ISSN 0002-9920 (print)ISSN 1088-9477 (online)

Volume 63, Number 7

of the American Mathematical SocietyAugust 2016

Tverberg's Theorem at 50

page 732

From Mathematical Physics to Analysis: A Walk in Barry Simon’s Mathematical Gardenpage 740

Denver Meetingpage 845

Minneapolis Meetingpage 849

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FEATURES

Noticesof the American Mathematical Society

August 2016

732Tverberg's Theorem at 50: Extensions and Counterexamples by Imre Bárány, Pavle V. M. Blagojevic, and Günter M. Ziegler

P

P + P = 2 P

P * P

772WHAT IS... a Diophantine m-tuple?by Andrej Dujella

From Mathematical Physics to Analysis: A Walk in Barry Simon’s Mathematical GardenCoordinating Editor Fritz Gesztesy

Our August issue features the recent counterexample to the conjectured topological version of Tverberg's 1966 theorem and 2016 Steele Prize winner Barry Simon on the occasion of his 70th birthday conference this month. —Frank Morgan, Editor-in-Chief

ALSO IN THIS ISSUEReport on the 2014–2015 New Doctoral Recipients 754 William Yslas Vélez, Thomas H. Barr, and Colleen A. Rose

AMS Executive Director Donald E. McClure Retires 777 Allyn Jackson

Chinese Mathematics and ICCM 780 Lizhen Ji

Baa Hózhó Math: Math Circles for Navajo Students and Teachers 784 Dave Auckly, Bob Klein, Amanda Serenevy, and Tatiana Shubin

Doctoral Degrees Conferred 790 2014–2015

The Mathematics of Love 821 Reviewed by Marc Colyvan

THE GRADUATE STUDENT SECTIONHelen Moore Interview 768 Alexander Diaz-Lopez

FROM THE SECRETARY'S OFFICE

Voting Information for 2016 AMS Election 775

The background figures on the cover show reflections in the Poincaré Disk (left) and a spiraling line compac-tification that adds a circle at infinity (right) © Mihai Stoiciu, from Barry Simon's 5-volume set A Compre-hensive Course in Analysis.bookstore.ams.org/simon-set

740

From Mathematical Physics toAnalysis: A Walk in BarrySimon’s Mathematical Garden

Editor’s Note: Fritz Gesztesy kindly accepted our invitation to put together this feature in honor of Barry Simon on theoccasion of Simon’s 2016 AMS Leroy P. Steele Prize for Lifetime Achievement and his 70th birthday conference thisAugust 28–September 1.

Fritz Gesztesy

This is a collection of contributions by collabo-rators, postdocs, and students of Barry Simonof the California Institute of Technology on theoccasion of Simon’s receiving the 2016 AMSLeroy P. Steele Prize for Lifetime Achievement.

The citation for the award mentions his tremendous im-pact on the education and research of a whole generationof mathematical scientists, and we will underscore thisby demonstrating his penetrating influence on topicsranging from quantum field theory, statistical mechanics,the general theory of Schrödinger operators, spectral andinverse spectral theory to orthogonal polynomials.

But we should start at the beginning: Barry was born toparents Minnie and Hy Simon in 1946, and together withhis older brother, Rick, grew up in Brooklyn, New York.There he attended James Madison High School, obtaininga perfect score on the MAA’s American High SchoolMathematics Examinaton in 1962 and thus becoming thesubject of an article in the New York Times at the tenderage of sixteen. Under the influence of Sam Marantz, aninspiring physics teacher in high school, he applied toHarvard and was admitted. While at Harvard he was a topfive Putnam Competition Winner in 1965 and received hisBA summa cum laude in physics in 1966. George Mackeyat Harvard recommended Barry pursue a doctorate withArthur Wightman at Princeton because Wightman waswell known for advocating the application of rigorousmathematics in physics.

Fritz Gesztesy is Mahala and Rose Houchins Professor of Mathe-matics at the University of Missouri, Columbia. His email addressis gesztesyf@missouri.edu. Since August 1 he is Jean andRalph Storm Chair of Mathematics at Baylor University. His emailaddress there is Fritz_Gesztesy@baylor.edu.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1403

Barry Simon with his mother and brother, Rick (left),and with his father (right) (ca. 1950).

Barry Simon completed his doctorate in physics atPrinceton under Wightman’s supervision in 1970. Thebody of his work during the time of his doctoral re-search was of such importance that he was immediatelyappointed to assistant professor, jointly in mathematicsand physics, at Princeton. He rapidly rose to the rank offull professor by 1976. Several contributions below willattest to the electric atmosphere at Princeton in thosedays, making it a thriving center for quantum field theory,statistical mechanics, and nonrelativistic quantum me-chanics. Barry joined Caltech in 1981, holding the positionof IBM Professor of Mathematics and Theoretical Physicssince 1984. At Caltech, Barry’s interests further broad-ened into areas such as random and ergodic Schrödingeroperators, exotic spectra, inverse spectral theory, and theanalytic theory of orthogonal polynomials.

740 Notices of the AMS Volume 63, Number 7

Honorary doctorate, Ludwig-Maximilians-Universityof Munich, 2014.

… rigorousmathematicsin physics

Since many of Barry’s ma-jor research accomplishmentsare discussed in depth and putinto proper context in the vari-ous contributions to follow, weitemize only a few brief com-ments at this point, focusingon some key results he proved,fundamental concepts he ad-

vocated, and some of the important terms he and hiscollaborators first coined with lasting impact:

• a rigorous framework for resonances, complexand exterior scaling, Fermi’s golden rule, proofof the Oppenheimer formula for the Stark ef-fect, convergence of time-dependent perturbationtheory;

• constructive (Euclidean) quantum field theory intwo space-time dimensions, connections to sta-tistical mechanics, lattice approximations andcorrelation inequalities, 𝑃(𝜙)2 spatially cutofffield theories;

• hypercontractive and ultracontractive semi-groups;

• magnetic fields, diamagnetic inequality, Kato’sinequality;

• a proof of continuous symmetry breaking inclassical and quantum statistical models;

• Thomas–Fermi theory, semiclassical bounds, non-Weyl asymptotics;

• asymptotic perturbation theory of eigenvalues:Borel and Padé summability, Zeeman effect, anhar-monic oscillators, instanton tunneling, Birman–Schwinger principle, coupling constant thresh-olds;

• general theory of Schrödinger operators: essen-tial self-adjointness, pointwise bounds on eigen-functions, path integral techniques, absence ofsingular continuous spectrum in 𝑁-body sytems;

• Berry’s phase and holonomy, homotopic interpre-tation of the Thouless integers and topologicalstructure in the integer quantum Hall effect;

• random and almost periodic Schrödinger andJacobi operators, exotic spectral phenomena (Can-tor, singular continuous, and dense pure pointspectra) and their transition to becoming a cen-tral object in mathematical physics (the singu-lar continuous revolution), Wonderland theorem,Thouless formula, almost Mathieu equation;

• trace formulas for potential coefficients in termsof the Krein–Lifshitz spectral shift function,uniqueness theorems in inverse spectral theoryfor Schrödinger and Jacobi operators, oscillationtheory in gaps of the essential spectrum, inversespectral analysis with partial information on thepotential;

• a new approach (the analog of the continuedfraction method) to inverse spectral theory ofSchrödinger operators, his local Borg–Marchenkotheorem;

• a systematic application of operator theory tech-niques to orthogonal polynomials on the realline (OPRL) and on the unit circle (OPUC), CMVmatrices, Verblunsky coefficients;

• sum rules for Jacobi matrices and applications tospectral theory (Killip–Simon theorem), perturba-tions of OPRL and OPUC with periodic recursioncoefficients;

• Szegő asymptotics, a proof of Nevai’s conjectureand its finite gap extension, the finite gap analogof the Szegő-Shohat-Nevai theorem, the fine struc-ture of zeros of orthogonal polynomials (clockbehavior), higher-order Szegő theorems.

Barry Simon’s influence on our community by fartranscends his approximately four hundred papers, par-ticularly in viewof126coauthors, 50mentees, 31graduatestudents, and about 50 postdocs mentored. In this con-text, one must especially mention his twenty books, thefirst fifteen of which have educated scores of mathemati-cians and mathematical physicists, two generations bynow, and continuing into the foreseeable future.

One cannot overestimate the influence of Reed andSimon’s four-volume series, Methods of Modern Mathe-matical Physics, I–IV (1972–79). It took on the same levelof importance that Courant–Hilbert’s two volumes hadfor previous generations, and it continues to fill that role

August 2016 Notices of the AMS 741

all over the globe to this day. To gauge the importanceof Reed and Simon behind the Iron Curtain, we contactedAlbrecht Böttcher (TU Chemnitz, Germany). Like so manyin our generation, Albrecht has a very personal relation-ship with Volume I and underscored (his words) “the trulyingenious selection and presentation of the mathematicaltopics.” The latter sentiment, however, is by no meansunique to colleagues who read Reed-Simon in Russiantranslation; it is just as prevalent in the West.

One cannotoverestimate theinfluence of Reed

and Simon’sMethods ofModern

MathematicalPhysics, I–IV

Cumulative salesfigures for all fourvolumes to date top37,000 copies. Barry’sother books, most no-tably Functional Inte-gration and QuantumPhysics (1979), TraceIdeals and Their Appli-cations (1979, 2005),Orthogonal Polynomi-als on the Unit Circle,Parts 1, 2 (2005), andSzegő’s Theorem andIts Descendants (2011),profoundly influencedresearch in these ar-eas. Finally, we have

not even begun to understand the legacy which will becreated with his newest five-volume set, A ComprehensiveCourse in Analysis (2015), which offers a panorama fromreal to complex and harmonic analysis all the way tooperator theory.

In short, Barry has been a phenomenal force in math-ematical physics, encyclopedic in his knowledge and agrand master of mathematical structure and abstractanalysis. Above all, he truly enjoys collaboration and thehuman interactions that come with it. As a sign of histremendous influence on our community we note that todate MathSciNet lists 15,325 citations by 6,602 authors,and Google Scholar lists 61,680 citations and an h-indexof 104.

Barry has been recognized with honorary degrees fromthe Technion, Haifa; the University of Wales–Swansea,and LMU–Munich. He was awarded the Stampacchia Prizein 1982 with M. Aizenman, the Poincaré Prize of theInternational Association ofMathematical Physics in 2012,

The books Barry Simon has authored …thus far.

and the Bolyai Prize of theHungarianAcademyof Sciencesin 2015. He is a fellow of the American Physical Society(1981), theAmericanAcademyofArts and Sciences (2005),and the AMS (2013). He has served as vice president ofthe AMS and of IAMP.

Finally, on a personal note: Having been a frequentcollaborator of Barry’s, I have often been approachedwith the assumption of being one of Barry’s students orpostdocs, but this is not the case. On the other hand, likeso many of my generation, I learned about the tools of ourtrade through his writings and especially from Reed andSimon I–IV, so of course it feels like I was Barry’s student,even though the proper term appears to be that I’m one ofhis many mentees. Barry has been a constant inspirationtome for about forty years now; I feel incredibly fortunatethat he became my mentor and friend.

Evans M. HarrellSingular Perturbation Theory and ResonancesThe very first article in Barry Simon’s publication list,which appeared in Il Nuovo Cimento when he was a 22-year-old graduate student, was concerned with singularperturbation theory. This paper showed that a certainregularized, renormalized perturbation expansion for atwo-dimensional quantum field theory model convergeswith a positive radius of convergence. As Barry candidlyadmitted in that article, in itself the result was of limitedsignificance, but in a subject for which at that time “allthe mathematically suitable results…are of a negativenature,” it announced a new, more constructive era.

To the reader familiar with Barry Simon’s works onmathematical physics of the 1970s, it is striking howmany of the hallmarks of his technique are alreadyapparent in this first article. Before entering deeply intothe research, Barry first carried out a thorough andpenetrating review of the entire literature on the subject.This signature of his method was something those ofus who were students at Princeton in the 1970s wouldwitness every time Barry began a new research project:Seeing him emerge from the library shared by Jadwinand Fine Halls with a mountain of books and articles,it was humbling to realize that Barry was not merelybrave enough to collect all of the knowledge about thenext subject he wished to study, but seemingly overnighthe would absorb it in detail and carefully assess eachcontribution for its mathematical appropriateness. Trueto form, in that first article, Barry laid out which claims inthe literature were established with mathematical rigor,which were plausibly to be believed, perhaps with someextra attention to assumptions, and which were franklydubious. Finally, Barry’s own way of formulating theproblem was sparse and clear, and his reasoning incisive.

The perturbation theory that applies to nonrelativisticquantummechanics is a linear theory, allowingstraightfor-ward calculations of systematically corrected eigenvalues

Evans M. Harrell is professor emeritus at the School of Mathemat-ics at the Georgia Institute of Technology. His email address isharrell@math.gatech.edu.

