Mathematicians From The United States

"... She is an academic mathematician turned Wall Street quant turned data scientist who has been involved in and recently started an algorithmic auditing company. She is one of the strongest voices speaking out for limiting the ways we allow algorithms to influence our lives and against the notion that an algorithm, because it is implemented by an unemotional machine, cannot perpetrate bias or injustice."

"... Eventually, I became a tenure-track professor at , which had a combined math department with . And then I made a big change. I quit my job and went to work as a quant for , a leading hedge fund. In leaving academia for finance, I carried mathematics from abstract theory into practice. The operations we performed on numbers translated into trillions of dollars sloshing from one account to another. At first I was excited and amazed by working in this new laboratory, the global economy. But in the autumn of 2008, after I'd been there for a bit more than a year, it came crashing down The crash made it all too clear that mathematics, once my refuge, was not only deeply entangled in the also fueling many of them. The housing crisis, the collapse of major financial institutions, the rise of unemployment—all had been aided and abetted by mathematicians wielding magic formulas."

"The influence on mathematics of its two neighbors, physics and logic, is sometimes opposite or, at least, complementary. Whereas the entropy theorems of probability theory and mathematical physics imply that, in a large universe, disorder is probable, certain combinatorial theorems imply that complete disorder is impossible."

"The theory of s in approximately one century old, although its origin may be traced back much further. As originally formulated by and subsequently used throughout his work, the theory was intended as a tool to be used in the study of geometric problems. After two periods of theoretical development, one in the 1930s and the other in the 1960s, there has recently been a renewed interest in exterior differential systems as providing a systematic framework for the study of geometric problems. It is my opinion that this development is just beginning, and that exterior differential systems should become a standard tool for geometers, especially for questions where the differential equations expressing the problem are overdetermined systems, and for global questions. When used properly, the theory has a marvelous ability to reveal the underlying geometry in a complicated problem."

"One of the points I have tried to make is that mathematics is extremely useful to our society. If this is true, one would think that we as a society would vigorouly support the research that leads to new uses and that students would be at an all time high. Today that is not the case. The mathematics community has yet to effectvely demonstrate to the public and their elected representatives that our subject is dfferent from the sciences. We do not design widgets or cure diseases, yet our impact on engineering and medicine is enabling and significant. But the community has dwelled so long in splendid isolation that the public poorly understands what we do."

"for a smooth algebraic curve includes both the Hodge structure (period matrix) on cohomology and the use of that Hodge structure to study the geometry of the curve, via the . extended the theory of the period matrix to smooth algebraic varieties of any dimension, defining in general a Hodge structure on the cohomology of the variety. He gave a few applications to the geometry of the variety, but these did not attain the richness of the Jacobian variety. In recent years, Hodge theory has been successfully extended to arbitrary varieties, and to families of varieties."

"In the papers published by Hill in the American Journal of Mathematics there is introduced for the first time a very radical and important idea. Up to this time the orbits of the moon and planets were considered as being ellipses which continually change. The problem was to find the changes in the ellipses, or the deviations from the initial ellipses. That is, the ellipse was taken as a first approximation to the orbit of the body under consideration. Hill proposed to take a certain simple type of periodic orbit as a first approximation. He proved the existence of the periodic orbits by numerically integrating the differential equations in numerous special cases by a process known as mechanical quadratures. These were the first periodic orbits of the problem of three bodies having a practical use, and the first ones known to exist beyond the simple ones which were discovered by Lagrange. It should be added that Hill omitted a small part of the disturbing action of the sun, viz., that which is said to depend upon the solar parallax; but his method would have applied without sensible modification to the rigorous problem. In fact, in all his researches, on the problem of three bodies, Darwin used methods which differ from those of Hill only in the variables employed and in inconsequential details."

"The application of mathematics to the solution of the problems presented by the motion of the heavenly bodies has had a larger degree of success than the same application in the case of the other departments of physics. This is probably due to two causes. The principal objects to be treated in the former case are visible every clear night, consequently the questions connected with them received earlier attention; while, in the latter case, the phenomena to be discussed must ofttimes be produced by artificial means in the laboratory; and the discovery of certain classes of them, as, for instance, the property of magnetism, may justly be attributed to accident. A second cause is undoubtedly to be found in the fact that the application of quantitative reasoning to what is usually denominated as physics generally leads to a more difficult department of mathematics than in the case of the motion of the heavenly bodies. In the latter we have but one independent variable, the time; while in the former generally several are present, which makes the difference of having to integrate ordinary differential equations or those which are partial."

