2011 Bôcher Prize Assaf Naor and Gunther Uhlmann received the Lindenstrauss. He was a postdoctoral researcher 2011 AMS Maxime Bôcher Memorial Prize at the at the Theory Group of Microsoft Research in 117th Annual Meeting of the AMS in New Orleans Redmond, Washington, from 2002 to 2004, and in January 2011. a researcher at the Theory Group from 2004 to 2006. Since 2006 he has been a faculty member at Citation: Assaf Naor the Courant Institute of Mathematical Sciences of The Bôcher Prize is awarded to Assaf Naor for New York University, where he is a professor of introducing new invariants of metric spaces and mathematics and an associated faculty member for applying his new in computer science. He received the Bergmann understanding of the Memorial Award (2007), the European Mathemati- distortion between cal Society Prize (2008), the Packard Fellowship various metric struc- (2008), and the Salem Prize (2008), and he was an tures to theoretical invited speaker at the International Congress of computer science, es- Mathematicians (2010). Naor’s research is focused pecially in the papers on analysis and metric geometry and their interac- “On metric Ramsey tions with approximation algorithms, combinator- type phenomena” (with ics, and probability. Yair Bartal, Nathan Lin- ial, and Manor Mendel, Response: Assaf Naor Annals of Math. (2) 162 I am immensely grateful for the Bôcher Memorial (2005) no. 2, 643–709); Prize. Above all, I wish to thank my collaborators, “Metric cotype” (with especially Manor Mendel, with whom a decade-long Assaf Naor Manor Mendel, Annals collaboration and friendship has resulted in excit- of Math. (2) 168 (2008), ing, sometimes unexpected, discoveries. I thank my no. 1, 247–298); and “Euclidean distortion and the sparsest cut” (with Sanjeev Arora and James advisor Joram Lindenstrauss for his inspiration R. Lee, J. Amer. Math. Soc. 21 (2008), no. 1, 1–21). and encouragement. I am also grateful to Gideon The prize also recognizes Naor’s remarkable work Schechtman for being my unofficial advisor, col- with J. Cheeger and B. Kleiner on a lower bound in laborator, and friend, and for the mentoring and the sparsest cut problem. The bound follows from friendship of Keith Ball. a quantitative version of Cheeger and Kleiner’s In 1964 Lindenstrauss published a seminal beautiful differentiation theorem for Lipschitz paper entitled “On nonlinear projections in Ban- functions with values in L1, which was itself mo- ach spaces”. This paper contains results showing tivated by this application, as envisioned by Naor that Banach spaces exhibit unexpected rigidity and J. R. Lee. phenomena: the existence of certain nonlinear mappings that are quantitatively continuous (e.g., Biographical Sketch: Assaf Naor uniformly continuous or Lipschitz) implies that Assaf Naor was born in Rehovot, Israel, on May 7, certain linear properties are preserved. Subsequent 1975. He received his Ph.D. from the Hebrew Uni- work of Enflo on Hilbert’s fifth problem in infinite versity in Jerusalem under the direction of Joram dimensions and then a beautiful 1976 theorem of Ribe ushered in a new era in metric geometry. APRIL 2011 NOTICES OF THE AMS 603 Ribe’s theorem in particular showed that local such as nearest neighbor search and clustering, linear properties of Banach spaces are preserved are by definition questions on metric spaces, while under uniform homeomorphisms and can thus metric structures sometimes appear in problems be conceivably formulated in a way that involves that do not have a priori geometry in them, such only distance computations, ignoring entirely the as the structures that arise from continuous re- linear structure. Of course, if this could be done, laxations of combinatorial optimization problems. one could then investigate which metric spaces This general philosophy of embedding theory is have these properties, even if the geometric spaces responsible for many important approximation al- in question have nothing to do with linear spaces gorithms and data structures. The work contained (e.g., Riemannian manifolds, graphs, or discrete in the two other papers that are mentioned in the groups). Since at roughly the same time a deep citation has been partially inspired by these appli- theory of quantitative local linear invariants of Banach spaces was flourishing, this raised the cations: sharp nonlinear Dvoretzky theorems are possibility that this powerful linear theory could applied to online algorithms and data structures; be applied in much greater generality to geometric and new embedding methods manage to asymp- problems that are ubiquitous in mathematics. Un- totically determine (up to lower order terms) the doubtedly many mathematicians noticed the great most nonEuclidean finite subset of L1 (completing potential here, but to the best of my knowledge the 1969 work of Enflo), and at the same time they this program was first formulated in writing by yield the best known approximation algorithm for Bourgain in 1986. the sparsest cut problem