Mathematicians From The United States

"It was the work of... Friedmann, Robertson and Walker, which resulted in the general mathematical framework that is still used today when discussing relativistic cosmological models of a homogeneous and isotropic universe."

"The basic cause and nature of cosmic expansion, along with its recently-observed acceleration, are significant problems of the standard model; so, condisering the evidence that the acceleration is best described by pure \Lambda, there is strong motivation to search for an alternative big bang model that would respect the pioneering concept of expansion, as a direct consequence of the ‘de Sitter effect’ in the modified . It is therefore worth investigating the axiomatic basis of the Robertson-Walker (RW) line-element."

"[I]nvestigations of cosmological issues, whose only observational basis was the observations. ...involved ...mostly of mathematicians like Howard Robertson and and astronomers with strong mathematical training such as Eddington, Lemaître, George McVittie, and William McCrea. They were mainly interested in how to apply general relativity to cosmological problems, which involved not only understanding cosmic dynamics but also solving the intricate problem of interpreting cosmological solutions to the Einstein equations, in particular separating time (which determined the evolution of the universe) from space (to which simplified assumptions concerning the structure of the universe, such as homogeneity and , were to be applied)."

"This... is an outrageously over-simplified account of the assumptions and procedures..."

"[I]n deriving the general line-element for the background geometry of FLRW [Friedmann–Lemaître–Robertson–Walker, sometimes called the Standard Model] cosmology, Robertson required four basic assumptions: i. a congruence of s, ii. , iii. homogeneity, and iv. . i. and ii. are required to satisfy of a causal coherence amongst s in the entire Universe, by which every single event in the bundle of fundamental world lines is associated with a well-defined three-dimensional set of others with which it ‘really’ occurs simultaneously. However, it seems that ii. is therefore mostly required to satisfy the concept that synchronous events in a given inertial frame should have occurred simultaneously, against which I’ve argued..."

"The search for the curvature K indicates that, after making all known corrections, the number N seems to increase faster with d than the third power, which would be expected in a Euclidean space, hence K is positive. The space implied thereby is therefore bounded, of finite total volume, and of a present "radius of curvature" R = \frac{1}{K^\frac{1}{2}} which is found to be of the order of 500 million light years. Other observations, on the "red shift" of light from these distant objects, enable us to conclude with perhaps more assurance that this radius is increasing..."

"In what respect... does the general theory of relativity differ...? The answer is: in its universality; the force of gravitation in the geometrical structure acts equally on all matter. There is here a close analogy between the gravitational mass M...(Sun) and the inertial mass m... (Earth) on the one hand, and the heat conduction k of the field (plate)... and the coefficient of expansion c... on the other. ...The success of the general relativity theory... is attributable to the fact that the gravitational and inertial masses of any body are... rigorously proportional for all matter."

"What is the true geometry of the plate? ...Anyone examining the situation will prefer Poincaré's common-sense solution... to attribute it Euclidean geometry, and to consider the measured deviations... as due to the actions of a force (thermal stresses in the rule). ...On employing a brass rule in place of one of steel we would find that the local curvature is trebled—and an ideal rule (c = 0) would... lead to Euclidean geometry."

"The field equation may... be given a geometrical foundation, at least to a first approximation, by replacing it with the requirement that the mean curvature of the space vanish at any point at which no heat is being applied to the medium—in complete analogy with... the general theory of relativity by which classical field equations are replaced by the requirement that the Ricci contracted curvature tensor vanish."

"What is needed is a homely experiment which could be carried out in the basement with parts from an old sewing machine and an Ingersoll watch, with an old file of Popular Mechanics standing by for reference! This I am, alas, afraid we have not achieved, but I do believe that the following example... is adequate to expose the principles..."

"Let a thin, flat metal plate be heated... so that the temperature T is not uniform... clamp or otherwise constrain the plate to keep it from buckling... [and] remain [reasonably] flat... Make simple geometric measurements... with a short metal rule, which has a certain coefficient of expansion c... What is the geometry of the plate as revealed by the results of those measurements? ...[T]he geometry will not turn out to be Euclidean, for the rule will expand more in the hotter regions... [T]he plate will seem to have a negative curvature K... the kind of structure exhibited... in the neighborhood of a ".""