742 Notices of the AMS Volume 63, Number 7

Lorenzo Sadun, Yosi Avron, Evans Harrell, BarrySimon (ca. 1988).

and eigenfunctions. Schrödinger’s adaptation of the pro-cedures of Rayleigh to calculate the shifts in hydrogenemission spectrum in the presence of an electric field(known as the Stark effect) went a long way in establish-ing the validity of his new quantum theory. This success isironic, given that the series for which Schrödinger and Ep-stein calculated the first terms has radius of convergencezero, and the spectrum of the Stark Hamiltonian for anynonzero value of the electric field is purely continuous,containing no discrete eigenvalues at all. The instant theinteraction is switched on, the nature of the spectrumchanges radically, and the perturbed eigenvalue becomesa resonance state! The Stark effect is known today tobelong to the realm of singular perturbation theory.

Somephysicists

were “usingmethods ofunknownvalidity”

In the two decadesafter Schrödinger’s work,mathematicians created a com-prehensive theory of linearoperators on Hilbert space.In the 1930s and 1940s Rel-lich and Kato produced amathematically rigorous the-ory of regular perturbations oflinear operators and some as-pects of singular perturbations.In their hands, perturbationtheory was concerned with ana-lytic operator-valued functions

of a complex variable, defined initially as convergentpower series with operator coefficients, which are mosttypically self-adjoint in applications to quantum mechan-ics. These functions exhibit the range of behavior ofordinary scalar analytic functions of a complex variable,with manageable complications when the spectrum con-sists of discrete eigenvalues of finite multiplicity, andsome new phenomena when there is an essential spec-trum. Physicists andchemists at this time solvedproblemsand developed perturbative techniques that sometimesfell into the domain of regular perturbation, but justas frequently produced series that could be calculatedterm by term while exhibiting singular features, as in a

The Stark effect: The spectrum of the hydrogen atomunder an increasingly intense electric field, from theCourtney-Spellmeyer-Jiao-Kleppner article appearingin Physical Review A, vol. 51 (1995).

zero radius of convergence or, worse, convergence but tothe wrong answer. (For instance, a resonance eigenvalueassociated with tunneling may have a nonzero imaginarypart that, typically, is represented in perturbation theoryby a series of the form ∑𝑛 𝑐𝑛𝛽𝑛 with 𝑐𝑛 = 0 for all 𝑛.)

At the time Barry Simon hit the scene, an industrywas thriving in attempts to get information from suchexpansions, whether by replacing the series by otherexpressions, especially Padé approximates𝑃[𝑚,𝑛], whichare ratios of polynomials of𝑚 and 𝑛 degrees, or by the useof analytic-function techniques like Borel summability tomake sense of divergent series. Many of Barry’s earlyworks addressed these topics.

One of the important models in quantum mechanicswith a singular perturbation is the quartic anharmonicoscillator. Its Hamiltonian is(1) 𝑝2 + 𝑥2 +𝛽𝑥4,and it was the subject of a landmark study by Benderand Wu in 1968-69 in which, “using methods of unknownvalidity”—in Barry’s memorable phrase—they painted afascinating and largely correct picture of the analyticstructure of the eigenvalues of the anharmonic oscilla-tor as functions of the coupling constant 𝛽, consideredas a complex variable. Among other things, Bender andWu conjectured that the power series expansion for theground-state eigenvalue of (1) had radius of convergence0. Then-new computational capabilities in symbolic alge-bra had allowed Bender and Wu and others to calculateperturbation series at high orders, and with the firstseventy-five coefficients 𝑎𝑛 for the ground-state eigen-value in hand, Bender and Wu specifically conjecturedthat

(2) 𝑎𝑛 ∼ 𝜋− 32√63𝑛Γ(𝑛+ 1

2) .

Barry’s response to the explorations of Bender and Wuwas to pen the definitive rigorous analysis of the analytic

August 2016 Notices of the AMS 743

Some of Barry’s coauthors, SimonFest 2006, on theoccasion of Barry Simon’s sixtieth birthday.

properties of the anharmonic oscillator in an extendedarticle in the Annals of Physics in 1970. This work is atimeless classic, a textbook model for how to do singularperturbation theory, and it remains one of Barry’s mosthighly cited works. Most of the claims of Bender and Wuwere put on a firm footing, and many further facts wereestablished. For example, it was shown that 𝛽 = 0 is athird-order branch point and an accumulation point ofsingularities. Moreover, Barry obtained sufficient controlon the growth rate of perturbation coefficients to showthat both the Padé and Borel methods were valid todetermine the eigenvalues for nonzero values of 𝛽. A fewyears later, with the aid of a dispersion relation derivedusing this understanding of the Riemann surfaces of theeigenvalues, Barry and coauthors proved the formula (2).

Later, several other models of singular perturbationtheory, including the Stark and Zeeman effects, receivedsimilar treatments in thehandsof Barry andhis associates,especially Yosi Avron and IraHerbst. These gemsprovideda foundation for developments in singular perturbationtheory and deepened the understanding of many of thetouchstones of quantum mechanics.

Barry has always been quick to recognize others’ goodideas when they appear and generous in promoting themin the community of mathematical physicists and beyond.A very pretty method introduced in that era by Aguilar,Balslev, and Combes in 1971 came to be called complexscaling. The original version made use of the dilatationsymmetry and complex analysis to move the essentialspectrum of a Schrödinger operator into the complexplane, while leaving discrete eigenvalues unaffected. The

unitary group of dilatations can be defined via(3) [𝑈(𝜃)𝑓] (x) ∶= 𝑒𝜈𝜃/2𝑓 (𝑒𝜃x)in terms of a real parameter 𝜃. If the scaled potentialenergy depends in an analytic way on 𝜃, then the pa-rameter can be complexified, and it is easy to see thatwith a compactness condition on the potential energythe essential spectrum of the complex-scaled Laplacianfor nonreal 𝜃 is simply rotated in the complex plane.Meanwhile, isolated eigenvalues are analytic as functionsof 𝜃, but since they are constant for real variations in𝜃, by unique continuation they are also constant for anyvariation of 𝜃, except that they can appear or disappearwhen they collide with the essential spectrum, at whichpoint analytic perturbation theory ceases to apply.

With this procedure the ad hoc tradition in physicsof treating resonances as nonreal eigenvalues somehowassociated with a self-adjoint Hamiltonian became math-ematically solid and canonical. I cannot do better indescribing complex scaling further than to point to Chap-ter XIII.10 of Reed-Simon, to which the reader is referredfor further details and context. Barry assiduously pro-moted this excellent tool for understanding resonances,evangelizing the technique to physicists and chemists, bywhom it was adopted and used in realistic problems.

I recall in particular when Barry took a delegation ofmathematical physicists to the 1978 Sanibel Workshopon Complex Scaling, organized by the noted quantumchemist Per-Olof Löwdin, at which the discussions be-tween the chemists and the believers in mathematicalmethods “of known validity” were quite fruitful and infor-mative on all sides. Of course, Barry not only recognized,clarified, and promoted the idea of complex scaling butmade his own fundamental advances in the subject, espe-cially by greatly extending the set of problems to whichit applied by his discovery that it suffices to performcomplex scaling on an exterior region.

A similar tale could be told of Barry’s recognition ofthe importance of the microlocal analysis of tunnelingphenomena by Helffer and Sjöstrand in the 1980s, whichBarry again promoted, clarified, and in certain waystransformed. But space here is limited, and besides, forthe reader interested in learning more about singularperturbation theory and resonances, there is an excellent1991 review article entitled “Fifty years of eigenvalueperturbation theory,” written by a master of the genre,Barry Simon himself.

Percy A. DeiftPrinceton in the 1970s; Exponential Decay ofEigenfunctions and Scattering TheoryThe 1970s were a very special time for mathematicalphysics at Princeton. One can read a lively account ofthose days, written by Barry himself, in the July 2012edition of the Bulletin of the International Association ofMathematical Physics. The main thrust of the activity

Percy A. Deift is Silver Professor of Mathematics at the CourantInstitute, NYU. His email address is deift@cims.nyu.edu.

744 Notices of the AMS Volume 63, Number 7

Some of Barry Simon’s students, SimonFest 2006, onthe occasion of Barry Simon’s sixtieth birthday.

was in statistical mechanics, quantum field theory, andnonrelativistic quantum mechanics. The list of peoplewho participated in math-phys at Princeton Universityin those years as students, postdocs, junior faculty orsenior faculty, or just visitors for a day or two reads likea who’s who of mathematical physics. Leading the chargewere Arthur Wightman, Elliott Lieb, and Barry Simon. Butthere were also Eugene Wigner, Valentine Bargmann, andEd Nelson. And in applied mathematics, there was MartinKruskal, still flush with excitement from his seminal workon the Korteweg-de Vries equation, and across the wayat the Institute were Tulio Regge and Freeman Dyson,doing wonderful things. Barry was a dynamo, challengingus with open problems, understanding every lectureinstantaneously, writing paper after paper, often at theseminars themselves, all the while supervising seven oreight PhD students.

I was one of those students. I had an appointment tomeet with Barry once every two weeks. I would work veryhard preparing a list of questions that I did not know howto answer. Say there were ten questions; by the end of thefirst ten minutes in Barry’s office, the first six questionswere resolved. Regarding questions seven and eight, Barrywould think about them for about two or three minutesand then tell me how to do them. Regarding questionsnine and ten, Barry would think about them, also forabout two or three minutes, and say, “I don’t know how todo them. But if you look in such and such a book or paper,you will find the answer.” Invariably he was right. So inless than half an hour, all my questions were resolved,and as I walked out of the door there was the next studentwaiting his turn!

Barry’s first PhD student was Tony O’Connor.O’Connor’s thesis concerned exponential decay foreigenfunctions 𝜓 of 𝑁-body Schrödinger operators 𝐻,𝐻𝜓 = 𝜆𝜓, for 𝜆 below the essential spectrum of 𝐻. Here𝐻 = 𝐻0 +𝑉, where 𝐻0 is the kinetic energy and 𝑉 is theinteraction potential. For Schrödinger operators in onedimension, such results go back to the nineteenth century,but for 𝑁 > 1 particles moving in three dimensions,

completely different techniques were necessary. Overthe years many people have worked on the problem,including, to name a few, Stanislav Merkuriev in theformer Soviet Union, and John Morgan and Thomas andMaria Hoffmann-Ostenhof in the West. O’Connor had theidea of using the analyticity of the Fourier transform andobtained results in the 𝐿2 sense (i.e., 𝑒𝛼| ⋅ |𝜓 ∈ 𝐿2, 𝛼 > 0).Such bounds are optimal for isotropic decay.

O’Connor’s paper motivated Jean-Michel Combes andLarry Thomas to introduce an approach that has nowbecome standard under the general rubric of “boostanalyticity,” and in a set of three papers in the mid-1970s, Barry further developed these ideas to obtainpointwise exponential bounds on eigenfunctions undervarious assumptions on the asymptotic behavior of theinteraction potential 𝑉, proving eventually that if 𝑉(𝑥)was bounded below by |𝑥|2𝑚, say, then one obtainedsuperexponential decay for 𝜓(𝑥),

|𝜓(𝑥)| ≤ 𝑐 𝑒−𝛼|𝑥|𝑚+1 𝑐, 𝛼 > 0.Schrödinger operators typically involve interaction poten-tials 𝑉 which are sums of two-body interactions

𝑉(𝑥) = ∑1≤𝑖<𝑗≤𝑁

𝑉𝑖𝑗(𝑥𝑖 − 𝑥𝑗)

for particles 𝑥𝑖 in ℝ3, 𝑖 = 1,… ,𝑁. Although typically𝑉𝑖𝑗(𝑦) → 0 as |𝑦| → ∞,

𝑉(𝑥) clearly does not decay if |𝑥| → ∞ in such a waythat 𝑥𝑖 − 𝑥𝑗, say, remains bounded for some 𝑖 ≠ 𝑗. Suchnonisotropy in the potential 𝑉 suggests that isotropicbounds of the form

|𝜓(𝑥)| ≤ 𝑐𝑒−𝛼|𝑥|

are not optimal amongst all possible bounds.In a fourthpaperonexponential decay in1978, together

with Deift, Hunziker, and Vock, Barry constructed optimalnonisotropic bounds for eigenfunctions of the form

|𝜓(𝑥)| ≤ 𝑐𝑒(𝛼,𝑥)

for suitable 𝛼 = (𝛼1,… ,𝛼𝑁), 𝛼𝑖 ∈ ℝ3. The 𝛼’s reflect thegeometry of the channels where

𝑉(𝑥) ↛ 0 as |𝑥| → ∞.Eventually, in 1982, Agmon showed that for a very

general class of elliptic operators 𝐻 in ℝ3𝑁, there was anaturally associated Riemannian metric on ℝ3𝑁 such thateigenfunctions 𝜓,

𝐻𝜓 = 𝜆𝜓,with 𝜆 below the essential spectrum of 𝐻, satisfied thebound

|𝜓(𝑥)| ≤ 𝑐𝜖 𝑒−(1−𝜖) 𝜌(𝑥) ∀𝜖 > 0,where 𝜌(𝑥) is the geodesic distance from 𝑥 to the origin inℝ3𝑁 with respect to Agmon’s metric. The Agmon metriccan be used to derive, and so explain, the bounds in the1978 work of Deift, Hunziker, Simon, and Vock.