"For more than sixty years after the publication of the Principia astronomers were puzzled to account for the motion of the lunar perigee, simply because they could not conceive that terms of the second and higher orders, with respect to the disturbing force, produced more than half of it. For a similar reason, the great inequalities of Jupiter and Saturn remained a long time unexplained."

"It was not until 1748 that any computation of the perturbations of Jupiter and Saturn, in accordance with the theory of gravitation, was undertaken. This was by . He appears to have limited himself to the terms which have the mean elongation of the planets of the planets from each other as their argument. Later the terms factored by the simple power of the eccentricities were added by himself, , , , and . But these terms not bringing about a reconciliation between observation and theory, and were led to make their notable researches on the possibility of secular equations in the mean motions of the planet. At length the whole difficulty with Jupiter and Saturn was removed by discovery of the great inequalities in 1786. almost immediately constructed tables which far exceeded in accuracy any previously possessed. They are those that appear in the third edition of Astronomie."

"About half a century after Thales came Pythagoras. Under his inspiration geometry was first pursued as a study for its own sake. A man of great ability and a most interesting and magnetic mystic, he finally settled at Crotona on the southeastern coast of Italy."

"The idea of constructing a table in which the logarithm of unity was zero originated with Napier. Napier and Briggs never thought of logarithms as exponents of a base. ... It was not till considerably later that our modern definition of a logarithm as an exponent was put forward by such mathematicians as , 1684; , 1742; , 1748, 1770."

"Our knowledge of Babylonian mathematics is derived mainly from tablets in the British Museum, the Prussian State Museum of Berlin, the Ottoman Museum of Constantinople, the University of Strasbourg, the University of Pennsylvania and the Palais du Cinquantenaire of Brussels."

"In yesteryears there were two gloriously inspiring centers of mathematical study in America. One of these was at the University of Chicago, when , and and were in their prime. The other center was at The Johns Hopkins University, 1876–83, where scholars were " Led by soaring-genius'd Sylvester," as has expressed it in his " Ode to The Johns Hopkins University.""

"... in America before the end of 1888 there had been appreciable amount of mathematical research, some of it of first importance, even according to recent standards. There had been centers of mathematical inspiration. Such universities as Yale, The Johns Hopkins, and Harvard, had been sending out doctors in mathematics for a number of years, and many Americans had been getting degrees in Europe. The time was ripe for an organization to draw together many people scattered throughout the country who were especially interested in mathematical pursuits."

"Grain boundaries and surfaces of crystalline materials have a surface free energy which in general depends on the normal direction of the interface relative to the crystal lattice(s). Determining the surface energy minimizing configurations of such interfaces, for a given surface free energy function, is an interesting mathematical problem; it reduces in the case of isotropic (i.e. constant) surface energy to the minimal surface problem. A first step is to classify minimizing cones, since they can arise as tangent cones to minimizing or asymptotically minimizing surfaces. In the isotropic case for two-dimensional surfaces in R^3, the only minimizing cones are planes. For anisotropic surface energy functions, we give here a catalog of 12 types of embedded minimizing cones, and prove that it is a complete catalog among embedded minimizing crystalline cones ..."

"Item: Retirement party for Joanne Elliott. One of my (male) colleagues reminisced about seeing Joanne as an attractive young woman in the common room at Princeton surrounded by young men eager to be near her. The comment made me very uncomfortable, since it placed emphasis on her attractiveness in a setting where conversations are often mathematical. If only the men had been clustered around her because they were eager to hear her theorems and conjectures! But at least as the story was related, that was not the case."

"The subject of motion by crystalline curvature is of interest for three quite distinct reasons. One is that some physical surface energies and physical models of crystal growth simply do give rise to such motion. Another is its use as a way to approximate motion of curves by curvature, both for computation and possibly for proving theorems. The third is that this motion simply is interesting and beautiful in its own right, having results that sometimes parallel those for ordinary curvature and sometimes are strikingly different."

