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Home , Andrew Strominger, Breakthrough Prize, Cumrun Vafa, Dirac Medal, Edward Witten, Ian Agol

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2017 Breakthrough Prizes Awarded

Jean Bourgain

Breakthrough Prize in Mathematics be obtained from the results of Weil and Deligne on the of the Institute for Advanced Study, Riemann hypothesis over finite fields), as well as the de- Princeton, New Jersey, has been selected as the recipient velopment (with Gamburd and Sarnak) of the affine sieve of the 2017 in Mathematics by the that has proven to be a powerful tool for analyzing thin Breakthrough Prize Foundation. Bourgain was honored for groups. Most recently, with Demeter and Guth, Bourgain “major contributions across an incredibly diverse range of has established several important decoupling theorems areas, including harmonic analysis, functional analysis, er- in Fourier analysis that have had applications to partial godic theory, partial differential equations, mathematical differential equations, combinatorial incidence geometry, , combinatorics, and theoretical computer science.” and analytic number theory, in particular solving the Main The prize carries a cash award of US$3 million. Conjecture of Vinogradov, as well as obtaining new bounds The Notices asked of the University of Cal- on the .” ifornia Los Angeles to comment on the work of Bourgain (Tao was on the Breakthrough Prize committee and was Biographical Sketch: Jean Bourgain also one of the nominators of Bourgain). Tao responded: Jean Bourgain was born in 1954 in Oostende, Belgium. He “Jean Bourgain is an unparalleled problem-solver in anal- received his PhD (1977) and his habilitation (1979) from ysis who has revolutionized many areas of the subject by the Free University of Brussels. He was a professor at the introducing new techniques and ideas. Some of his earliest University of Brussels and at the University of Illinois at contributions include his solution of the Λ(p) problem in Urbana–Champaign, moving in 1985 to the Institut des Fourier analysis, his advances on the Kakeya and restric- tion problems in harmonic analysis and the application Hautes Études Scientifiques in Paris. Ten years later, he of related estimates to nonlinear partial differential became a professor at the Institute for Advanced Study equations, and his numerous works in high dimensional in Princeton, where he is currently the IBM von Neumann geometry, for instance in establishing the reverse Santaló Professor of Mathematics. In 1994, he received the Fields inequality with Milman or the restricted invertibility the- Medal. His other distinctions include the Salem Prize orem with Tzafriri. In the last decade, Bourgain’s inves- (1983), the Élie Cartan Prize of the French Academy of tigations into the sum-product phenomenon have led to Sciences (1990), the Ostrowski Prize (1991), the Shaw deep new exponential sum estimates in number theory Prize in Mathematical Sciences (2010), and the Crafoord (which in some regimes can even improve on what can Prize in Mathematics of the Royal Swedish Academy of Sciences (2012). DOI: http://dx.doi.org/10.1090/noti1509