"Now it is the practice of astronomers to assume that brightness falls off inversely with the square of the "distance" of an object—as it would do in Euclidean space, if there were no absorption... We must therefore examine the relation between this astronomer's "distance" d... and the distance r which appears as an element of the geometry."

"We have merely (!) to measure the volume V of a sphere of radius r or the sum \sigma of the angles of a triangle of measured are \delta, and from the results to compute the value of K."

"These formulae [in (1) and (2) above] may be shown to be valid for a circle or a triangle in the hyperbolic plane... for which K < 0. Accordingly here the perimeter and area of a circle are greater, and the sum of the three angles of a triangle are less, than the corresponding quantities in the Euclidean plane. It can also be shown that each full line is of infinite length, that through a given point outside a given line an infinity of full lines may be drawn which do not meet the given line (the two lines bounding the family are said to be "parallel" to the given line), and that two full lines which meet do so in but one point."

"[T]he space constant K... "" may in principle at least be determined by measurement on the surface, without recourse to its embodiment in a higher dimensional space."

"The value of the intrinsic approach is especially apparent in considering 3-dimensional congruence spaces... The intrinsic geometry of such a space of curvature K provides formulae for the surface area S and the volume V of a "small sphere" of radius r, whose leading terms are 3)S = 4 \pi r^2 (1 - \frac{Kr^2}{3} + ...), V = \frac{4}{3} \pi r^3 (1 - \frac{Kr^2}{5} + ...)."

"All the light which is radiated... will, after it has traveled a distance r, lie on the surface of a sphere whose area S is given by the first of the formulae (3). And since the practical procedure... in determining d is equivalent to assuming that all this light lies on the surface of a Euclidean sphere of radius d, it follows...4 \pi d^2 = S = 4 \pi r^2 (1 - \frac{K r^2}{3} + ...);whence, to our approximation 4)d = r (1- \frac{K r^2}{6} + ...), or r = d (1 + \frac{K d^2}{6} + ...)."

"[W]e propose... to deal exclusively with properties intrinsic to the space... measured within the space itself... in terms of... inner properties."

"Measurements which may be made on the surface of the earth... is an example of a 2-dimensional congruence space of positive curvature K = \frac{1}{R^2}... [C]onsider... a "small circle" of radius r (measured on the surface!)... its perimeter L and area A... are clearly less than the corresponding measures 2\pi r and \pi r^2... in the Euclidean plane. ...for sufficiently small r (i.e., small compared with R) these quantities on the sphere are given by 1):L = 2 \pi r (1 - \frac{Kr^2}{6} + ...), A = \pi r^2 (1 - \frac{Kr^2}{12} + ...)"

"In all these congruence geometries, except the Euclidean, there is at hand a natural unit of length R = \frac{1}{K^\frac{1}{2}}; this length we shall, without prejudice, call the "radius of curvature" of the space."

"In the sum \sigma of the three angles of a triangle (whose sides are arcs of s) is greater than two right angles [180&deg;]; it can... be shown that this "spherical excess" is given by 2)\sigma - \pi = K \deltawhere \delta is the area of the spherical triangle and the angles are measured in s (in which 180&deg; = \pi [radians]). Further, each full line (great circle) is of finite length 2 \pi R, and any two full lines meet in two points—there are no parallels!"

"is a congruence geometry, or equivalently the space comprising its elements is homogeneous and isotropic; the intrinsic relations between... elements of a configuration are unaffected by the position or orientation of the configuration. ...[M]otions of are the familiar translations and rotations... made in proving the theorems of Euclid."

"[O]nly in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"The solution of (1), which represents a homogeneous manifold, may be written in the form:ds^2 = \frac{d\rho^2}{1 - \kappa^2\rho^2} - \rho^2 (d\theta^2 + sin^2 \theta \; d\phi^2) + (1 - \kappa^2 \rho^2)\; c^2 d\tau^2, \qquad (2)where \kappa = \sqrt \frac{\lambda}{3}. If we consider \rho as determining distance from the origin... and \tau as measuring the proper-time of a clock at the origin, we are led to the de Sitter spherical world..."