"Surface tension is commonly thought of as a fluid phenomenon; the mere mention of the term brings to mind bugs skimming over water, liquids rising or falling in capillary tubes—and soap films and soap bubbles. But there is in fact a notion of surface tension (which is surface energy per unit surface area) for the interface between any two substances, or even between one substance and a vacuum. This surface energy arises from the fact that atoms (or molecules, or ions) of a given substance have a different environment at the interface between that substance and another than those in the bulk of the substance. (Sometimes even the composition of the surface is different from the bulk; this occurs for instance in soapy water having an interface with air.)"

"A surface free energy function is defined to be crystalline if its Wulff shape (the equilibrium crystal shape) is a polyhedron. All the questions that one considers for the area functional, where the surface free energy per unit area is 1 for all normal directions, can be considered for crystalline surface free energies. Such questions are interesting for both mathematical and physical reasons. Methods from the geometric calculus of variations are useful for studying a number of such questions; a survey of some of the results is given."

"Notices: Can you tell me your memories of Julia Robinson, what she was like as a person? Davis: Very nice, very straightforward. Broad in her interests, mathematical and otherwise. And great power—there is no question in my mind that she was a much more powerful mathematician than I. We worked together on a problem on which we didn’t get anywhere. We were trying to prove the unsolvability of the decision problem for word equations. It turned out that we wouldn’t have been able to do that because the problem is solvable. Makanin solved it positively."

"In Julia Robinson we find a mathematician who was a heroine in her own time and a role model for all time. It is a story of childhood, illness, love, marriage, disappointment, obsession, and triumph."

"The analysis of algorithmic process that emerged from the work of Gödel, Church, Turing, and Post has been of great importance not only for theoretical investigations but also for practice, by providing an expansive framework for computer science. The discussion of computation-like processes that transcend the limits imposed by the Church–Turing thesis can likewise be framed either in terms of theory or of practice."

"Davis became one of the earliest computer programmers when he began programming on the ORDVAC computer at the University of Illinois in the early 1950s. His book Computability and Unsolvability ... first appeared in 1958 and has become a classic in theoretical computer science."

"Takeuti has studied models of axiomatic set theory in which the “truth values” are elements of a complete Boolean algebra of projections on closed subspaces of a Hilbert space, and has found that the real numbers of such a model can be taken to be self-adjoint operators which can be resolved in terms of projections belonging to the Boolean algebra. It is suggested that this is the mathematical source of the replacement of real quantities by operators in quantizing a classical description, and that quantum theory involves a relativity principle with Takeuti's Boolean algebras serving as reference “frames.”"

"A partially computable function may be thought of as one for which we possess an algorithm which enables us to compute its value for elements of its domain, but which will have us computing forever in attempting to obtain a functional value for an element not in its domain, without ever assuring us that no value is forthcoming. In other words, when an answer is forthcoming, the algorithm provides it; when no answer is forthcoming, the algorithm has one spend an infinite amount of time in a vain search for an answer."

"Nonstandard analysis is a technique rather a subject. Aside from theorems that tell us that nonstandard notions are equivalent to corresponding standard notions, all the results we obtain can be proved by standard methods. Therefore, the subject can only be claimed to be of importance insofar as it leads to simpler, more accessible expositions, or (more important) to mathematical discoveries."

"In this paper, we shall show the validity of an iterative procedure suggested by George W. Brown ... This method corresponds to each player choosing in turn the best pure strategy against the accumulated mixed strategy of his opponent up to then."

"We say a mathematical theory is decidable if there is an effective method of determining the validity of each statement of the theory. If there is no such method, the theory is undecidable. It is clear that if there is a mechanical way of transforming each statement of an undecidable theory into an equivalent statement of another theory, the second theory is also undecidable. This principle, together with the fact that the arithmetic of natural numbers is undecidable, enables us to solve the decision problem for fields of finite degree over the rationals."

"And I continued to struggle with the Tenth Problem. In 1961 Martin Davis, Hilary Putnam, and I published a joint paper, "The undecidability of exponential diophantine equations," which used ideas from the papers Martin and I had presented at the International Congress along with various new results. The paper contains what is sometimes referred to as the Robinson hypothesis (or, as Martin calls it, "J.R.") to the effect that if there were some diophantine relation that grew faster than an exponential but not too terribly fast—less than some function could be expressed in exponentials—then we would be able to define exponentiation. It would follow from the definition that exponential diophantine equations would be equivalent to diophantine equations and that, therefore, the solution to Hilbert's tenth problem would be negative. At the time many people told Martin that this approach was misguided, to say the least. They were more polite to me."