The scattering problem in quantum chemistry, goingback to the 1920s, can be stated informally as follows: In achemical reaction, do molecules go to molecules? In otherwords, suppose in the distant past that the particle systemis described by a collection of noninteracting molecules.

August 2016 Notices of the AMS 745

SimonFest in 2006 for Barry Simon’s sixtieth birthday.Percy Deift is in the front row, second from the right.

As time goes on, the atoms in the different moleculesbegin to interact with each other, and the molecules breakup. At large positive times, is the particle system againdescribed by molecules?

To be mathematically precise, consider a collection ofatoms

𝑥𝑖 ∈ ℝ3, 𝑖 = 1,… ,𝑁,with Hamiltonian

𝐻 = 𝐻0 +𝑉 = 𝐻0 + ∑1<𝑗≤𝑁

𝑉𝑖𝑗 (𝑥𝑖 − 𝑥𝑗) ,

in the center of mass frame. Let 𝐶(1),… ,𝐶(𝑚) be adecomposition of the atoms into 𝑚 clusters,

𝑚

∑𝑗=1

#(𝐶(𝑗)) = 𝑁.

Now suppose that for large negative times the clusters{𝐶(𝑗)} are far apart and in each cluster 𝐶(𝑘) the atomsare in an eigenstate 𝜓𝑘 of the cluster (read “molecular”)Hamiltonian, that is,

𝐻(𝑘)𝜓𝑘 = 𝐸𝑘𝜓𝑘, where 𝐻(𝑘) = 𝐻(𝑘)0 +𝑉(𝑘).

Here 𝐻(𝑘)0 is the kinetic energy for the cluster and

𝑉(𝑘) = ∑1≤𝑖≤𝑗≤# (𝐶(𝑘))

𝑥𝑖,𝑥𝑗∈𝐶(𝑘)

𝑉𝑖𝑗 (𝑥𝑖 − 𝑥𝑗)

is the interaction potential for the atoms in the cluster. Sothe scattering problem of quantum chemistry becomesthe following: At large positive times, is the particlesystem again described by molecules, that is, by somedecomposition of well-separated clusters ̃𝐶(1),… , ̃𝐶(�̃�)

with the atoms in each cluster ̃𝐶(𝑘) in an eigenstate ofthe cluster Hamiltonian �̃�(𝑘) (or more precisely, in somelinear combination of such molecular configurations)?

In the physical and mathematical literature, the scat-tering problem is known as the problem of “asymptoticcompleteness” or the “unitarity of the 𝑆-matrix.” Muchwork has been done on this problem by many peopleover the years, including the time-independent approachof Faddeev and his school, leading up to the eventualresolution of the problem in 1987 by Israel Michael Sigaland Avy Soffer using time-dependent methods pioneeredby Enss.

At the mathematical level, the first task in resolvingthe problem is to prove that such molecular states indeedexist. This boils down to proving that the so-calledwave operators 𝑊(𝐻,𝐻𝑐) exist, where 𝐻 is again theHamiltonian for the full system and𝐻𝑐 is the Hamiltonianfor the molecular system corresponding to the clusterdecomposition 𝐶 = {𝐶(1),… ,𝐶(𝑚)}. At the technical levelthis is a relatively easy thing to do. To prove asymptoticcompleteness, one must show that all states orthogonalto bound states of the full Hamiltonian 𝐻 are in the linearspan of these molecular wave operators 𝑊(𝐻,𝐻𝑐). Thisis a range question, and range questions in mathematicsare generically hard.

Barry is oneof the most

prolific math-ematicians of

hisgeneration

In 1978, together withDeift, Simon introduced anew class of wave operators,𝑊(𝐻𝑐, 𝐽𝑐,𝐻), and showed thatasymptotic completeness wasequivalent to proving the exis-tence of these wave operators.Here 𝐽𝑐 is an auxiliary func-tion reflecting the geometry ofthe cluster decomposition 𝐶.In this way the problem ofasymptotic completeness wastransformed froma rangeprob-lem to a potentially simpler

existence problem, and this is the path that Sigal andSoffer eventually followed in their resolution of asymp-totic completeness. As the ranges of the wave operators𝑊(𝐻,𝐻𝑐) lie in the absolutely continuous space of 𝐻, akey ingredient in proving asymptotic completeness wasto show that the singular continuous space for 𝐻 wastrivial. This key component was established by BarrySimon, together with Peter Perry and Israel Michael Sigal,in seminal work in 1980 using remarkable ideas of EricMourre [2].

On a Personal NoteBarry is one of the most prolific mathematicians ofhis generation. It was in the late 1970s, around thetime that we were working on nonisotropic bounds foreigenfunctions, that I got a glimpse of the speed withwhich Barry did things. Soon after Volker Enss introducedhis seminal time-dependent ideas on spectral theory andscattering theory, a few of us went to Barry’s house inEdison, New Jersey, to discuss a potential project inspiredby Enss’s work. We spent the afternoon laying out indetail a list of problems that needed to be addressed andleft in the late afternoon. The next morning Barry came

746 Notices of the AMS Volume 63, Number 7

From about 1995 to 2000, Barry taught the first termof Caltech’s required freshman calculus, where hedeclared that “epsilon and delta are a calculusstudent’s finest weapons.” One year, on the last day,the students presented him with the boxing glovesshown.

into the office: Not only had he solved all the problemson our list, but he had in his hand the first draft of hissubsequent paper [3]! We were overwhelmed. For a youngperson like me, this was most discouraging. And I wasdoubly discouraged: Barry was younger than I was!

Barry has many fine qualities as a colleague and asa researcher, but I would like to focus on just one ofthem, viz., Barry’s keen sense of fairness and correctattribution of results. People in orthogonal polynomialsknow well Barry’s insistence on calling the recurrencecoefficients for orthogonal polynomials on the circleVerblunsky coefficients, in recognition of the almostforgotten seminal work of Samuel Verblunsky. But Iwould like to tell a different story. In the early 1980s Barrywas in Australia, where he met up with Michael Berry,who was also visiting. Berry began telling Barry aboutsome curious and puzzling calculations he had beenmaking in quantum adiabatic theory. Barry immediatelyunderstood that what was really going on was a matterof holonomy, and with characteristic speed he wrote andsent off a paper to Physical Review Letters, pointedly titled“Holonomy, the Quantum Adiabatic Theorem, and Berry’sphase.” In this way, a major discovery that could quiteeasily have become known as “Barry’s phase” was fixedin the literature as “Berry’s phase,” and justly so.

References[1] P. Deift and B. Simon, A time-dependent approach to the

completeness of multiparticle quantum systems, Commun.Pure Appl. Math. 30 (1977), 573–583. MR 0459397

[2] E. Mourre, Absence of singular continuous spectrum forcertain selfadjoint operators, Commun. Math. Phys. 78(1980/81), 391–408. MR 0603501

[3] B. Simon, Phase space analysis of simple scattering systems.Extensions of some work of Enss, Duke Math. J. 46 (1979),119–168. MR 0523604

Lon RosenBarry and I were both young pups when we collaboratedin the 1970s. It was an intense and exciting experiencefor me, one which I cherish and now take pleasure inrecalling for you. Warning: these personal recollectionshave no scientific content. If that’s what you’re lookingfor, please see my contribution to the Festschrift in honorof Barry Simon’s sixtieth birthday.

I must confess that my first meeting with Barry wasfar from auspicious. In 1967 I was in my first year ofdoctoral studies at the Courant Institute. Feeling isolated,I was reconsidering my decision not to have chosenPrinceton for graduate school. I asked a friend to arrangea lunch meeting for me with a typical student of ArthurWightman’s. I knew little about the “typical student” whowas chosen (Barry Simon), although I was familiar with hisname because Barry and I had both been Putnam Fellowsin the 1965 competition.

Some typical student! He practically tore my head off.Whatever I said about my interests or ideas, Barry wouldtrump it. I’d never met anyone else with such extensiveknowledge, amazing recall, and proofs at the ready. Istill haven’t. Thanks to Barry, I stayed put at Courant.Fortunately, James Glimm, who was to be my terrificthesis advisor, soon joined the faculty there. I learnedlater that Barry had been going through a rough patchin his personal and professional life around the time wemet and that the fire-breathing dragon who had me for

Lon Rosen is professsor emeritus at the Department of Mathemat-ics, University of British Columbia. His email address is rosen@math.ubc.ca.

Barry and Martha Simon, LMU Munich, 2014.

August 2016 Notices of the AMS 747

lunch was actually a gentle prince in disguise, if I may bepermitted a fairy tale metaphor.

Three years later I gave a seminar at Princeton on thesubject of higher-order estimates for the 𝑃(𝜙)2 model. Atthe conclusion of the seminar Barry showed me a cleverbootstrap trick that quickly established my most difficultestimate—or at least a weaker but perfectly acceptableversion of it. I was grateful and revised the publishedpaper accordingly.

Barry’simperative tounderstandeverything inthe simplestpossible way

This experience was notunique to me. As manyspeakers know, Barry’srapid-strike ability couldbe unnerving at seminars.He would sit front rowcentre, working on a pa-per, only to surface withastute observations, coun-terexamples, or shorterproofs. This penchant for“tricks” arises, it seemsto me, from Barry’s im-perative to understandeverything in the simplest possible way.

Barry and I both attended the Les Houches SummerSchool of 1970. It was there that I gained an appreciationfor Barry’s sense of humour. In particular, we had a lot offun putting on a skit which satirized the lecturing stylesand idiosyncrasies of the various celebrated speakers. Forexample, when Barry began an impersonation by firstbreaking a half dozen chalk sticks into small pieces,everyone roared, knowing that “Arthur Jaffe” was aboutto deliver his next lecture.

Sometimes the humour was (possibly) unintentional.Here’s a little story which Ed Nelson told me. Barry hadreturned from a trip to the former Soviet Union, where hehad great difficulty in arranging for kosher food. It wasapparently necessary for him to haul a suitcase filled withedibles. “Oh well,” sighed Barry, “I guess everyone has hiscross to bear.”

He would oftencome in with a

twinkle in his eye

In 1971 when Ijoined Barry in Prince-ton, we began ourjoint research by work-ing on what I wouldcall “incremental stuffusing available tech-niques,” things likecoupling constant an-

alyticity of the 𝑃(𝜙)2 Hamiltonian. One day ArthurWightman called us together for a presentation by aquiet visitor from Italy, Francesco Guerra, whom I barelyknew. Francesco proceeded to the chalkboard and madesome extraordinary claims about the vacuum energy den-sity in the 𝑃(𝜙)2 model. Barry and I gave each other asideways look as if to say, “he’s got to be kidding.” Hewasn’t! Francesco’s short proofs were stunning. The ironywas not lost on us that they were based on the Euclideanapproach of Ed Nelson of Princeton University. In anycase, that was the moment that the Euclidean Revolution

Barry Simon in Bangkok (ca. 2003).

began for Barry and me. The three of us (GRS) entered along and fruitful collaboration exploring and exploitingthe parallels between the 𝑃(𝜙)2 field theory model andclassical statistical mechanics. As usual, Barry snatchedup the new ideas like a dog with a bone.

During the GRS period, Barry had numerous otherprojects on the go, such as his research in mathematicalquantum mechanics and the Reed-Simon magnum opus.Thank goodness he was devoting only a fraction of histime to GRS, whereas I was on it full time. Otherwise,I would never have been able to keep up with him andcontribute my fair share. Barry’s joy in doing researchwas infectious. He would often come in with a twinkle inhis eye and say something like, “While I was standing inthe supermarket line, look what I discovered!” Barry wasalways appreciative of my efforts and extremely generousto others. I learned a tremendous amount from him bothdirectly during our collaboration and in subsequent yearsfrom his prodigious published output.

Jürg FröhlichBarry Simon and Statistical MechanicsWell, this is aboutmymentor and friendBarry Simon! I firstmet Barry at a summer school on constructive quantumfield theory and statistical mechanics at Les Houches,France, in 1970. We were only twenty-four years old atthe time, and I had just started my life as a PhD studentof Klaus Hepp, while Barry, a former PhD student of thelate Arthur S. Wightman, was already a “Herr Doktor”and—if my memory is correct—an assistant professorat Princeton University. As I wrote on the occasion ofhis sixtieth birthday celebrations [1], Barry would usuallybeat me in almost everything! To start with, he was borntwo and a half months before me.