with general demands. The 1980s witnessed a remarkable surge of The methods behind these results are in essence activity in this area, yielding several theorems on multiscale analysis, combined with a variety of general metric spaces that are inspired by previ- probabilistic techniques. ously known linear results: these include metric The quantitative structure theory of metric space versions of the theory of Rademacher type, a spaces is flourishing, with important new results nonlinear Dvoretzky theorem, and extension theo- appearing frequently. New algorithmic applica- rems for Lipschitz functions, as well as important tions are constantly being discovered, and deep results such as Bourgain’s embedding theorem connections are being made with harmonic analy- and Bourgain’s metrical characterization of su- sis, group theory, and geometric measure theory. perreflexivity. Key players in this 1980s “golden There is still a lot that we do not know in the origi- age” include Ball, Bourgain, Gromov, Johnson, nal program that is motivated by Ribe’s theorem, Lindenstrauss, Milman, Pisier, Schechtman, and Talagrand. including missing metric invariants that have yet to One reason that might explain why the 1980s be defined. Experience shows that future progress golden age ended is that there were several stub- will be difficult but fruitful, and the fact that many born problems that led to an incomplete theory, talented mathematicians and computer scientists and one needed new definitions of concepts, are working in this field suggests that exciting such as a notion of Rademacher cotype for metric developments are yet to come. I view the Bôcher spaces, in order to proceed. One of the papers that Memorial Prize as a recognition of the achieve- is mentioned in the citation formulated a definition ments of this field and as an encouragement to the of metric cotype and proved several accompany- many researchers who developed the theory thus ing theorems and applications showing that it is far to continue their efforts to uncover the hidden a satisfactory notion of cotype for metric spaces. structure of metric spaces. This work was a result of years of intensive effort and required insights that involve geometry, com- Citation: Gunther Uhlmann binatorics, and harmonic analysis. The Bôcher Prize is awarded to Gunther Uhlmann The quantitative structural theory of metric for his fundamental work on inverse problems spaces also received renewed impetus due to the and in particular for the solution to the Calderón important 1995 paper of Linial, London, and Rabi- problem in the papers “The Calderón problem with novich that exhibited the usefulness of these prob- partial data” (with Carlos E. Kenig and Johannes lems (specifically Bourgain’s embedding theorem) Sjöstrand, Annals of Math. (2) 165 (2007) no. 2, to the design of efficient approximation algorithms for NP-hard problems. The general theme here is 567–591) and “The Calderón problem with partial that a natural way to understand the structure of data in two dimensions” (with Oleg Yu. Imanuvilov a metric space is by representing it as faithfully and Masahiro Yamamoto, J. Amer Math. Soc. 23 as possible as points in a well-understood normed (2010), no. 3, 655–691). The prize also recognizes space, or as a superposition of simpler structures Uhlmann’s incisive work on boundary rigidity with such as trees. The relevance of such problems to L. Pestov and with P. Stepanov and on nonunique- computer science ranges from “obvious” to round- ness (also known as cloaking) with A. Greenleaf, about and surprising: many algorithmic tasks, Y. Kurylev, and M. Lassas. 604 NOTICES OF THE AMS VOLUME 58, NUMBER 4 Biographical Sketch: Gunther Uhlmann kindest person I have ever met—a wonderful role Gunther Uhlmann was born in Quillota, Chile, in model for anybody to follow. I also treasured the 1952. He studied mathematics as an undergradu- friendship I started with Cathleen Morawetz dur- ate at the Universidad de Chile in Santiago, gaining ing my stay at Courant. Most of all I have had the his Licenciatura degree in 1973. He continued his unwavering support of my family, Carolina, Anita, studies at MIT, where and Eric. Without them this would not have been he received a Ph.D. in possible. Carolina would have been so proud. This 1976 under the direc- prize is for her. tion of Victor Guille- About the Prize min. He held postdoc- toral positions at MIT, Established in 1923, the prize honors the memory Harvard, and the Cou- of Maxime Bôcher (1867–1918), who was the So- rant Institute. In 1980 ciety’s second Colloquium Lecturer in 1896 and he became assistant who served as AMS president during 1909–1910. professor at MIT and Bôcher was also one of the founding editors of then moved in 1985 Transactions of the AMS. The original endowment to the University of was contributed by members of the Society. The Washington, where he prize is awarded for a notable paper in analysis was appointed Walker published during the preceding six years.