"When you're really thinking hard about mathematics, you're in your own world. And you're cushioned from other things."

"... Be guided by beauty ... Beauty is an aesthetic. There is beauty in things that work really well — the way a company is run, or the way a theorem comes out. … Don’t give up easily. Stick to something. Not to the point where it’s clearly insane, but be persistent. … Getting fired once can be a good experience. You just don’t want to make a habit of it. … Work with the smartest people you can. Hopefully smarter than you even. It amplifies your effect. … Hire the smartest people you possibly can … Work collaboratively, and let everyone know what everyone else is researching, so people aren’t wasting their time. … You never know where good science is going to take you."

"A former math professor, Simons is arguably the most successful trader in the history of modern finance. Since 1988, Renaissance's flagship Medallion hedge fund has generated average annual returns of 66 percent, racking up trading profits of more than $100 billion ... No one in the investment world comes close. Warren Buffett, George Soros, Peter Lynch, Steve Cohen, and Ray Dalio all fall short ..."

"Generally speaking, I favor increased levels of support for mathematics and physical and life science. Aside from the inevitable positive impact on the world’s economy, the advancement of knowledge lifts humanity’s spirits, opening our eyes to the wonders of the universe."

"Bers' paper [15], in which he shows that a C-linear algebra isomorphism between the rings of holomorphic functions on two surfaces is induced by a conformal map between them, appeared in 1948 and started a whole industry. ..."

"Minimal surfaces are of interest in various branches of mathematics. In the calculus of variations they appear as surfaces of least area, in differential geometry as surfaces of vanishing mean curvature. In gas dynamics the equation of minimal surfaces,"

"The theory of analytic functions of a complex variable occupies a central place in analysis and it is not surprising that mathematical literature abounds in generalizations. In some generalizations one extends the domain of the functions considered, or their range, or both (functions of several complex variables, analytic functions with values in a vector space or an algebra, analytic functions of hyper-complex variables, analytic operators, etc.) If we restrict ourselves to functions from plane domains to plane domains, or, more generally, from Riemann surfaces to Riemann surfaces, we encounter two well known and very useful generalizations of analytic functions: interior functions and quasi-conformal functions. Interior functions ... have all topological properties of analytic functions and no others. As a matter of fact, they may be defined as functions which can be made analytic by a homeomorphism of the domain of definition. Quasi-conformal functions ... are interior functions subject to an additional metric condition. If the functions are assumed to be continuously differentiable mappings, this additional condition requires that infinitesimal circles be taken into infinitesimal ellipses of uniformly bounded eccentricity."

"The Italian geometers have erected, on somewhat shaky foundations, a stupendous edifice: the theory of algebraic surfaces. It is the main object of modern algebraic geometry to strengthen, preserve, and further embellish this edifice, while at the same time building up also the theory of algebraic varieties of higher dimension. The bitter complaint that Poincaré has directed, in his time, against the modern theory of functions of a real variable cannot be deservedly directed against modern algebraic geometry. We are not intent on proving that our fathers were wrong. On the contrary, our whole purpose is to prove that our fathers were right. ... In helping geometry, modern algebra is helping itself above all. We maintain that abstract algebraic geometry is one of the best things that happened to commutative algebra in a long time."

"The idea of topologizing an algebraic variety V by choosing as closed sets the algebraic subvarieties of V can be used with good effect in order to topologize the set M* of all homomorphic mappings of any abstract field A into another abstract field K. In this general case we are dealing essentially with a generalization of the concept of the Riemann manifold of a field of algebraic functions ..."

"The well known classical treatise by Krazer on the theory of θ functions contains several beautiful chapters dealing with the applications of this theory to algebraic geometry (in the largest sense of the word), but on the whole this treatise is more analytic than geometric in character. In it, page after page, swarms of complicated formulas and relations follow each other, rarely illuminated by a geometric interpretation. One can say, without fear of exaggeration, that all the geometric applications of this theory to algebraic curves and varieties made since Riemann and Weierstrass (Hurwitz, Poincaré, Schottky, Wirtinger, etc.) are absent in Krazer's treatise, or at most are only mentioned in short historical notes."

"In 1869 Dini solved the problem of geodesic representation of two surfaces upon one another. In 1896 Levi-Civita extended the problem to spaces of any order."

