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Faltings and Iwaniec Awarded Shaw Prize

Faltings and Iwaniec Awarded Shaw Prize

Faltings and Iwaniec Awarded

On June 1, 2015, the Shaw Foun- The Work of dation announced the awarding of “A polynomial equation of degree n in one variable the 2015 Shaw Prize in Mathemat- with coefficients which are rational numbers has ical Sciences to Gerd Faltings just n complex numbers as solutions. Such an of the Max Planck Institute for equation has a symmetry , its Galois group, in , , that describes how these complex solutions are and of Rutgers related to each other. University “for their introduction “A polynomial equation in two variables with and development of fundamental rational coefficients has infinitely many complex tools in , allowing solutions, forming an algebraic curve. In most

Max Planck Institute for Mathematics, Bonn, Germany. them as well as others to resolve cases (that is, when the curve has genus 2 or more) Gerd Faltings some long-standing classical prob- only finitely many of these solutions are pairs of lems.” The prize carries a cash rational numbers. This well-known conjecture of award of US$1 million. Mordell had defied resolution for sixty years before Faltings proved it. His unexpected proof provided The Shaw Prize in Mathematical fundamental new tools in Arakelov and arithmetic Sciences Committee released the , as well as a proof of another fundamen- following statement about the tal finiteness theorem—the Shafarevich and Tate prizewinners’ work. Conjecture—concerning polynomial equations in “Number theory concerns many variables. Later, developing a quite different whole numbers, prime numbers, method of Vojta, Faltings established a far-reach- and polynomial equations involv- ing higher dimensional finiteness theorem for ing them. The central problems rational solutions to systems of equations on are often easy to state but ex- Abelian varieties (the Lang Conjectures). In order traordinarily difficult to resolve. to study rational solutions of polynomial equations Nick Romanenko, . Henryk Iwaniec Success, when it is achieved, re- by geometry, one needs arithmetic versions of the lies on tools from many fields of tools of complex geometry. One such tool is Hodge mathematics. This is no coincidence since some of theory. Faltings’s foundational contributions to these fields were introduced in attempts to resolve Hodge theory over the p-adic numbers, as well as classical problems in number theory. Faltings and his introduction of other related novel and pow- Iwaniec have developed many of the most powerful erful techniques, are at the core of some of the modern tools in algebra, analysis, algebraic and recent advances connecting Galois groups (from , automorphic forms, and the polynomial equations in one or more variables) theory of zeta functions. They and others have and the modern theory of automorphic forms (a used these tools to resolve long-standing problems vast generalization of the theory of periodic func- in number theory.” tions). The recent striking work of concerning Galois representations is a good exam- DOI: http://dx.doi.org/10.1090/noti1273 ple of the power of these techniques.”

1074 Notices of the AMS Volume 62, Number 9 The Work of Henryk Iwaniec Mathematics. He was awarded the in “Iwaniec’s work concerns the analytic side of di- 1986. He held a Guggenheim Fellowship in 1988 ophantine analysis, where the goal is usually to and is also the recipient of the Gottfried Wilhelm prove that equations do have integral or prime (1996) and the King Faisal Interna- solutions, and ideally to estimate how many there tional Prize for Science (2014). are up to a given size. Henryk Iwaniec was born in 1947 in Elblag, Po- “One of the oldest techniques for finding primes land. He received his PhD from the University of is , originating in Eratosthenes’s Warsaw in 1972. He held positions at the Institute description of how to list the prime numbers. of Mathematics of the Polish Academy of Sciences Iwaniec’s foundational works and breakthroughs until 1983, when he left Poland. He held visiting in sieve theory and its applications form a large positions at the Institute for Advanced Study in part of this active area of mathematics. His proof Princeton, at the , and the (with ) that there are infinitely University of Colorado at Boulder. He became pro- many primes of the form X2 + Y 4 is one of the most fessor of mathematics at Rutgers in 1989, where he is currently New Jersey Professor of Mathematics. striking results about prime numbers known; the He has been honored with the Ostrowski Prize techniques introduced to prove it are the basis (2001), the Frank Nelson in Number of many further works. The theory of Riemann’s Theory (2002), and the Leroy P. Steele Prize for zeta function—and more generally of L-functions Mathematical Exposition (2011). He was a member associated with automorphic forms—plays a of the inaugural class of AMS Fellows (2012). central role in the study of prime numbers and diophantine equations. Iwaniec invented many of About the Prize the powerful techniques for studying L-functions The Shaw Prize is an international award es- of automorphic forms, which are used widely tablished to honor individuals who are currently today. Specifically, his techniques to estimate the active in their respective fields and who have Fourier coefficients of modular forms of half-inte- achieved distinguished and significant advances, gral weight and for estimating L-functions on their who have made outstanding contributions in critical lines (the latter jointly with William Duke culture and the arts, or who have achieved excel- and John Friedlander) have led to the solution of lence in other domains. The award is dedicated to a number of long-standing problems in number furthering societal progress, enhancing quality of theory, including one of Hilbert’s problems: that life, and enriching humanity’s spiritual civilization. quadratic equations in integers (in three or more Preference is given to individuals whose significant variables) can always be solved unless there is an work was recently achieved. ‘obvious’ reason that they cannot. The Shaw Prize consists of three annual awards: “In a series of papers remarkable both in terms the Prize in Astronomy, the Prize in Science and of its concept and novel techniques, Iwaniec to- Medicine, and the Prize in Mathematical Sciences. gether with different authors (Étienne Fouvry and Established under the auspices of Run Run Shaw then and John Friedlander) es- in November 2002, the prize is managed and ad- tablished results about the distribution of primes ministered by the Shaw Prize Foundation based in arithmetic progressions which go beyond the in Hong Kong. notorious Riemann hypothesis. This opened the Previous recipients of the Shaw Prize in Math- door to some potentially very striking applications. ematical Sciences are (2014), Yitang Zhang’s much celebrated recent result on David L. Donoho (2013), (2012), bounded gaps between primes relies heavily on the and Richard S. Hamilton works of Iwaniec et al. Iwaniec’s work mentioned (2011), (2010), Simon K. Donaldson above, together with his many other technically and Clifford H. Taubes (2009), brilliant works, have a central position in modern and Ludwig Faddeev (2008), analytic number theory.” and Richard Taylor (2007), and Wen-Tsun Wu (2006), (2005), and Biographical Sketches Shiing-Shen Chern (2004). Gerd Faltings was born in 1954 in Gelsenkirch- en-Buer, . He obtained his PhD in —From Shaw Foundation announcements mathematics from the University of Münster in 1978. He then spent a year doing postdoctoral work as a research fellow at from 1978 to 1979. He was an assistant professor at the University of Münster from 1979 to 1982. From 1982 to 1984, he was professor at the Uni- versity of Wuppertal. He was professor at from 1985 to 1994. Since 1995 he has been a director of the Max Planck Institute for

October 2015 Notices of the AMS 1075

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