# School of Mathematics

## John Forbes Nash, Jr., 1928–2015

On May 19, 2015, King Harald V of Norway presented the Abel Prize from the Norwegian Academy of Science and Letters to John Forbes Nash, Jr., Member (1956–57, 1961–62, 1963–64) in the School of Mathematics, and long-time member of the Princeton University Department of Mathematics, for his contributions to the theory of nonlinear partial differential equations, which are used to describe the basic laws of phenomena in physics, chemistry, biology, and other sciences. Returning to Princeton from the prize ceremony in Oslo, Nash and his wife Alicia died together in an automobile accident. “I hope one thing will become clear when we look back on Dr. John Nash’s life,” observed Robbert Dijkgraaf, Director of the Institute and Leon Levy Professor. “There are many brilliant minds, but he was a very special kind. . . . He was always going in directions that were either thought to be impossible, or actively discouraged.”

## How Topology Detects Certain Phases of Matter

### by Daniel S. Freed

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.

## Geometric Langlands, Khovanov Homology, String Theory

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 (www.ams.org/notices/201505/ rnoti-p491.pdf). ^{1}*

## Curiosities: Permutation Puzzles: From Archimedes to the Rubik’s Cube

### by Matthew Kahle

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.

## Lens of Computation on the Sciences

*What do quantum interference, flocking of birds, Facebook communities, and stock prices have in common?*

Many natural and social phenomena may be viewed as inherently computational; they evolve patterns of information that can be described algorithmically and studied through computational models and techniques. A workshop on the computational lens, organized by Avi Wigderson, Herbert H. Maass Professor in the School of Mathematics, highlighted the state-of-art and future challenges of this interaction of computational theory with physics, social sciences, economics, and biology.