"In these days, when the number of papers in mathematics published each year is almost without limit and the ramifications are no less perplexing in their variety, one is delighted to find here and there a digest of the work in a particular field."

"The layman thinks that mathematics deals with facts and that thus there can possibly be no differences of opinion among mathematicians. We know that this is not the case."

"... Lie algebras have a significance reaching beyond the domain of algebra, because they play such an important role in the theory of Lie groups. Thus, classical Lie algebra theory is strongly dominated by the fact that the finite-dimensional analytic representations of a simply connected analytic group are identifiable with the finite-dimensional representations of its Lie algebra. In the theory of infinite-dimensional representations, the connection with Lie algebra representations is somewhat tenuous, but it is nevertheless at the core of the major advances made in that theory during the last 30 years."

"... the theory of valuations may be viewed as a branch of topological algebra. In fact, historically speaking, it represents the first invasion of topology, more precisely, of early metric topology, into the domains of algebra. The introduction of metric methods into algebra has been so fruitful that today many of the deeper algebraic theories carry their mark. In this regard, one should distinguish between the classical use in algebra of the natural metric of the real or complex number fields, such as in proving the "fundamental theorem of algebra," and the much more recent use of the far less evident metrics which are derived from arithmetic notions of divisibility and which constitute the principal notion of valuation theory. Such a metric occurs for the first time in Hensel's construction of the p-adic numbers ..."

"Let F be a field of characteristic 0, and let F be a finite dimensional vector space over F. Let E denote the algebra of all endomorphisms of V, and let L be any Lie subalgebra of E. Among the algebraic Lie algebras contained in E and containing L, there is one that is contained in all of them, and this is called the algebraic hull of L in E. Here, an algebraic Lie algebra is defined as the Lie algebra of an algebraic group. It is an easy consequence of the definitions that if A and B are algebraic groups of automorphisms of V such that A⊂B then the Lie algebra of A is contained in the Lie algebra of B. Hence the existence of the algebraic hull of L is an immediate consequence of the following basic result: let G be the intersection of all algebraic groups of automorphisms of V whose Lie algebras contain L."

"A Lie algebra is said to be algebraic if it is isomorphic with the Lie algebra of an affine algebraic group. In view of the fact that entirely unrelated affine algebraic groups (typically, vector groups and toroidal groups) may have isomorphic Lie algebras, this notion of algebraic Lie algebra calls for some clarification. The most relevant result in this direction is due to M. Goto. It says that a finite- dimensional Lie algebra L over a field of characteristic 0 is algebraic if and only if the image of L under the adjoint representation is the Lie algebra of an algebraic subgroup of the group of automorphisms of L ..."

"Yet long before 1937 people had suggested a non-Greek origin to Greek mathematics; and long before 1943 people had pointed out that sacred books of the East contain the ‘Pythagorean numbers’ [and indeed the ‘theorem’]. Such numbers are mentioned in the Sulvasutras; ancient Indian works on altar constructions. […] Neugebauer does not mention the Sulvasutras in his book Vorlesungen über Geschichte der Antiken Mathematischen Wissenschaften, nor does B. L. van der Waerden them in his book Science Awakening. Why this omission?"

"Seidenberg (1983), "regard[s] it as certain that knowledge of Pythagoras' Theorem was known to the Satapatha Brahmana, which mentions calculations connected with the purusa bird altar, and to the Taittiriya Samhita, which showed similar geometrical awareness" (106). Since these texts are generally dated to around 1000-800 B.C.E., "Greek geometry did not somehow make its way into Vedic geometry, as Greek geometry is only supposed to have started about 600 B.C." (108). Scholars no longer consider Vedic geometry to have been borrowed from the Greeks, so Seidenberg's more controversial claim in the modern context is his rejection of the possibility that the algebra either of the Indians or the Greeks was derived from Babylonia, since the former were aware of aspects of the theorem to which the Babylonians make no reference. He also rejects the possibility that these aspects could have been discovered by the Indians after receiving the basic theorem from Babylonia and then transforming it, and concludes that either "Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source" (Seidenberg 1983, 121)."

"[…] nor did he [Thibaut] formulate the obvious conclusion, namely, that the Greeks were not the inventors of plane geometry, rather it was the Indians. At least this was the message that the Greek scholars saw in Thibaut’s paper. And they didn’t like it."