At Les Houches, the late Oscar E. Lanford III lecturedon general functional analysis, including measure theory

Jürg Fröhlich is professor emeritus of the Department of Physicsat the ETH, Zürich. His email address is juerg@phys.ethz.ch.

748 Notices of the AMS Volume 63, Number 7

and the theory of operator algebras, tools rightly thoughtto be essential to understanding quantum theory andstatistical mechanics. Besides Barry andme, Alain Conneswas a student at that school. Barry and Alain were soonengaged in a competition to simplify Oscar’s proofs; Barrywould usually win—not only in mathematics but also inthe consumption of food.

Our common mentor and friend, the late EdwardNelson, wrote about Barry [1]: “In the late 1960s, Barrywas a graduate student in physics at Princeton andattended some courses I taught. I soon learned that Idid not need to prepare with great thoroughness; it wasenough to get things approximately right and Barry fromwhere he was sitting would tell us how to get themprecisely right. I miss Barry.” Well, Barry and I miss Ed!

My next encounter with Barry was in 1972 when hetaught a graduate course on quantum field theory inthe “3ième cycle” of French-speaking Switzerland. Thatcourse was the basis for his 𝑃(𝜙)2-book, which is stillremembered in the community. This is perhaps because itcontains those famous “Fröhlich bounds” or, more likely,because it is written in a very pedagogical manner—indeed, one of Barry’s outstanding strengths was and stillis to being able to write mathematical prose in a veryclear, pedagogical style.

In the fall of 1974 I accepted the offer of an assis-tant professorship at the mathematics department ofPrinceton University which had been prepared by ArthurWightman and Barry Simon. Major benefits of having hadBarry as a colleague were that I never had to submita grant proposal to the NSF; Barry’s proposal not onlycovered his own needs but also the ones of the lateValja Bargmann and me, and should one successfullycollaborate with him, he would always write the paper(except for appendices on tedious technicalities, such ascluster expansions, which he gracefully assigned to hiscollaborators).

Barry needed onlyroughly 5 percent

of the timeordinary mortalsneed to write a

paper

Barry needed onlyroughly 5 percent ofthe time ordinary mor-tals need to writea paper. He did hiswriting while listeningto seminar talks. Al-though he appeared tobe absorbed in his ac-tivity (carried out bymoving his left handalong rather peculiartrajectories), he wouldnevertheless be able to

point out errors to the lecturer or ask relevant questionsat the end. Barry was simply brilliant in “multitasking.”

Letme briefly describe two of our joint papers. The firstone is entitled “Pure states for general 𝑃(𝜙)2-theories:Construction, regularity and variational equality” andwas published in the Annals of Mathematics in 1977.In this paper, ideas and concepts from classical sta-tistical mechanics were transferred to Euclidean fieldtheory with the purpose of learning something new about

the latter. General concepts were illustrated on simpleexamples of Euclidean field theory in two dimensions,which, mathematically, may be defined as generalizedstochastic processes—more precisely, Markovian randomfields—over ℝ2 constructed as perturbations of Gaussianprocesses by local multiplicative functionals. From suchprocesses interacting scalar quantum field theories ontwo-dimensional Minkowski space can be reconstructed,furnishing examples of what has become known as con-structive quantum field theory. CQFT was first advocatedby Arthur Wightman in the early 1960s with the purposeof showing that quantum theory and the special theoryof relativity are compatible with each other, and wassubsequently pursued by Edward Nelson, James Glimm,Arthur Jaffe, and their followers.

The modern mathematically rigorous approach to sta-tistical mechanics was developed by, among many otherpeople, Roland L. Dobrushin, Oscar E. Lanford, and, mostimportantly, David Ruelle. Our paper is unthinkable with-out their work andwithout the discoveries of K. Symanzik,E. Nelson, F. Guerra, L. Rosen, and B. Simon in Euclideanfield theory, some of whose works are classics. They hadshown that in the Euclidean region (time purely imaginary)of complexified Minkowski space, a quantum field theoryof Bose fields looks like a model of classical statisticalmechanics. In my paper with Barry this fact is exploitedin an essential way.

We were to “win the jackpot” with the paper “Infraredbounds, phase transitions and continuous symmetrybreaking,” which was the result of joint work with ourfriend Tom Spencer and was published in Communica-tions in Mathematical Physics in 1976.1 Barry, Tom, andI decided to attempt to understand phase transitionsaccompanied by the spontaneous breaking of continuoussymmetries and long-range correlations (i.e., a divergentcorrelation length), in models of classical lattice spin sys-tems and lattice gases. We exploited ideas from quantumfield theory; in particular, we discovered an analogue ofthe so-called Källen-Lehmann spectral representation oftwo-point correlation functions in quantum field theory.For this representation to hold true it is necessary that themodel under scrutiny satisfy the Osterwalder-Schraderpositivity, also called reflection positivity, a property origi-nating in axiomatic quantum field theory (as described inwell-known books by Streater and Wightman and by Jost).

Here is an example: With each site 𝑥 of the latticeℤ𝑑 we associate a random variable 𝑆𝑥 ∈ ℝ𝑁, a classical“spin,” whose a priori distribution is given by a probabilitymeasure, 𝑑𝜇(⋅) on ℝ𝑁, invariant under rotations of ℝ𝑁;for instance,(4) 𝑑𝜇(𝑆) = const. 𝛿(|𝑆|2 − 1)𝑑𝑁𝑆.Let Λ be a finite cube in ℤ𝑑. The energy of a configuration,𝑆Λ ∶= {𝑆𝑥}𝑥∈Λ, of “spins” is given by a functional (called“Hamiltonian”),(5) 𝐻(𝑆Λ) ∶= − ∑

𝑥,𝑦∈Λ𝐽(𝑥 − 𝑦)𝑆𝑥 ⋅ 𝑆𝑦.

1At the beginning of my career I was privileged to have severalmentors, among whom Tom was undoubtedly the most importantone!

August 2016 Notices of the AMS 749

Birmingham, AL, Meeting on Differential Equations,1983. Back row: Fröhlich, Yajima, Simon, Temam, Enss,Kato, Schechter, Brezis, Carroll, Rabinowitz. Front row:Crandall, Ekeland, Agmon, Morawetz, Smoller, Lieb,Lax.

Here 𝐽(𝑥) is a function in ℓ1(ℤ𝑑) assumed to be reflection-positive and invariant under permutations of latticedirections. By a theorem of Bochner, these propertiesimply that it has an integral representation,

(6) 𝐽(𝑥1, 𝑥) = ∫1

−1𝜆|𝑥1|−1𝑒𝑖𝑘⋅𝑥𝑑𝜌(𝜆, 𝑘), 𝑥1 ≠ 0,

with 𝑥 = (𝑥2,… , 𝑥𝑑), where 𝑑𝜌(𝜆, 𝑘) is a positive measureon [−1, 1] × 𝕋𝑑−1, for example, 𝐽(𝑥) = 𝛿|𝑥|,1. In (5) we

impose periodic boundary conditions at the boundaryof Λ. The distribution of configurations 𝑆Λ of “spins” inthermal equilibrium at inverse temperature 𝛽 is given bythe Gibbs measure(7) 𝑑𝑃𝛽(𝑆Λ) ∶= 𝑍−1

𝛽,Λ𝑒𝑥𝑝[−𝛽𝐻(𝑆Λ)]∏𝑥∈Λ

𝑑𝜇(𝑆𝑥),

where 𝑍𝛽,Λ is a normalization factor (called “partitionfunction”). Let ⟨ ⋅ ⟩𝛽,Λ denote an expectation with respectto 𝑑𝑃𝛽. For “wave vectors” 𝑘 in the lattice dual to Λ,we define 𝜔𝛽,Λ(𝑘) to be the Fourier transform of thecorrelation function ⟨𝑆0 ⋅ 𝑆𝑥⟩𝛽,Λ, 0, 𝑥 ∈ Λ.

Simon, Spencer, and I proved that

(8) 0 ≤ 𝜔𝛽,Λ(𝑘) ≤𝑁

2𝛽( ̂𝐽(0) − ̂𝐽(𝑘))for 𝑘 ≠ 0,∀𝛽,

where ̂𝐽(𝑘) is the Fourier transformof 𝐽(𝑥), 𝑥 ∈ Λ. This so-called “infrared bound” is inspired by the Källen-Lehmannrepresentation of two-point functions in canonical rela-tivistic quantum field theory. The realization that (6)implies the upper bound in (8) is the basic result in ourwork. It is then an exercise to show that if the “couplingfunction” 𝐽(⋅) is such that(9) |Λ|−1 ∑

𝑘≠0[ ̂𝐽(0) − ̂𝐽(𝑘)]−1 ≤ const.,

uniformly in Λ, then in the thermodynamic limit Λ ↗ ℤ𝑑,phases with broken 𝑂(𝑁)-symmetry coexist and arepermuted among themselves under the action of the

symmetry group 𝑂(𝑁), provided 𝛽 is large enough. (By(4), ⟨|𝑆0|2⟩𝛽,Λ = 1; this and (8), (9) imply that the weightof the mode at 𝑘 = 0 in 𝜔𝛽,Λ(𝑘) is ∝ |Λ|.) For small 𝛽,however, the Gibbs state is well known to be unique.

It turns out that the bound (8) hasmany further applica-tions. It is an important ingredient in a beautiful analysisof critical behavior in the Ising model (by Aizenman,Duminil-Copin, and Sidoravicius) and in showing that thelarge-distance scaling limit of the nearest-neighbor Isingand the classical XY-model isGaussian in dimension 𝑑 > 4(“triviality of 𝜆𝜙4

𝑑-theory” in 𝑑 ≥ 4 dimensions); see [2],[3].

In 1980 Barry proved a correlation inequality, some-times referred to as the “Simon-Lieb inequality,” usefulto establish decay of correlations in models of classicalferromagnetic lattice spin systems, e.g., the one sketchedabove with 𝑁 = 1 or 2. Two years later, the basic idea ex-pressed in his inequality became a very useful ingredientin the analysis of multiscale problems, such as Andersonlocalization (as in work by Tom Spencer and me).

To conclude, letme draw the reader’s attention to BarrySimon’s book The Statistical Mechanics of Lattice Gases.

References[1] math.caltech.edu/SimonFest/stories.html[2] M. Aizenman, Geometric analysis of 𝜙4-fields and Ising

models, Commun. Math. Phys. 86 (1982), 1–48.[3] J. Fröhlich, On the triviality of 𝜆𝜙4

𝑑-theories and the approachto the critical point in 𝑑 ≥ 4 dimensions, Nucl. Phys. B 200(1982), 281–296.

Mike ReedOn Barry SimonWhen people ask, “How long did it take for you andBarry Simon to write those four volumes of Methods ofModern Mathematical Physics?” I usually say, “About tenyears,” since we started in the late 1960s when Barrywas a graduate student and I was a lecturer at Princeton,and we finished in the late 1970s. Writing those bookstook 50 percent of my research time for ten years butonly 10 percent of Barry’s research time, and that wasn’tbecause I contributed more—far from it. The reason isthat no one works faster than Barry. He instantly sees thethe significance of new ideas (whether in mathematics orphysics), understands the technical structures necessaryto bring the ideas to fruition, and immediately startswriting.

Barry’s legendaryspeedsometimesgothim into trouble.I remember going to seminars at Princeton with Barrycarrying new preprints frommore senior mathematicians.As the seminar proceeded, Barry would read the preprint,absorb the idea, understand the correct machinery toprove a stronger result, and begin writing. No one is moregenerous than Barry at giving credit to others; he alwaysdoes and did. Nevertheless, when Barry’s paper with astronger result and a better proof would appear before

Mike Reed is professor and Bass Fellow in Mathematics at DukeUniversity. His email address is reed@math.duke.edu.

750 Notices of the AMS Volume 63, Number 7

Barry Simon and Michael Reed, Durham, NC, 2007.

the original result, the preprint’s author would sometimeshave hard feelings. These feelings would usually dissipatewhen he or she actually met Barry and discovered howopen and generous he is.

We wrote the books because we saw that the physicsof quantum mechanics and quantum field theory raiseddeep and interesting analysis questions. Of course ourancestors knew this too. But we saw how hard it wasfor mathematicians to understand the issues and fullyengage because it was so difficult to read the physicsliterature and translate the ideas and computations intomathematical questions. Originally, we were going towrite one small volume that would give the functionalanalysis background followed by short chapters introduc-ing mathematicians to problems in modern physics. Butwe were driven by Barry’s deep knowledge and intuitionabout physics and our shared enthusiasm to do and saymore, and the result is the four volumes that we wrote.

