Earth, it is clear that neither the sun nor, much less, the sun in the company of five planets can revolve around an immobile Earth. Thus is shown not only the falsity but the impossibility of the ."

"[W]ith this one problem... three years ago, after the above was published, Gottfried Leibniz... produced a construction... in terms of his system—not without the blemish of paralogism."