mathematicians-from-china

"聪明在于勤奋，天才在于积累。"

"I have no doubt that future historians of differential geometry will rank Chern as the worthly successor of Elie Cartan in that field."

"The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Théorie générale des surfaces, Tome 1 (1887), 2 (1888), 3 (1894), 4 (1896), and later editions and reprints. L. Bianchi. Lezioni di Geometria Differenziale, Pisa 1894; German translation by Lukat, Lehrbuch der Differentialgeometrie, 1899. The subject is basically local surface theory."

"Integral geometry, started by the English geometer M. W. Crofton, has received recently important developments through the works of W. Blaschke, L. A. Santaló, and others. Generally speaking, its principal aim is to study the relations between the measures which can be attached to a given variety."

"The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves."

"Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan."

"It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind."

"In 1917 Levi-Civita discovered his celebrated parallelism which is an infinitesimal transportation of tangent vectors preserving the scalar product and is the first example of a connection. The salient fact about the Levi-Civita parallelism is the result that it is the parallelism, and not the Riemannian metric, which accounts for most of the properties concerning curvature."

"Recently, having refreshed my understanding of the mathematics of relativity theory, I called one of my old Berkeley professors to ask him some questions about the geometry of general relativity. S. S. Chern is arguably the greatest living geometer. We spoke on the phone for a long time, and he patiently answered all my questions. When I told him I was contemplating writing a book about relativity, cosmology, and geometry and how they interconnect to explain the universe, he said, "It's a wonderful idea for a book, but writing it will surely take too many years of your life ... I wouldn't do it." Then he hung up."

"S. S. Chern revolutionized differential geometry with the use of moving frames, the invention of characteristic classes, the modern concept of a connection and so much more, but he’ll probably always be most remembered for the yellowing University of Chicago mimeographed lecture notes from the 1950s. An entire generation of geometers learned the elements of differentiable manifolds from those notes."

"Gauss... created one of the first mathematical data sets. He computed the first [~]100,000 prime numbers. ...He found a statistical pattern ...they get sparser and sparser, but the drop-off in the density was inversely proportional to the of the range of numbers. So he conjectured... the the number of primes up to x is... x divided by the natural log of x, and he had no way to prove this. It was a conjecture. It was revolutionary because it was maybe the first important conjecture of math that was statistical in nature. ...It just gave you an approximation that got better... as you went further... out. ...It started the field of... . ...[I]t was the first of many ...which ...started consolidating the idea that the prime numbers ...really didn't have a pattern, that they behaved like random sets of numbers with a certain density. They have some patterns. They're almost all odd. ...They're not actually random. They're... pseudorandom."

"Understand the problem. What kind of problem is it? There are three main types of problems: ‘Show that ...’ or ‘Evaluate ...’ questions, in which a certain statement has to be proved true, or a certain expression has to be worked out; ‘Find a...’ or ‘Find all...’ questions, which requires one to find something (or everything) that satisfies certain requirements; ‘Is there a ...’ questions, which either require you to prove a statement or provide a counterexample (and thus is one of the previous two types of problem)."

"Relying on intelligence alone to pull things off at the last minute may work for a while, but generally speaking at the graduate level or higher it doesn't. One needs to do a serious amount of reading and writing, and not just thinking, in order to get anywhere serious in mathematics."

"[P]rofessional mathematics is not a sport (in sharp contrast to mathematics competitions). The objective in mathematics is not to obtain the highest ranking, the highest score, or the highest number of prizes and awards; instead, it is to increase understanding of mathematics (both for yourself, and for your colleagues and students), and to contribute to its development and applications."

"For the third law.., Kepler had... 6 data points..: every planet.., the length of the orbit and the distance to the sun.., and he did... regression. ...He could fit a curve to these 6 data points and he got a square cube law. Amazing, but he was... lucky. That's not enough data to be... reliable."

"There was a later astronomer, Johann Bode who took the same data... and... had a prediction that the distances to the planets formed a shifted ... He also fit a curve, except there was one point missing, a... gap... His law predicted... a missing planet. ...[W]hen Uranus was discovered by Herschel, the distance fit... this pattern. Then Ceres was discovered... in the astroid belt, and it also fit the pattern. ...But then Neptune was discovered, and it was... way off. ...[I]t was a numerical fluke."

"Over time it became more... productive to think of the primes as if they were generated by some... random set, and this allowed us to make... other predictions. ...[B]ecause of this statistical random model.., we are ...absolutely convinced it's true. ...[W]e have, over time, developed this very accurate conceptual model of what primes should behave like.., but it's mostly and non-rigorous, but extremely accurate. It's the same reason we believe the Riemann hypothesis is true, and why we believe that cryptography based on the primes is... mathematically secure."

"[O]ne reason why we care about the Riemann hypothesis is that if... we knew it was false, ...it would mean there was a secret pattern to the primes that we were not aware of, and I think we would very rapidly abandon any cryptography based on the primes, because if there was one pattern... there are probably more, and these... can lead to exploits in crypto."

"I do not plan to come back. I have no reason to come back.. I plan to do my best to help the Chinese people build up the nation to where they can live with dignity and happiness."

"Qian Xuesen... didn't like being called the father of China's guided-missile program: he felt that the title didn't give credit to his fellow researchers. Indeed, while the Chinese-born, U.S.-educated rocket scientist was technically brilliant, he also realized that legions of bright thinkers can do far more than one genius ever could."

"In 1948, the MIT mathematician Norbert Wiener gave a widely read, albeit completely nonmathematical, account of cybernetics. A more mathematical treatment of the elements of engineering cybernetics was presented by H.S. Tsien in 1954, driven by problems related to control of missiles. Together, these works and others of that time form much of the intellectual basis for modern work in robotics and control."

"That the government permitted this genius, this scientific genius, to be sent to Communist China to pick his brains is one of the tragedies of this century."

"It was the stupidest thing this country ever did. He was no more a Communist than I was, and we forced him to go."

"An engineering science aims to organize the design principles used in engineering practice into a discipline and thus to exhibit the similarities between different areas of engineering practice and to emphasize the power of fundamental concepts. In short, an engineering science is predominated by theoretical analysis and very often uses the tool of advanced mathematics."

"The purpose of "Engineering Cybernetics" is then to study those parts of the broad science of cybernetics which have direct engineering applications in designing controlled or guided systems. It certainly includes such topics usually treated in books on servomechanisms. But a wider range of topics is only one difference between engineering cybernetics and servomechanisms engineering. A deeper - and thus more important - difference lies in the fact that engineering cybernetics is an engineering science, while servomechanisms engineering is an engineering practice."

"The celebrated physicist and mathematician A.M. Ampere coined the word cybernetique to mean the science of civil government (Part II of "Essai sur la philosophic des sciences", 1845, Paris). Ampere's grandiose scheme of political sciences has not, and perhaps never will, come to fruition. In the meantime, conflict between governments with the use of force greatly accelerated the development of another branch of science, the science of control and guidance of mechanical and electrical systems. It is thus perhaps ironical that Ampere's word should be borrowed by N. Wiener to name this new science, so important to modern warfare. The "cybernetics" of Wiener ("Cybernetics, or Control and Communication in the animal and the Machine," John Wiley & Sons, Inc., New York, 1948) is the science of organization of mechanical and electrical components for stability and purposeful actions. A distinguishing feature of this new science is the total absence of considerations of energy, heat, and efficiency, which are so important in other natural sciences. In fact, the primary concern of cybernetics is on the qualitative aspects of the interrelations among the various components of a system and the synthetic behavior of the complete mechanism."