We had a terrific time! This was long ago, so ourhandwritten manuscripts were typed. Then we wouldtake the typed manuscript, usually 500–700 pages, andread it aloud. This was the only way to go slowly enough tocheck the English, the mathematics, and the physics. Onewould read, both would think, and the other would writedown corrections. Typically it took three weeks full timeto read a manuscript. For a couple of them, I lived with

He respects others,whatever theirtalent, whatevertheir station in life

Barry. We’d get upin the morning, getto work, and give upwhen we were tired inthe evening. We wereblessed by the toler-ance and good cookingof Barry’s wife, Martha.It was very rare thatwe’d be irritated or

angry at each other, because we both have strong person-alities that are not easily troubled and we were completelyfocused on the mathematics and the science. We did allthe problems in all the volumes, except the starred onesthat we sure were correct but couldn’t immediately seehow to do.

Of course, we were pleased and proud that so manycolleagues and students found our books useful. We bothstill teach out of them and field email questions aboutthe problems. Since we were so young when they werewritten, we got lots of funny remarks at conferences frommathematicians who didn’t know us, such as, “You can’tbe the Simon who wrote those books; you’re too young,”and, “Hah! I always thought that Reed was Simon’s firstname.”

There are lots of things to celebrate about Barry Simon:his stupendous achievements, his many students, hissense of humor, his generosity to colleagues. I celebratehis deep sense of common humanity with other humanbeings. He respects others, whatever their talent, whatevertheir station in life, and this sense of common humanitymakes him very special.

This article will continue next month with contribu-tions by S. Jitomirskaya, Y. Avron, D. Damanik, J. Breuer,Y. Last, and A. Martinez-Finkelshtein.

The Leroy P. Steele Lifetime Achievement2016 Barry Simon2015 Victor Kac2014 Phillip A. Griffiths2013 Yakov G. Sinai2012 Ivo M. Babuška2011 John W. Milnor2010 William Fulton2009 Luis Caffarelli2008 George Lusztig2007 Henry P. McKean2006 Frederick W. Gehring, Dennis P. Sullivan2005 Israel M. Gelfand2004 Cathleen Synge Morawetz2003 Ronald Graham, Victor Guillemin2002 Michael Artin, Elias Stein2001 Harry Kesten2000 Isadore M. Singer1999 Richard V. Kadison1998 Nathan Jacobson1997 Ralph S. Phillips1996 Goro Shimura1995 John T. Tate1994 Louis Nirenberg1993 Eugene B. Dynkin

August 2016 Notices of the AMS 751

Barry Simon’s Students at PrincetonAnthony O’Connor, 1972Jay Rosen, 1974Robert Israel, 1975Percy Deift, 1976Evans Harrell II, 1976George Hagedorn, 1978Mark Ashbaugh, 1980Antti Kupiainen, 1980Steven Levin, 1980Peter Perry, 1981Keith Miller, 1982

Barry Simon’s Students at the California Instituteof Technology

Byron Siu, 1984Nestor Caticha Alfonso, 1985Barton Huxtable, 1987Kristiana Odencrantz, 1987Clemens Glaffig, 1988Askell Hardarson, 1988John Lindner, 1989Vojkan Jaksic, 1992Yunfeng Zhu, 1996Alexander Kiselev, 1997Andrei Khodakovsky, 1999Rowan Killip, 2000Andrej Zlatos, 2003Irina Nenciu, 2005Mihai Stoiciu, 2005Manwah Wong, 2009Rostyslav Kozhan, 2010Anna Maltsev, 2010Milivoje Lukic, 2011Brian Zachary Simanek, 2012

Creditsp. 740 Childhood photo of Simon with family, courtesyof Barry Simon.p. 741 Photo of Simon receiving his honorary doctorate,photographer Heinrich Steinlein. Used with permission.p. 742 Photo of Simon’s books, courtesy of Barry Simon.p. 743 Photo of Sadun, Avron, Harrell, and Simon, courtesyof Evans Harrell.p. 743. The Stark Effect, used with permission of TheAmerican Physical Society.p. 744 Photo of Simon’s coauthors, courtesy of The Cali-fornia Institute of Technology.p. 745 Photo of Simon’s students, courtesy of The Califor-nia Institute of Technology.p. 746 Photo of 2006 SimonFest, courtesy of The Califor-nia Institute of Technology.p. 747 Photo of Simon in boxing gloves, courtesy of BarrySimon.p. 747 Photo of Barry and Martha Simon, photographerHeinrich Steinlein. Used with permission.p. 748 Photo of Simon in Bangkok, courtesy of BarrySimon.p. 750 Photo of Birmingham Meeting on Differential Equa-tions, courtesy of Barry Simon.p. 751 Photo of Simon and Reed, courtesy of Barry Simon.

752 Notices of the AMS Volume 63, Number 7

From Mathematical Physicsto Analysis: A Walk in BarrySimon’s Mathematical Garden, IIFritz Gesztesy

Editor’s Note: This is a continuation of the August feature in honor of Barry Simon on the occasion ofhis 2016 AMS Leroy P. Steele Prize for Lifetime Achievement and his seventieth birthday conferenceAugust 28–September 1, 2016. The authors of Part I were P. A. Deift, J. Fröhlich, E. M. Harrell, M. Reed,L. Rosen, and F. Gesztesy, who coordinated Parts I and II.

Joseph (Yosi) AvronBarry and PythagorasAdmirationI passionately admired Barry in the years that shaped me:he seemed to know everything that was worth knowing,be it math, physics, history, or literature; he could thinkfaster thananyone else I knew; he couldwritemathematics

Barry [is] biggerthan life

so it read like beautifulpoetry, and he did iteffortlessly; he was awonderful teacherwhocould give a perfectlyorganized proof of anytheoremon the spur of

the moment and he could multitask like a superhumanbeing. Barry was bigger than life. He was my idol and hassince been an important part of my life.

Of Walks and TracesBarry used to visit Israel regularly. He always set up baseat the Hebrew University in Jerusalem and came for a dayor two to give a seminar at the Technion. Barry made hisitinerary early, which meant that I had plenty of time toget ready for his visit, which really meant that I had plentyof time to worry what worthwhile observation I had toimpress Barry with. Barry’s visits were like my annualdriving tests: if Barry simply shrugged and lost interest,

Fritz Gesztesy is Jean & Ralph Storm Chair of Mathematics atBaylor University. His email address is Fritz_Gesztesy@baylor.edu.

Joseph (Yosi) Avron is professor of physics at the Technion, Haifa.His email address is avron@physics.technion.ac.il.

For permission to reprint this article, please contact:reprint-permission@ams.org.DOI: http://dx.doi.org/10.1090/noti1412

Barry and Yosi, QMath7, Prague, 1998.

this meant that I flunked. Here is the story of a visit thateventually led to a joint paper.

Sometime in 1990, following Barry’s seminar at theTechnion, we were strolling through campus. This timeI came prepared. Ruedi Seiler and I were trying tounderstand Jean Bellissard’s noncommutative geometryof the quantum Hall effect, where comparison of infinite-dimensional projections plays a role. I told Barry what Ithought was an amusing identity about a pair of finitedimensional projections:

(1) 𝑇𝑟(𝑃−𝑄) = 𝑇𝑟(𝑃−𝑄)3.You can verify equation (1) using 𝑃2 = 𝑃 and 𝑄2 = 𝑄 andthe cyclicity of the trace. But this does not really explainwhy the relation is true.

Memories are fragmented and treacherous. I cannottell today if I found the trace identity on my own or ifI learned it and conveniently forgot who taught it to me.Bellissard taught me how many wonderful facts about

878 Notices of the AMS Volume 63, Number 8

traces and identities similar to the trace identity play arole in his theory of the quantum Hall effect [2], so itis possible that he taught me this identity and I simplyforgot.

Anticommutative PythagorasThe following day Barry showed me two identities involv-ing a pair of orthogonal projections that in one fell swoopexplained the trace identity and put it in a much broadercontext. My favorite mnemonic for these identities isanticommutative Pythagoras

(2) 𝐶2 +𝑆2 = 1, 𝐶𝑆+ 𝑆𝐶 = 0,where the “cosine” and “sine” are differences of projec-tions:(3) 𝐶 = 𝑃−𝑄, 𝑆 = 𝑃⟂ −𝑄 = 1−𝑃−𝑄.

SupersymmetryHere is how equations (1) and (2) are related: Suppose𝜆 ≠ ±1 is an eigenvalue of (the self-adjoint) 𝐶:

𝐶|𝜓⟩ = 𝜆 |𝜓⟩ .Then −𝜆 is also an eigenvalue of 𝐶, with eigenvector|𝜙⟩ = 𝑆 |𝜓⟩. This follows from

𝐶|𝜙⟩ = 𝐶𝑆 |𝜓⟩ = −𝑆𝐶|𝜓⟩ = −𝜆𝑆 |𝜓⟩ = −𝜆 |𝜙⟩ .The proviso 𝜆 ≠ ±1 comes about because one needs tomake sure |𝜙⟩ ≠ 0. Indeed, since 𝑃,𝑄 are orthogonalprojections, 𝑆 = 𝑆∗, and

⟨𝜙|𝜙⟩ = ⟨𝜓|𝑆∗𝑆 |𝜓⟩ = ⟨𝜓|𝑆2 |𝜓⟩ = ⟨𝜓|1 −𝐶2 |𝜓⟩= (1 − 𝜆2) ⟨𝜓|𝜓⟩ .

It follows that if 𝐶 is trace class, then the trace of all oddpowers of 𝐶 coincide:

𝑇𝑟(𝑃−𝑄) = 𝑇𝑟(𝑃−𝑄)2𝑛+1

= dimker(𝐶− 1) − dimker(𝐶+ 1) ∈ ℤ.(4)

This is illustrated in Figure 1.

C-plane

1-1

Figure 1. The spectrum of 𝐶: The paired eigenvalues−1 < ±𝜆𝑗 < 1 are marked in blue. The eigenvalue at 1is unpaired and is marked in red.

If 𝑃 − 𝑄 is compact, then the right-hand side ofequation (4) gives a natural “regularization” of the traceand shows that it is always an integer.

The Quantum Hall EffectPairs of projections play a role in the theory of thequantumHall effect. Letmeonlypointouthowphysics andmath shed light on each other in the case of equation (1).

For three projections, 𝑃,𝑄,𝑅, the trace identity impliesthat(5) 𝑇𝑟(𝑃−𝑄)3 = 𝑇𝑟(𝑃−𝑅)3 +𝑇𝑟(𝑅−𝑄)3,which follows from

𝑇𝑟(𝑃−𝑄) = 𝑇𝑟(𝑃−𝑅) +𝑇𝑟(𝑅−𝑄).This makes one wonder: Why should cubic powers ofdifferences of projection behave linearly upon tracing?

A physical insight into the linearity comes frominterpretation of 𝑇𝑟(𝑃 − 𝑄)3 as the Hall conduc-tance. The linearity of equation (5) may then be viewedasaversionofOhm’s lawof theadditivityof conductances.

Slow ScriptBarry had the reputation of being the fastest pen in theWest. So, writing these memoirs, I was actually surprisedto find out that our paper [1] came out only four yearslater. It was written during one of Barry’s subsequentvisits to Israel in his tiny cramped office at the EinsteinInstitute at the Hebrew University.

References[1] J. Avron, R. Seiler, and B. Simon, The index of a pair of

projections, J. Funct. Anal. 120 (1994), 220–237. MR1262254[2] J. Bellissard, A. van Elst, and H. Schulz-Baldes, The non-

commutative geometry of the quantum Hall effect, J. Math.Phys. 35 (1994), 5373–5451. MR1295473

Martha and Barry Simon, Ludwig-Maximilians-University of Munich, 2014.

September 2016 Notices of the AMS 879

Svetlana JitomirskayaQuasiperiodic Schrödinger Operators“In many years, flu sweeps the world. The actual strainvaries from year to year; some years it has been Hong

“In many years, flusweeps the

world…In 1981, itwas the almostperiodic flu!”

Kong flu, some yearsswine flu. In 1981, itwas the almost peri-odic flu!” So startsBarry Simon’s paper[3], also known as “theflu paper,” publishedin 1982 and cited overfour hundred times.

In this paper, Barryreviewed a series ofworks by himself,

Avron, Bellissard, Johnson, Moser, Sarnak, and others—important contributions to a newly emerging topicdemonstrating a sudden burst of strong worldwideinterest.

Some thirty-five years later, the “flu” is still here in fullswing, and while Barry was not the one who started it nordid he have students of his own working in this field, itis fair to say that he has been largely responsible for thespread of this disease in the mathematical world.