"An "empty world," i.e., a homogeneous manifold at all points at which equations (1) are satisfied, has, according to the theory, a constant Riemann curvature, and any deviation from this fundamental solution is to be directly attributed to the influence of matter or energy."

"Every one knows that the fine phrase "God geometrizes" is attributed to Plato, but few know where this famous passage is found, or the exact words in which it was first expressed. Those who, like the author, have spent hours and even days in the search of the exact statements, or the exact references, of similar famous passages, will not question the timeliness and usefulness of a book whose distinct purpose it is to bring together into a single volume exact quotations, with their exact references, bearing on one of the most time-honored, and even today the most active and most fruitful of all the sciences, the queen-mother of all the sciences, that is, mathematics."

"Within mathematics itself, Voevodsky's proposal, if adopted, will create a new paradigm. In his “fairy tale” and some of his other papers, Langlands made deft use of categories and even 2-categories, but number theory is only superficially categorical, and so is the Langlands program. In the event that Univalent Foundations could shed light on a guiding problem in number theory — the Riemann hypothesis or the Birch Swinnerton-Dyer conjecture, which is not so far removed from Voevodsky's motives — then we could easily see Grothendieck's program absorbing the Langlands program within Voevodsky's new paradigm."

"We should, of course, expect that any universe which expands without limit will approach the empty de Sitter case, and that its ultimate fate is a state in which each physical unit—perhaps each nebula or intimate group of nebulae—is the only thing which exists within its own observable universe."

"That the existence of these motions (the "axiom of free mobility") is a desideratum, if not... a necessity, for a geometry applicable to physical space, has been forcefully argued on a priori grounds by von Helmholtz, Whitehead, Russell and others; for only in a homogeneous and isotropic space can the traditional concept of a rigid body be maintained."

"The general theory of relativity considers physical space-time as a four-dimensional manifold whose line element coefficients g_{\mu \nu} satisfy the differential equationsG_{\mu \nu} = \lambda g_{\mu \nu} \qquad .\;.\;.\;.\;.\;.\; (1)in all regions free from matter and electromagnetic field, where G_{\mu \nu} is the contracted Riemann-Christoffel tensor associated with the fundamental tensor g_{\mu \nu}, and \lambda is the ."

"Langlands' life has been by no means as extravagant as Grothendieck's, but his romanticism is evident to anyone who reads his prose; the audacity of his program, one of the most elaborate syntheses of conjectures and theorems ever undertaken, has few equivalents in any field of scholarship."

"In considerations involving the nature of the world as a whole the irregularities caused by the aggregation of matter into stars and stellar systems may be ignored; and if we further assume that the total matter in the world has but little effect on its macroscopic properties, we may consider them as being determined by the solution of an empty world."

"Literametrics' may become to philology, what 'Biometrics' has already become to the biological sciences."

"Since I was a teenager I had an interest in number theory. Fortunately, I came across a good number theory book by L. E. Dickson, so I knew a little number theory. Also I had been reading Bell’s histories of people like Gauss. I liked number theory. It’s natural, in a way, because many wonderful problems and theorems in number theory can be explained to any interested high-school student. Number theory is easier to get into in that sense. But of course it depends on one’s intuition and taste also."

"It doesn't matter how long it takes, if the end result is a good theorem."

"Moreover, according to Alexander, the “troubled mathematical martyr,” exemplified not only by Galois but also by Abel, János Bolyai, Riemann, Cantor, Gödel, Turing, John Nash, Grothendieck, Perelman, and even, in a certain sense, Cauchy, remains to this day the dominant image of the “ideal mathematician,” long after the romantic paradigm was exhausted in the arts."

"A “good theorem,” as Tate puts it, lasts forever. Once proved, it will always stay proved, and other mathematicians are free to use it and build on it as they please, sometimes to great effect."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"Langlands and Grothendieck are both (at least) Giants by any measure, and both were consciously successors of Galois."

