School of Mathematics

Beautiful Mathematics

by Daniel Rockmore 

(Credit: Andrea Kane)

My earliest mathematical memories involve my father. One is of a walk from home to the edge of downtown Metuchen (the tiny central Jersey town where I grew up), to a little luncheonette called The Corner Confectionary. This wasn’t a frequent or regular event, but from time to time on a weekend morning we’d make our way there. It was about a mile as we first walked to the corner of Rose Street and Spring Street and then strolled up Spring—a beautiful leafy street with huge oak trees on which our friends the Kahns lived—to finally reach Main Street where we made a quick left, crossed the bridge over the railroad tracks to arrive at the store. I can still see its layout, even in the cluttered neural attic that holds my childhood memories: cash register by the door, rack filled with newspapers, magazines, and comic books, ice cream treats in the back corner, and of course, the long counter, lined with stools on which we would sit and spin—until told to stop.

The Ree Group Formula

by Enrico Bombieri 

The Ree Group Formula (Credit: Enrico Bombieri/Parasol Press, Ltd/Yale Art Gallery/Harlan & Weaver, Inc)

Does beauty exist in mathematics? The question concerns mathematical objects and their relations, the real subject of verifiable proofs. Mathematicians generally agree that beauty does exist in the structural beauty of theorems and proofs, even if most of the time it is largely visible only to mathematicians themselves.

The concept of group beautifully expresses symmetry in mathematics. What is a group? Consider any object, concrete or abstract. A symmetry of the object—mathematically, an automorphism—is a mapping of the object onto itself that preserves all of its properties. The product of two symmetries, one followed by the other, also is a symmetry, and every symmetry has an inverse that undoes it. Mathematicians consider continuous Lie groups, such as the rotations of a circle or of a sphere, to be a beautiful foundation for a great portion of mathematics, and for physics as well. Besides continuous Lie groups there are noncontinuous finite and discrete groups; some are obtainable from Lie groups by reduction to a finite or discrete setting.

John Forbes Nash, Jr., 1928–2015

John Nash, reading in the Fuld Hall Common Room (2011)

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 

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.

Geometric Langlands, Khovanov Homology, String Theory

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 (www.ams.org/notices/201505/ rnoti-p491.pdf).1

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