Quasiperiodic Schrödinger operators naturally arise insolid state physics, describing the influence of a weakexternal magnetic field on the electrons of a crystal. Inparticular, for a two-dimensional crystalline layer withmagnetic flux 𝛼 per unit cell exerted perpendicular to thelattice plane, a certain choice of gauge reduces the modelto

(𝐻𝜆,𝛼,𝜃𝑢)(𝑛) = 𝑢(𝑛+ 1) + 𝑢(𝑛− 1)+2𝜆 cos2𝜋(𝜃+ 𝑛𝛼)𝑢(𝑛),(6)

the almost Mathieu operator, with 𝜆 determined by theanisotropy of the lattice. First proposed in the work ofPeierls back in the 1930s, this model did not becomepopular in physics until the work of Peierls’s studentHarper in the 1950s. The popularity further increaseddramatically after the numerical study of Hofstadter in1976. The famous Hofstadter’s butterfly in Figure 2, aplot of the spectra of (6) for fifty rational values of 𝛼, wasthe first numerically produced fractal before the wordfractal was even coined. That gave a significant boostto a conjecture first formulated by Azbel in the 1960sthat the spectrum of (6) must be a Cantor set. Alongsidethe pioneering Dinaburg-Sinai work from the 1970s—thefirst application of KAM to show Bloch waves in a simi-lar model—and further conjectures formulated by Aubryand Andre, it pointed to very unusual features of thismodel: metal-insulator transition, dense point spectrum,and Cantor spectrum. All of the above made this sub-ject particularly appealing to mathematicians who wereused to disproving rather than proving such phenomena.

Svetlana Jitomirskaya is professor of mathematics at theUniversity of California at Irvine. Her email address isszhitomi@math.uci.edu.

Meanwhile, Avron and Simon noted that earlier work ofGordon implied that this model also provides an easyexample of a singular continuous spectrum. No wonderthat the “flu” started spreading also in the math world,where of course it was only natural to consider the moregeneral class of almost periodic operators.

Figure 2. The colored Hofstadter butterfly.

Over the years, the field has seen a number offundamental advances by many contributors. Sinai’sand Fröhlich-Spencer-Wittwer’s KAMs, Helffer-Sjóstrand’ssemiclassical analysis, Eliasson’s reducibility/perfect an-alytic KAM, Bourgain’s analytic revolution that madenonperturbative methods robust and allowed them to gomultidimensional, deep results by Goldstein-Schlag andothers all kept adding to the excitement. Then there werefurther physics discoveries making almost periodic mod-els, particularly the almost Mathieu operator, relevantin new contexts. The most remarkable of those was thetheory of Thouless et al. that explained the quantizationof charge transport in the integer quantum Hall effect—aNobel Prize winning discovery by von Klitzing in 1980—as connected with certain topological invariants (Chernnumbers). Central to their theory is the use of the almostMathieu operator. Moreover, predictions of Thouless etal. were verified experimentally by Albrecht, von Klitzing,et al. in 2001. Three further Nobel Prizes—quasicrystals,graphene, and topological insulators—were also linked tothis field, playing a role in the unceasing spread of the“flu.”

Barry’s flupaper, alongwith his further papers from the1980s, besides making some fundamental contributions,defined the foundations of this field in a way that madeit very appealing for new students to come in. In fact,that’s the way the field, despite many major advances, isseen to this day, with Chapter 9 of Barry’s 1982 ThurnauSummer School lecture notes [1] still being the best quickintroduction to the subject.

To give but one small illustration of how Barry’s workcontributed to the worldwide flu spread, one can lookat Moscow in the 1980s. My advisor, Yasha Sinai, hada significant preprint problem. With no office or eventable space at the Moscow State University, he kept allthe preprints people had been sending him from all overthe world on a big desk in his two-room apartment. Thepreprints were piling up, so by the late 1980s, when I was

880 Notices of the AMS Volume 63, Number 8

Barry Simon, Marseille, 2007.

Svetlana Jitomirskaya

his student, there was no spaceat all left to work on that desk,and Sinai was using a tiny bu-reau for writing. At some pointhe declared that he wouldthrow away an old preprintfor every new one he received.However some preprints weretoo precious to throw away,especially since it was almostimpossible to get ahold ofmostarticles by other means. ThenSinai had the idea to give the-

matic bunches of preprints to his various students. That’show I got ahold of all of Barry’s quasiperiodic preprintsfrom the 1980s. My role was to be a librarian for thebunch: I had them catalogued and was checking them outfor two weeks at a time to various readers (and followingup with the undisciplined ones who tried to hold ontothem for a longer period). In Moscow it was still pre-Xeroxtime, but the flu found its way through the Iron Curtainnevertheless. The popularity of those preprints led to myknowing them very well, so that I could check out theone with the requested fact rather than the whole bunch.Also, I spent most of my time in graduate school as astay-at-home mom, which gave me more time alone withthose preprints. They quickly got me hooked, both by thesubject and also the clarity, elegance, and freshness ofBarry’s writings. The never-boring style required a level ofmental workout that seemed just right. The style seemedso “textbook classic” to me that when I came to UCI inthe early 1990s and Abel Klein offered to take me alongto Caltech “to meet Barry Simon,” my first reaction was,literally, “Is he still alive?”

After the 1980s, aside from the Avron-van Mouche-Simonpaper that came 𝜀-close to proving one of theAubry-Andre conjectures, our joint work on singular continuousspectra, and important results with Gesztesy and Lastthat came as corollaries of more general developments,Barry seemingly got cured himself and moved on to otherareas, yet the damage to the world was already done.

Arguably, even more important for the spread wasBarry’s fifteen problems paper [4]. There he gave a listof fifteen (according to the title, but in reality thirty-five) important problems in mathematical physics, where,alongwithmost fundamental questions suchas “existenceof crystals,” he threw in the mix a couple of problems onthe spectral theory of the almost Mathieu operator, listedas conjectures. Well, is there anything that could betterentice a talented young person to enter the field thanan attractive and accessible conjecture by Barry Simonappearing in a list like that? The answer is “Of course! It isa wrong such conjecture by Barry Simon.” Indeed, that’show Yoram Last entered the area, disproving in his thesiswritten under the direction of Yosi Avron a wrong part ofthe almost Mathieu conjecture. Despite a lot of progressin the 1990s, some of the correct parts were not yetfully solved, and then Barry did something even bolder.In his list of (now only) fifteen problems in “Schrödingeroperators in the twenty-first century” [5], Barry devotedthree(!) to some of the more delicate remaining issuesin the spectral theory of the almost Mathieu operator.This did not go unnoticed by the new young generation. Afresh PhD, J. Puig, solved an almost-everywhere version ofthe Ten Martini problem, with the enticing name coined,of course, by Barry. At about the same time, anotherfresh PhD, Artur Avila, set out to fully solve all threealmost Mathieu problems of [5], which he methodicallydid, some with coauthors. This got him infected enoughto devote his Fields Medal talk in 2014 entirely to thefield of quasiperiodic operators, despite having otheraccomplishments.

It is particularly remarkable that two of the problemswere unresolved only for zeromeasure sets of parameters,and including those in the list of fifteen for the twenty-first century highlighted the fact that the field wasmoving frommeasure theory/probability towardsanalyticnumber theory, with recent advancesmaking it possible toseek very precise information for all values of parameters.This defined a significant trend in the later development:interplay of spectral theory with arithmetics, sometimesimportant only for the proofs,1 but at times showingfascinating arithmetic phase transitions.

For example, one of the Aubry-Andre conjecturespredicted a metal insulator transition for (6): absolutelycontinuous spectrum for 𝜆 < 1 and pure point for 𝜆 > 1,based on Fourier-type duality of the family (6) betweenthese two regions, called subcritical and supercritical.Barry’s corrected conjecture acknowledged the possibilityof the singular continuous spectrum and dependence onthe arithmetics. It turns out that as far as the subcriticalregime 𝜆 < 1 goes, Aubry and Andre were right afterall, with the final result obtained by Avila in 2008, andthis is a reflection of a more general phenomenon betterunderstood in the framework of Avila’s global theory andalmost reducibility theorem. However, in the supercritical

1For example, the celebrated Ten Martini proof was dealing, afterPuig’s work, only with the remaining measure zero set of non-Diophantine frequencies, and while the end result is arithmetics-independent, the proof centers around delicate arithmetic issues.

September 2016 Notices of the AMS 881

regime the dependence on the arithmetics is even moresubtle than originally anticipated. Namely, let 𝑝𝑛

𝑞𝑛 be thecontinued fraction approximants of 𝛼 ∈ ℝ\ℚ. For any𝛼,𝜃 we define 𝛽(𝛼), 𝛿(𝛼,𝜃) ∈ [0,∞] as

𝛽 = 𝛽(𝛼) = limsup𝑛→∞

ln𝑞𝑛+1𝑞𝑛

,

𝛿 = 𝛿(𝛼,𝜃) = limsup𝑛→∞

− ln |||2𝜃 + 𝑛𝛼||||𝑛| .

(7)

We say that 𝛼 is Diophantine if 𝛽(𝛼) = 0 and that 𝜃 is𝛼-Diophantine if 𝛿(𝛼,𝜃) = 0. Lebesgue almost all 𝛼,𝜃 areDiophantine. Thenwe have the following pair of transitionresults [2]:

(1) For Diophantine 𝛼 and any 𝜃 the spectrum under-goes a transition from purely singular continuousfor 1 < 𝜆 < 𝑒𝛿 to pure point for 𝜆 > 𝑒𝛿.

(2) For 𝛼-Diophantine 𝜃 and any 𝛼 the spectrumundergoes a transition from purely singular con-tinuous for 1 < 𝜆 < 𝑒𝛽 to pure point for 𝜆 >𝑒𝛽.

This confirms a conjecture I made in 1994, also re-cently partially solved by Avila, You, and Zhou. Here theconjecture-making was definitely just an attempt to em-ulate Barry. In fact, this particular conjecture was partlymotivated by Barry’s work on the Maryland model, wherehe was the first to go so deep into the interplay betweenthe spectral theory and arithmetic. That program wasfinally completed recently in our paper with Liu, present-ing a full description of spectral transitions for all valuesof parameters. Moreover, [2] contains the descriptionof exact asymptotics of corresponding eigenfunctionsand transfermatrices, opening up a number of excitingpossibilities for further analysis.

The almost periodic flu is currently in full strength,and new vistas—and new conjectures—constantly keepcoming. With his fundamental contributions and themany cases in which he was responsible for the originalinfection, Barry deserves significant blame!

The Cantor function, or Devil’s Staircase.

References[1] H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schrödinger

Operators with Application to Quantum Mechanics and GlobalGeometry, Texts and Monographs in Physics, Springer-Verlag,Berlin, 1987. MR883643

[2] S. Jitomirskaya and W. Liu, Universal hierarchial structureof quasiperiodic eigenfunctions, preprint, 2016.

[3] B. Simon, Almost periodic Schrödinger operators: A review,Adv. in Applied Math. 3 (1982), 463–490. MR682631

[4] , Fifteen problems in mathematical physics, in Per-spectives in Mathematics: Anniversary of Oberwolfach 1984,W. Jäger, Jürgen Moser, and R. Remmert (eds.), Birkhäuser,Boston, 1984, pp. 423–454. MR779685

[5] , Schrödinger operators in the twenty-first century, inMathematical Physics 2000, A. Fokas, A. Grigoryan, T. Kibble,and B. Zegarlinski (eds.), Imperial College Press, London, 2000,pp. 283–288. MR1773049

David DamanikMathematical Physics at Caltech around the Turnof the Century; from Schrödinger Operators withExotic Spectra to Orthogonal Polynomials on theUnit CircleLike many outstanding mathematicians, Barry haschanged his research area focus from time to time. Thiswas on display at his sixtieth birthday conference at

…his moving fromexotic spectra toOPUC was mostnatural and

perhaps almostunavoidable

Caltech in 2006, whereeach of the fivedays was devotedto one major areato which he hasmade substantial con-tributions. Each daycorresponded roughlyto one decade of work;the talks on the fourthday presented work onSchrödinger operatorswith exotic spectra,which were the focus

of much of Barry’s research in the 1990s, while the talkson the fifth day presented work on orthogonal polynomi-als, an area to which Barry devoted most of his attentionin the early 2000s.

I was extremely fortunate to be a member of Barry’sresearch group for most of the period 1996–2006. Ihad joined his group primarily due to my interest inexotic spectra, but seeing his transformation into anOPUC (orthogonal polynomials on the unit circle) gurugave me a front-row experience of witnessing somethingspecial. The ease and speed with which Barry absorbedan enormous amount of material and turned into one ofhistory’s foremost experts in an area which had initiallybeen quite foreign to him was truly amazing.

However, inhindsight hismoving fromexotic spectra toOPUC was most natural and perhaps almost unavoidable.Let me explain…

David Damanik is Robert L. Moody Sr. Chair in Mathematics atRice University. His email address is damanik@rice.edu.