"[T]he astronomical data give the number N of nebulae counted out to a given inferred "distance" d, and in order to determine the curvature... we must express N, or equivalently V, to which it is assumed proportional, in terms of d. ...from the second of formulae (3) and... (4)... to the approximation here adopted, 5)V = \frac{4}{3} \pi d^2 (1 + \frac{3}{10} K d^2 + ...);...plotting N against... d and comparing... with the formula (5), it should be possible operationally to determine the "curvature" K."

"Euclidean geometry is only one of several congruence geometries... Each of these geometries is characterized by a real number K, which for Euclidean geometry is 0, for the hyperbolic negative, and for the spherical and elliptic geometries, positive. In the case of 2-dimensional congruence spaces... K may be interpreted as the ' of the surface into the third dimension—whence it derives its name..."

"Lefschetz made a brief trip to Rome, and I asked Severi what he thought of Lefschetz's work. È bravo,' Severi told me, meaning more or less 'he is talented.' The word bravo,' as far as I can tell, has no equivalent in other languages. 'He is no Poincaré,' Severi added. Poincaré was an eagle, un'aquila—and at this, he raised his hand high. Lefschetz was a sparrow, un passero,' and he lowered his hand halfway. But Lefschetz was talented, è bravo però, è bravo."

"I would read Lefschetz's book on analysis situs and algebraic geometry - the book about which Hodge used to say that all the important statements were true and all the others false."

"Solomon Lefschetz had a reputation for “kibitzing” during the lectures of colleagues; at perhaps the first public talk on computing that von Neumann ever gave, von Neumann said, halfway in, “Well, so far so good” — and Lefschetz added, “and so trivial.” (The students passed down an impious faculty song about their elders; the verse for Lefschetz ended, “When he’s at last beneath the sod, he’ll then begin to heckle God.”)"

"It was my lot to plant the harpoon of algebraic topology into the body of the whale of algebraic geometry."

"As is well known, when one endeavors to pass from one-dimensional birational geometry to the higher dimensions, the difficulties multiply enormously. Many results do not extend at all, or if they do, they are apt to assume a far more complicated aspect or else to demand most difficult proofs."

"In its early phase (Abel, Riemann, Weierstrass), algebraic geometry was just a chapter in analytic function theory. ... A new current appeared however (1870) under the powerful influence of Max Noether who really put "geometry" and more "birational geometry" into algebraic geometry. In the classical mémoire of Brill-Noether (Math. Ann., 1874), the foundations of "geometry on an algebraic curve" were laid down centered upon the study of linear series cut out by linear systems of curves upon a fixed curve ƒ{x, y) = 0. This produced birational invariance (for example of the genus p) by essentially algebraic methods."

"It is quite obvious that he was strongly influenced by the similar problems in algebraic geometry, and in particular by the theory of correspondences, studied since the middle of the nineteenth century by Chasles and the school of “enumerative geometry” (de Jonquières, Zeuthen, Schubert), then by Hurwitz in the theory of Riemann surfaces, and which had been thoroughly investigated by Severi in the first years of the twentieth century; this influence explains the rather unusual frame within which Lefschetz developed his theory."

"The numerical relations existing between ordinary or so-called Plückerian singularities of a plane curve were determined as early as 1834 by P, but the inverse question has been left almost untouched. It may be stated thus: To show the existence of a curve having assigned Plückerian characters; and is equivalent to the determination of the maximum of cusps κM that a curve of order m and genus p may have. V ... has solved the question for rational curves."

"In the development of the theory of algebraic functions of one variable the introduction by Riemann of the surfaces that bear his name has played a well-known part. Owing to the partial failure of space intuition with the increase in dimensionality, the introduction of similar ideas into the field of algebraic functions of several variables has been of necessity slow. It was first done by Emile Picard, whose work along this line will remain a classic. A little later came the capital writings of Poincaré in which he laid down the foundations of Analysis Situs, thus providing the needed tools to obviate the failure of space intuition."