Articles from the Institute Letter

Additional articles from new and past issues of the Institute Letter will continue to be posted over time and as they become available.

by Danian Hu

C. S. Wang Chang at IAS

Addressing an international audience in 2004, Professor Dong Guangbi, an erudite historian of science, summarized Chinese physics development over the previous century, and he argued that the country from which Chinese physicists and physics benefited most was the United States of America.2 Dong’s argument was supported by the background of the seven “most creative Chinese physicists.”

Five out of these seven received doctorates in America and four of the five— Chou, Wu, Yang, and Lee—were former Members of the Institute for Advanced Study, indicating the dominating American influence and the significant role of IAS in Chinese development. This essay supports Dong’s thesis with additional evidence revealed in my preliminary survey of Chinese physicists schooled in America during the first half of the twentieth century.

The first Chinese physicist to graduate from an American college was most likely Yuanli Hsia (夏元瑮, 1883–1944), one of a few in the first generation of Chinese physicists. Sponsored by the Guangdong Provincial Government, Hsia came to study at Yale University. Upon his graduation in 1907, Hsia went on to the University of Berlin where he studied with Max Planck and Heinrich Rubens before his return to China in 1912. He then served six years at Peking University as dean of the School of Sciences. Remarkably, Hsia did not accept Einstein’s theory of relativity before 1919 when he returned to Berlin and met Einstein through Planck. Hsia’s early resistance to relativity seemed to be partially influenced by his Yale professor Henry Bumstead. After studying with Einstein during 1919–1921, however, Hsia became an active and enthusiastic relativist who delivered numerous speeches and published many articles in China, expounding and advocating Einstein’s theories. He produced in 1921 the first Chinese translation of Einstein’s only popular book, Relativity: The Special and General Theories.


by Daniel S. Freed

Moduli space of gapped quantum systems​

Topology is the branch of geometry that deals with large-scale features of shapes. One cliché is that a topologist cannot distinguish a doughnut from a coffee cup: if a coffee cup were made of rubber, one could continuously deform it to a doughnut without tearing. A geometer, equipped with precision tools, can measure local quantities (distances, curvature) to distinguish the coffee cup from the doughnut. A topologist, seemingly handicapped by defective eyes, can only discern that each has one hole, so at least can distinguish both from a two-holed pretzel. But, after all, a topologist is a geometer too, and the lack of close vision can reveal a forest otherwise obscured by trees.

The problem described here—the classification of phases of matter—has great current relevance. Beyond that, the story I tell is one small illustration of the many wonders of mathematics: the abstract and artistic impulses, which guide the internal development of mathematical ideas, yield theorems with unanticipated powerful applications to scientific and technological problems far removed from the original source of and inspiration for those ideas.


by Johannes M. Henn

In systems where the conservation is only approximate, the elliptical orbits precess, meaning the orientation of the ellipses does not change. (Figure adapted from Wikipedia)

What do the motion of the planets in our solar system, the energy levels of the hydrogen atom, and the interactions between subatomic particles have in common? Surprisingly, they are all governed by the same hidden symmetry principles.

Symmetry is a very important notion in physics, for mainly two reasons. On the one hand, systems with a lot of symmetry are usually easier to solve and study, so that key properties can be understood analytically. On the other hand, and more fundamentally, in the development of physics, symmetry principles have often been a successful guiding principle toward theories relevant for describing nature. An example is Einstein’s equivalence principle that led to the development of general relativity.

What is the hidden symmetry underlying the motion of the planets, such as the Sun and the Earth? The answer to this question is important for the Kepler problem, i.e., the question of how to predict the position and velocity of two bodies, given some initial conditions. (It should be noted that physicists often use the word “problem” not in the standard meaning, which has a negative connotation; rather, it should be thought of in a positive sense, as an interesting challenge.) The motion is governed by Newton’s laws, which tell us, in particular, that the gravitational force between two objects depends only on their relative distance. From this, it follows that the orbits lie in a plane. However, observing the trajectories more closely, one sees that they form ellipses that do not precess with time. In other words, the orientation of the ellipses does not change, and hence the orbits are closed. This regularity is a hint for a hidden symmetry, which in turn implies a constant of motion. Indeed, a certain vector, named after Laplace-Runge-Lenz (LRL), does not change with time (see figure).1 It points toward the perihelion of the ellipse, i.e., the point of the orbit where the Earth comes closest to the Sun, and its conservation explains the regularity of the orbits that we observe.

Edward Witten at the 2013 Prospects in Theoretical Physics Program at the Institute (Dan Komoda)

In 2006, Edward Witten, Charles Simonyi Professor in the School of Natural Sciences, cowrote with Anton Kapustin a 225-page paper, “Electric-Magnetic Duality and the Geometric Langlands Program,” on the relation of part of the geometric Langlands program to ideas of the duality between electricity and magnetism.

Some background about the Langlands program: In 1967, Robert Langlands, now Professor Emeritus in the School of Mathematics, wrote a seventeen-page handwritten letter to André Weil, a Professor at the Institute at the time, in which he proposed a grand unifying theory that relates seemingly unrelated concepts in number theory, algebraic geometry, and the theory of automorphic forms. A typed copy of the letter, made at Weil’s request for easier reading, circulated widely among mathematicians in the late 1960s and 1970s, and for more than four decades, mathematicians have been working on its conjectures, known collectively as the Langlands program.

Witten spoke about his experience writing the paper with Kapustin and his thoughts about future directions in mathematics and physics in an interview that took place in November 2014 on the occasion of Witten’s receipt of the 2014 Kyoto Prize in Basic Sciences for his outstanding contributions to mathematical science through his exploration of superstring theory. The following excerpts are drawn from a slightly edited version of the interview conducted by Hirosi Ooguri, Member (1988–89) and Visiting Professor (2015) in the School of Natural Sciences, which was published in the May 2015 issue of Notices of the American Mathematical Society ( rnoti-p491.pdf).1


by Matthew Kahle

(Imtiyaz Quraishi)

The Rubik’s Cube is one of the most popular toys in history. It is also an example of a permutation puzzle, which have existed in mathematics in one form or another for at least 140 years. In hindsight, it is strange that the cube ever became so popular considering how hard it is. Very few ever solved it, and far fewer without a book or website as guide. (To his credit, the Hungarian professor of architecture who invented the cube took a few weeks but solved it.) It might be surprising that more than thirty years after the 1982 craze, people are still discovering new things about the Rubik’s Cube.

It might be surprising that more than thirty years after the 1982 craze, people are still discovering new things about the Rubik’s Cube. For example, mathematician Morley Davidson (Member 1995–96) and Tomas Rokicki recently showed that God’s number of the Rubik’s Cube is twenty-six. Their proof depends on a supercomputer calculation.

There is an almost inconceivably large number of positions for the cube––roughly 43 billion billion (4.3 x 1019). Although this number is enormous, it is finite, so there has to be in some sense “a worst-case scenario,” a position that is maximally mixed up.