498 Notices of the AMS Volume 64, Number 5 COMMUNICATION

Breakthrough Prize in Fundamental Physics physics and made many of the important contributions Joseph Polchinski of the Kavli Institute for Theoretical in this area. One milestone was his discovery, with Brian Physics, University of California Santa Barbara, and An- Greene and David Morrison, of a link between singularities drew Strominger and Cumrun Vafa, both of Harvard of a Calabi-Yau and phase transitions in physics. University, were awarded the Breakthrough Prize in Fun- Vafa was one of the originators of the concept of mirror damental Physics for their “serious advances in symmetry between a complex manifold and a symplectic theory, the alleged but still unproven , manifold and contributed many key ideas in this subject. and what it might mean for black holes and the universe.” He also introduced F-theory, an extension of They will share the cash award of US$3 million. using elliptic fibrations. With , Vafa The Notices asked of the Institute for introduced in 1998 what mathematically would be called Advanced Study in Princeton to comment on the work a categorification of the Gromov-Witten invariants; the of Polchinski, Strominger, and Vafa. Witten responded: following year, with Hirosi Ooguri, he developed an analog “Polchinski, Strominger, and Vafa are known for deep and of this for . far-reaching contributions to the understanding of quan- “Polchinski’s other work defies easy summary. He in- tum field theory and string theory, with implications for troduced a new point of view about the renormalization a variety of areas of physics and mathematics. group in condensed matter physics and relativistic field “Polchinski’s single most celebrated contribution, dat- theory. He has made many penetrating contributions to ing back to the early to mid-1990s, has been the theory the understanding of gauge/ and its impli- of D-. Certain string theories that semiclassically cations for our understanding of gauge theories. And in have closed strings only—meaning that the world history recent years, initially with Ahmed Almheiri, Donald Marolf, of the string is a two-manifold without boundary—turn and James Sully, he introduced and developed the baffling out in a full quantum treatment to also have open strings, and highly influential ‘’ paradox involving quantum corresponding to world histories with boundary. D-branes black holes, which still today presents a deep challenge to provided a crucial link between string theory and gauge our understanding of fundamental physics.” theory and have had dramatic consequences—developed in part in some of Polchinski’s later papers—for the under- Biographical Sketch: Joseph Polchinski standing of quantum ‘dualities’ among string theories and Joseph Polchinski was born May 16, 1954, in White Plains, field theories. For , these dualities are an essen- New York. He received his PhD from the University of Cali- tial part of understanding what happens when quantum fornia Berkeley in 1980 under . He was effects are strong and a semiclassical approximation is no professor at the University of Texas at Austin (1984–1992) longer useful. Mathematically, they have many much-stud- before joining U Cal Santa Barbara and the Kavli Institute. ied manifestations, ranging from mirror symmetry to the Among his honors are the Dannie Heineman Prize for geometric Langlands correspondence. (2007), the (2008), and “The single most celebrated contribution of Strominger the Physics Frontiers Breakthrough Prize (2013). He is the and Vafa involved the of black holes. author of the two-volume textbook String Theory (1998). In the early 1970s, and had interpreted the area of the horizon of a as Biographical Sketch: Andrew Strominger a form of entropy. This led many physicists to suspect Andrew Strominger was born in 1955 and received his PhD that when the black hole is studied quantum mechani- in 1982 from the Massachusetts Institute of Technology cally, its horizon area would equal the logarithm of the under the supervision of . He held a faculty number of its quantum states. For a long time, however, position at the University of California Santa Barbara be- it was impossible to demonstrate this through any sort fore joining the faculty of in 1997. His of concrete computation. At first, a suitable quantum honors include the Physics Frontiers Breakthrough Prize theory in which one might do the computation (with Cumrun Vafa, 2012), the Klein and Dirac Medals was not available, and later, when string theory emerged, (2014), and the Dannie Heineman Prize for Mathematical such a calculation was technically out of reach. The first Physics (2016). such calculation was accomplished by Strominger and Vafa in a dramatic 1996 paper. Their main technical tool Biographical Sketch: Cumrun Vafa was to interpret the quantum states of a black hole, in Cumrun Vafa was born in , , in 1960 and an appropriate context, in terms of the elliptic genus of received his PhD from under the the space of vector bundles on a . The direction of Edward Witten. His honors include the AMS connection between black holes and vector bundles was Leonard Eisenbud Prize (2008), the Dirac Medal of the ICTP deduced from the link between string theory and gauge (2008), the Breakthrough Prize in Fundamental Physics theory that is provided by D-branes. (with Andrew Strominger, 2012), and the Dannie Heineman “Many other contributions by Strominger and Vafa Prize for Mathematical Physics (2016). have been influential in mathematics as well as phys- ics. Only a few examples can be cited here. In the mid- About the Prize 1980s, Strominger was one of the initiators of the use of The Breakthrough Prize in Mathematics was created by Calabi-Yau to construct models of particle Mark Zuckerberg and in 2013. It aims to rec-

May 2017 Notices of the AMS 499 COMMUNICATION AmericAn mAthemAticAl Society ognize major advances in the field, to honor the world’s best mathematicians, to support their future endeavors, and to communicate the excitement of mathematics to the general public. The prize is accompanied by a cash award of US$3 million. Previous winners are , , , Terence Tao, Richard Let Your Taylor (2015), and (2016).

The Breakthrough Prize in Fundamental Physics was Voice founded in 2012 by Yuri Milner to recognize those indi- viduals who have made profound contributions to human knowledge. It is open to all physicists—theoretical, mathe- Be Heard! matical, experimental—working on the deepest mysteries of the Universe. The inaugural Breakthrough Prize cere- mony was held March 20, 2013, in Geneva, Switzerland, Do you enjoy what where the Prize was won by Alexander Polyakov. Two Special Fundamental Physics Prizes of $3 million were you’re reading in the awarded to Stephen Hawking and to seven CERN scientists with leading roles in the discovery of the the Higgs boson. Notices?

—Breakthrough Prize Committee Announcement Would you like to Photo Credits Photo of Jean Bourgain is courtesy of Cliff Moore. see an article on a Photo of Andrew Strominger is courtesy of Marina Werbeloff, Harvard University. certain topic? Person? Photo of Joseph Polchinski is courtesy of KITP. Photo of Cumrun Vafa is courtesy of Hayward Photography. Situation?

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500 Notices of the AMS Volume 64, Number 5