882 Notices of the AMS Volume 63, Number 8

Barry had for a long time been interested in themathematics of quantum mechanics and, in particular,the spectral analysis of Schrödingeroperators𝐻 = −Δ+𝑉.The most basic questions here concern the spectrum of𝐻, or the allowed energies of the system, and the spectralmeasures of 𝐻, from which one may glean informationabout the long-time behavior of the solutions of theSchrödinger equation 𝑖𝜕𝑡𝜓 = 𝐻𝜓. In the good old days,researchers in this field analyzed atomic models, where𝑉 vanishes reasonably rapidly at infinity, or crystallinemodels, where 𝑉 has translation symmetries forming afull-rank lattice. In both cases the spectrum, which isa subset of the real line, will consist of nondegenerateintervals plus possibly some isolated points outside theseintervals. The spectral measures, which are supportedby the spectrum, will in these cases have an absolutelycontinuous component, plus possibly some point masses.The latterwill sit at the isolatedpoints of the spectrum,buttheymay also sit inside the nondegenerate intervals.Whilethe standard decomposition of a measure on the real linewill also allow for a singular continuous component, in theearly days no nontrivial singular continuous componentswere known to occur for spectral measures of Schrödingeroperators 𝐻 with “reasonable” potentials 𝑉, and quite abit of effort was devoted to actually proving that theyindeed do not occur under suitable assumptions on 𝑉.

Both of these paradigms were severely challenged dueto discoveries in the 1970s and 1980s. Spectra contain-ing neither nondegenerate intervals nor isolated points(Cantor sets) were discovered in the context of almostperiodic 𝑉, and examples of potentials 𝑉 were foundfor which there actually did occur singular continuousspectral measures. The early results in these directionswere obtained for suitable examples. However, both phe-nomena were understood at a much deeper level in the1990s, and Barry was at the center of many of thesedevelopments. In fact, both phenomena turned out tobe generic in a suitable sense, and Barry contributedkey results. For example, in a series of seven papers inthe 1990s with a variety of coauthors, Barry studied theoccurrence of singular continuous measures in spectraltheory, discussing mechanisms leading to them, as wellas genericity questions about the applicability of thesemechanisms. It was this series of papers that drew me toBarry’s work and caused me to move from Germany tosunny Southern California.

One setting in which a strong effort was made in the1990s to clarify precisely which assumptions precludeor allow certain spectral phenomena was the case ofdecaying potentials. To be specific, consider Schrödingeroperators 𝐻 on the half-line, that is, in the Hilbertspace 𝐿2(0,∞), for which the potential 𝑉 is small atinfinity. For example, one may assume a power-law decaycondition |𝑉(𝑥)| ≤ 𝐶(1+|𝑥|)−𝛾, 𝛾 > 0, or an integrabilitycondition 𝑉 ∈ 𝐿𝑝(0,∞), 𝑝 ≥ 1. Under these assumptionsthe spectrum of 𝐻 will consist of the half-line [0,∞),plus possibly some isolated points below zero that canaccumulate only at zero. Thus the shape of the spectrumis classical in the sense described above. It is not clear,however, whether the spectral measures are classical as

David Damanik, Barry Simon, and Shinichi Kotani,Kyoto, Japan, 2006.

well, that is, whether they are absolutely continuous on[0,∞) and have only some additional point masses. Sincethe isolated points below zero will always correspondto point masses, the interesting question is about thenature of the spectral measures on [0,∞). It was alreadywell known at the time that power decay with 𝛾 > 1implies pure absolute continuity on (0,∞), that powerdecay with 𝛾 = 1 allows for eigenvalues inside (0,∞), andthat power decay with 𝛾 ≤ 1

2 allows for disappearanceof the absolutely continuous components of the spectralmeasures. Thus the central questions concerned the 𝛾-interval ( 1

2 , 1) and specifically whether the absolutelycontinuous part survives in all of (0,∞) and what type ofsingular components can occur in this energy region.

When I arrived at Caltech, these questions were themost pressing ones in Barry’s group. One of his manysuperb students, Alexander Kiselev, had written his 1997PhD thesis on this problem and was successively able toweaken the assumption on 𝛾 that ensured the survival ofthe absolutely continuous spectrumon (0,∞). At the time,the best result used the assumption 𝛾> 2

3 , but everyonewas betting on 𝛾> 1

2 being sufficient, so that on a powerscale there is indeed a sharp transition from the presenceof an absolutely continuous spectrum to the possibilityof its disappearance at 𝛾 = 1

2 . This was the so-called“ 12 -conjecture.” For a class of random decaying potentials,this spectral transition phenomenon was elucidated froma new angle, via modified Prüfer and EFGP transforms, ina 1998 paper Barry wrote together with Alexander Kiselevand Yoram Last.

In one of the major events in spectral theory in the1990s, the 1

2 -conjecture was proved in 1997 simulta-neously, using different methods, by Alexander Kiselevtogether with Michael Christ, and by Christian Remling,who was another German in Barry’s group in 1996–97.In fact, a stronger result was shown that is interestingin its own right: for Lebesgue almost all 𝐸 ∈ (0,∞), allsolutions 𝑢 of −𝑢′′(𝑥) +𝑉(𝑥)𝑢(𝑥) = 𝐸𝑢(𝑥) are bounded.

Recall that the smallness of 𝑉 near infinity can beexpressed through power decay bounds, as well as 𝐿𝑝

integrability statements. Given the results from the power-decaying case, one could reasonably conjecture that a

September 2016 Notices of the AMS 883

similar transition in spectral behavior takes place on the𝐿𝑝-scale at𝑝 = 2. So Barry, with his never-ending supply ofhypertalented students, suggested to Rowan Killip (whohad started his graduate studies at Caltech in 1996) thathe look at this question. It took only a very short visit ofPercy Deift to Caltech for this problem to fall. In the 1999paper by Deift and Killip, a very slick proof of 𝑉 ∈ 𝐿2

implying absolutely continuous spectrum on (0,∞) waspublished. Killip would then go on to extend this resultto periodic background and produce another outstandingthesis coming out of Barry’s group on the case of decayingpotentials, closing out the previous century in style.

With this spectral transition clarified, the other in-teresting question concerned the nature of the singularspectrum that may be embedded in [0,∞). For example,can there ever be an embedded singular continuous spec-trum, and, if so, under which assumptions on 𝑉 can itoccur? It turned out that the answer to these questionswas already implicitly contained in the approach to thefirst question used by Deift and Killip in terms of a moresophisticated use of sum rules. In two major events at thestart of this century this was uncovered. In the process ofuncovering what was really going on, Barry was naturallyled to learning the history of OPUC and examining indetail the very close connections between OPUC and thetheory of Schrödinger operators, which would then keephim busy for a number of years.

First was a 2001 preprint of Serguei Denissov, whoused Krein systems to construct embedded singularcontinuous spectra for some 𝐿2 potentials. Krein systemsare continuum analogs of OPUC, and, in understandingDenissov’s preprint, Killip and Simon were prompted tolearn new material to understand his proof, which in turnexposed them to the world of OPUC and the realizationthat the heart of the matter lay in an OPUC result fromthe early part of the previous century that puts 𝐿2 decayin 1-1 correspondence with a class of spectral measures.

Second, in the seminal 2003 Killip-Simon paper, quitepossibly one of Barry’s most influential papers ever, the

Serguei Denisov, Alexander Kiselev, Rowan Killip,David Damanik, Yoram Last, 2002.

Barry Simon, Andrei Martínez-Finkelshtein, andJonathan Breuer at Aarhus University, Denmark, 2014.

Jacobi matrix analog of this result was worked out. Semi-infinite Jacobi matrices are in many ways the discreteanalog of Schrödinger operators on the half-line, and asa consequence of this result, one could clearly see that𝐿2 decay not only allows the occurrence of embeddedsingular continuous spectra but also puts hardly anyrestrictions on the kind of embedded singular continuousspectra that can occur. The Schrödinger operator analogappeared later in a 2009 Killip-Simon paper, but themain thrust of the activity in Barry’s group following the2003 Killip-Simon paper was focused on digging into theexisting OPUC literature, clarifying what else it may teachus about Schrödinger operators and Jacobi matrices, and,more importantly, revolutionizing the OPUC theory byintroducing tools and ideas from the spectral analysis ofthe latter two classes of operators—and in essence payingback the favor.

In retrospect, the Killip-Simon papers laid bare whatthe Deift-Killip paper had only hinted at, namely, thatthe use of sum rules may connect coefficient/potentialinformation to spectral information and that this is infact a two-way street. This realization is what usheredin the new century in the mathematical physics group atCaltech and prompted Barry to move from exotic spectrato orthogonal polynomials.

Jonathan Breuer and Yoram Last

Barry between Caltech and JerusalemWe both consider ourselves (with pride) to be studentsof Barry Simon, although formally this is true of neitherof us. Aside from his books and papers, which were thebasic texts in our graduate education, he mentored usboth as postdocs, and we have collaborated, both jointlyand separately, with Barry. Each paper we have writtenwith him has been a significant learning experience, and

Jonathan Breuer is associate professor of mathematics at HebrewUniversity. His email address is jbreuer@math.huji.ac.il.

Yoram Last is professor of mathematics at Hebrew University. Hisemail address is ylast@math.huji.ac.il.

884 Notices of the AMS Volume 63, Number 8

Madrid, 2008.

our two joint collaborations with Barry, both dealing withasymptotics of Christoffel-Darboux (CD) kernels, are noexception.

The first project was started as one of us (JB) wasjust starting out as a postdoctoral scholar at Caltech withBarry as host. Barry was then writing his book on Szegő’stheorem [3] and during a visit to Israel told us aboutNevai’s delta convergence theorem and its connection tosubexponential growth of generalized eigenfunctions ofJacobi matrices. A discussion with one of us (YL) overlunch made it clear that examples could be constructedof regular measures (i.e., models with subexponentialgrowth) for which the delta convergence fails. The job offilling in the details for the construction naturally fell tothe most junior member (JB).

JB: “As I arrived at Caltech I was concentrating on fillingin the details for this example. However, mywife and I hadmade a promise to our son that when we got to Californiawe’d go visit Mickey Mouse in Disneyland as soon as wecould. Cherie Galvez, Barry’s late (and great!) secretary,suggested that since it was September and the academicyear had not yet started, we go there on a weekday andnot over the weekend when it’s crowded, a suggestion

This was my first,but not last,

encounter withBarry’s

mischievous side…

we gladly followed. AsI walked into Barry’soffice the followingweek, however, Barrylooked sternly at meand said he was givento understand I hadskipped a work dayto go to Disneyland.As I was stutteringmy response, his sternlook became a devilishsmile and he told me not to worry. Whatever Cherieapproves is fine with him. This was my first, but notlast, encounter with Barry’s mischievous side and myintroduction to the positive work atmosphere he creates.”

Eventually the paper [1] grew to be much more thana counterexample. We realized that the eigenfunctiongrowth condition was in fact equivalent to the delta-convergence, extendedNevai’s theorem, andmade severalconjectures. Throughout this project Barry was the clearleader. He formulated the problems, had the best under-standing of the context, realized the possible extensionof the theorem, and eventually wrote up the results. Thisis not uncommon with projects where he is involved.However, there are exceptions, one of which is our secondjoint paper with him.

This paper [2] deals with stability of the convergenceof the CD kernel to the sine kernel, and in this casethe motivation came more from our side. The generalmotivating problem behind this paper, which we considerimportant and largely unsolved, is that of stability ofasymptotic level spacing for Schrödinger operators underdecaying perturbations. There is an extensive body ofliterature on the stability of spectral properties. However,almost none of it deals with this type of “fine” spectralproperty.

One of the theorems in this paper is an illustration ofwhy it is beneficial to get Barry interested in a problem.This theorem says that universality at a point impliesthat this point is not an eigenvalue of the Jacobi matrix.After thinking about this problem for a little while, wepresented it to Barry at lunch on the first day of one ofhis visits to Jerusalem. By the end of lunch he had a clearand elegant proof and even thanked us for asking thequestion.

Barry’s mathematical prowess, his speed, his depth ofinsight, his unique ability to see directly to the heart of aproblemor a proof arewell known to his collaborators andhave become legendary through their stories. Slightly lessdiscussed, perhaps, is Barry’s leadership and, in particular,his dedication to the advancement of the mathematicalfields with which he is associated. The following story isan example.

The ninth OPSFA (orthogonal polynomials, specialfunctions, and applications) international conferencetook place in Marseille in July 2007. Two remarkableresults that were obtained just prior to the start of theconference were Lubinsky’s theorem on universal limitsof Christoffel–Darboux kernels and Remling’s theorem onright limits of Jacobi matrices with absolutely continuousspectrum. Neither Lubinsky nor Remling was speaking atthis conference. Nevertheless, Barry, who was a plenaryspeaker there, felt these two works had to be madeknown to the community. He thus asked the organizersfor two extra slots to discuss these results. Even thoughthe allotted slots were after the end of the daily schedule,both talks were very well attended and were clear andfascinating. There aren’t many mathematicians who willvolunteer to give two extra talks in a conference on resultsthat aren’t even theirs. Barry is not only a scholar but atrue leader in his field.

Congratulations Barry on this well-deserved honor. Wewish you (and ourselves) many more years of fruitfulcollaboration.

September 2016 Notices of the AMS 885

Andrei Martínez-Finkelshtein, Barry Simon, and MariaJose Cantero, Madrid, 2005.

References[1] J. Breuer, Y. Last, and B. Simon, The Nevai condition, Constr.

Approx. 32 (2010), 221–254. MR2677881[2] , Stability of asymptotics of Christoffel-Darboux

kernels, Commun. Math. Phys. 330 (2014), 1155–1178.MR3227509

[3] B. Simon, Szegő’s Theorem and Its Descendants: Spec-tral Theory for 𝐿2 Perturbations of Orthogonal Polynomials,M. B. Porter Lectures, Princeton University Press, Princeton, NJ,2011. MR2743058

Andrei Martínez-FinkelshteinOrthogonal Polynomials and Spectral Theory:Barry’s RevolutionQualifying exams were tough at Moscow State University,at least at the endof the 1980s. Those in analysis consistedof real and complex analysis, harmonic analysis, andoperator and spectral theory. In other words, basicallythe content of [2]. Looking for good textbooks, I wasadvised to read the first two volumes of the Russiantranslation of [1]. Books, especially scientific books, werecheap in the Soviet Union, affordable even by a graduatestudent, so I went to a bookstore to get my own copy. Allvolumes were out of print. Fortunately, there was a well-developed network of “Bukinists,” used bookstores whereI found all volumes except the most important one for me,Volume I, Functional Analysis. I checked unsuccessfullyin several places, leaving the 𝑛-th Bukinist disappointed,when I was called by a mysterious guy who in a low voiceoffered me the desired Volume I for several times itsofficial price! Indeed, Moscow at that time was a curiousplace, where smugglers made profit from Reed & Simon.Thus, my first and indirect encounter with Barry was notdeprived of a certain excitement. Whenmuch later I heardthat Barry had written a paper on orthogonal polynomials,I could not believe that it was the very same Barry Simon!I learned later how young Barry was when he wrote [1].

The role of Barry in the rapid development of thetheory of orthogonal polynomials in the last twenty years,

Andrei Martínez-Finkelshtein is professor of applied mathematicsat the University of Almería. His email address is andrei@ual.es.

especially in the use of spectral theory techniques, iswell known and documented. This exemplifies the oftendescribed and admired feature of Barry: how fast he canwork. I think it was very early in 2004 when I received amessage from Barry asking me to take a look at a paper.Before I had time to read it carefully, the small paper grewinto a much bigger one, and I got an updated version,which had the same fate. Days (I mean DAYS) later thepaper became a short book, then a longer book, then inApril 2004 he sent the message:

It’s done!! It’s done!! Well sort of. I have “es-sentially” completed my book on OrthogonalPolynomials on the Unit Circle.

Obviously, the book didn’t stop growing, with aboutbiweekly updates, until it was published in two volumesand more than one thousand pages!

It contained both classical and new results in orthog-onal polynomials, spectral theory, and complex analysis.For instance, it showed the central role played by thematrices, related to OPUC in the same way as Jacobimatrices are related to orthogonal polynomials on thereal line.

There were also higher-order analogues of Szegő’stheorem, that is, conditions on integrability of expressionscontaining the logarithm of the orthogonality weight interms of the recurrence (or Verblunsky) coefficients ofthe corresponding orthogonal polynomials.

About ten years after its publication, Orthogonal Poly-nomials on the Unit Circle is one of the most outstandingcontributions to the field, both in terms of scientificimpact and popularity, an indispensable reference for re-searchers, comparable to the influence that the classicalmonograph of Szegő [3] had in its time. It also stimulateda burst of activity in the area: “If Barry Simon is interestedin orthogonal polynomials, there should be something init!”

Barry and his collaborators also made numerouscontributions to the theory of orthogonal polynomials on

Aarhus University, Denmark,2014.

the real line, especiallyat the boundary withspectral theory.

Orthogonal polyno-mials on the real line arecharacterized by theirthree-term recurrencerelations, whose coeffi-cients can be assembledinto a three-diagonal(Jacobi) matrix 𝐽. Thespectral measure of thissemi-infinite matrix isprecisely the orthogo-nality measure for thepolynomials. Simon andhis collaborators madevery significant contri-butions to both direct(which try to read the properties of this measure from thebehavior of the entries of the Jacobi matrix) and inversespectral problems. For instance, they characterized when

886 Notices of the AMS Volume 63, Number 8

𝐽 is an ℓ2 perturbation of either a “free” or a periodicJacobi matrix, or they found higher-order analogues ofthe Szegő condition, developing for that purpose severaluseful technical tools.

Zeros of orthogonal polynomials have independentinterest and applications, and their behavior is beingactively studied. Their fine structure is strongly connectedto the so-called universality behavior of the Christoffel-Darboux kernels of the associated polynomials, relevantto statistics of eigenvalues of random matrices, a subjecton which there is an enormous amount of discussion inboth the mathematics and the physics literature. Avila,Last, and Simon showed in 2010 that universality andthe so-called “clock behavior” of zeros on the real line inthe absolutely continuous spectral region is implied byconvergence for the diagonal Christoffel-Darboux kerneland by boundedness of its analogue associated withsecond kind polynomials. They also showed that thesehypotheses are always valid for ergodic Jacobi matriceswith absolutely continuous spectra.

I am also interested in asymptotic problems for or-thogonal polynomials, and during 2004–05 some of mywork overlapped with Barry’s research; together with KenMcLaughlin and Ed Saff we were focusing on the asymp-totics of orthogonal polynomials on the unit circle withrespect to analytic weights. The zeros of such polyno-mials were also, and almost simultaneously, studied bySimon. But our techniques were very different: While wewere using the newly created tool of Riemann-Hilbertanalysis, Simon’s approach was more classical, obviouslyborrowing ideas from spectral theory. At a conferencein honor of Percy Deift’s birthday, Barry referred to theRiemann-Hilbert method (which yields impressive resultsbut invariably requires lengthy calculations) as “drivinga Caterpillar truck,” as opposed to his “using an ax” inorder to open a path through the jungle of the unknowntowards the desired goal. Later, looking at the exhaustive

Madrid, 2008.

Zeros of orthogonal polynomials on the unit circle ofdegrees 𝑛 = 1, 2,… , 150, with respect to the weight𝑊(𝑧) = |𝑧− 1|1/5|𝑧− 𝑎|−2/5|𝑧− 𝑎2|−1/5, for |𝑧| = 1, with𝑎 = exp(𝜋𝑖√2/2).

results that Barry was able to obtain, I compared hismethod to using not an ax, but napalm.2

Due to our mutual interest in these topics, I got aninvitation to visit Barry at Caltech in 2008 and was ableto see him in action. I actually visited Caltech severaltimes, with almost a year’s spacing. These visits werehighly enjoyable and stressful at the same time. I felt likea graduate student, and although it was a test for my self-esteem, it was a fantastic experience. We started to work,but it progressed slowly, partially due to my distractionwith so many other things I had to learn from Barry. Ourtypical interaction, on the rare occasions when it was Iwho came upwith a new idea, was like this: After spending

“Barry writesbooks in the time

others writepapers”

the whole weekendimmersed in lengthycomputations, I wouldask Cherie,3 Barry’slong-time secretary,for an appointment tosee him, and I wouldproudly scribble myformulas on the black-board. In the eventthey were right, Barry would look at them for a whilein silence, slightly squinting and playing with his beard,then murmur that it was a bit late, that he needed to drive

2This controversy between Caterpillar truck and ax went on for awhile. Barry has a notorious sense of humor, often reflected in hiswriting.3Sadly, Cherie passed away in July of 2013.

September 2016 Notices of the AMS 887

Olga Holtz, Herbert Stahl, Guillermo López-Lagomasino, Vilmos Totik, and Kathy Driver, SanAntonio, 2010.

home, but that he thought he could prove it in a few lines.An hour later (about the time it would take him to drivehome from Caltech!) I would receive a scan of Barry’s“doctor handwriting” containing a proof…in a few lines!

Here are a few more observations about Barry fromthat time:• Barry has a vast culture. Not only does his personal

toolbox contain so many mathematical results, theo-ries, formulas, and ideas, but he masterfully appliesthem elsewhere. He has quite wide interests: comput-ers and politics, just to mention two of them. Heknows a lot about these topics and discusses themwith passion. A preferred place for such discussionswas the so-called “brown bag meetings” at Caltech,right after his seminars. One day Barry was regrettingthat hewas spending toomuch time following politicalnews, and I wondered what more he could have donewithout “wasting” this time.

• Barry is so fast it sometimes looks unreal. I alreadytold how he would re-prove my laboriously obtainedresults when driving home. But I witnessed how hewould “spoil” somebody’s punchline at a seminar talk,exclaiming a few minutes into the talk, “Ah, you aregoing to do this and this, claiming that…!”

• On top of this, Barry is extraordinarily well organized.I mentioned that everybody visiting Barry neededan appointment to meet him, and his schedule wasstrictly respected.All these factors sum up to Barry’s legendary produc-

tivity: his five-volume Comprehensive Course in Analysishas 3,259 pages! Quoting Vilmos Totik, “Barry writesbooks in the time others write papers.”

I will finish by mentioning Barry Simon’s teaching,which has had a tremendous impact on the community.His lectures and review papers have had a great influenceon numerous people in a wide range of fields in physicsand mathematics and have served as an enormous sourceof inspiration. Barry is a passionate lecturer who mastersthe blackboard, something not so common in these daysof multimedia presentations. I remember that in June

of 2005 Barry gave a two-day seminar at the UniversityCarlos III de Madrid. It was bad timing: the main lecturehalls were closed for some reason, and we had to squeezeinto a small room with a tiny board, about 5 × 7 feet! Wewere all rather concerned about Barry’s reaction, but hemasterfully gave the whole course, using every single oneof those 35 square feet.

In contrast with that, the 9th International Symposiumon Orthogonal Polynomials, Special Functions, and Appli-cations took place in Marseille two years later, and Barryvolunteered to give an extra late-evening session on somehot topics on orthogonal polynomials. The main lectureroom in the International Center for Mathematical Meet-ings of the French Mathematical Society in Luminy wasspectacular, the blackboard made of nine large movingpanels. The use of this surface by Barry was masterfulagain; all blackboards were filled with formulas and theo-rems, going up and down in front of the audience in animpressive exhibition of his communication skills.

Throughout his scientific career Barry Simon has hada special concern for young (and not so young) scientists.The long list of people who have worked with BarrySimon is remarkable and includes many PhD students,postdoctoral fellows, and collaborators from many fields.It is a privilege and an honor for me to be part of thislist. There are still several open questions and unfinishedprojects with Barry, and I hope to be able to ask him foran appointment again in the near future and to scribblemy formulas on a blackboard, even risking to hear fromhim, “I think I can prove it in a few lines!”

References[1] M. Reed and B. Simon, Methods of Modern Mathemat-

ical Physics, Vols. I–IV, Academic Press, New York, 1sted., 1972–1979. MR0493419, MR0493420, MR0493421,MR0529429

[2] B. Simon, A Comprehensive Course in Analysis, Vols. 1–5, Amer. Math. Soc., Providence, RI, 2015. MR3408971,MR3443339, MR3364090, MR3410783

[3] G. Szegő, Orthogonal Polynomials, 4th ed., AMS Colloq. Publ.,Vol. 23, Amer. Math. Soc., Providence, RI, 1975. MR0372517

Shu Nakamura, Barry Simon, Peter Hislop, andFrédéric Klopp, Goa, India, 2000.

888 Notices of the AMS Volume 63, Number 8

CreditsPage 2 photo of Barry and Yosi, courtesy of Barry Simon.Page 3 photo of Martha and Barry, courtesy of HeinrichSteinlein.Figure 2, courtesy of Yosi Avron and Daniel Osadchy.Page 5 head shot, courtesy of Svetlana Jitomirskaya.Page 5 photo of Simon in Marseille, courtesy of AndreiMartínez-Finkelshtein.Page 6 illustration, courtesy of Barry Simon.Photos of Barry in Japan, and 2002 students and postdocs,courtesy of David Damanik.Page 8 photo of Barry in Denmark, 2014 and pages9–11 photos and artwork, courtesy of Andrei Martínez-Finkelshtein.Page 12 photo taken in San Antonio, 2010, courtesy ofVilmos Totik.Page 12 photo taken in Goa, India, 2000, courtesy of BarrySimon.

2 NOTICES OF THE AMS VOLUME 63, NUMBER 1

Twenty Years Ago in the Notices

September 1996

Finsler Geometry Is Just Rieman-nian Geometry without the Qua-dratic Restriction, by Shiing-Shen Chern.

The outstanding mathematician explains why Finsler geometry is a natural setting for Riemannian geometry in many and diverse situations.

www.ams.org/notices/199609/chern.pdf

September 2016 Notices of the AMS 889