sign in sign up
<<
Home , Charles Fefferman, Luis Caffarelli

v The Poetry of Analysis

Conference in honour of ANTONIO CÓRDOBA on the occasion of his 60th birthday

Madrid, June 22nd-26th, 2009

The Congress Antonio’s life Luis Vega Luis ÁngelSeco Ovidiu Savin Antonio Ros Rafael delaLlave Miguel ÁngelHerrero Andrew Granville Guy David Diego Córdoba Alice Chang Fernando Chamizo Xavier Cabré Joaquim Bruna Pilar Bayer Family tree List ofpublications An interviewwithAntonio Vita ...... Index 49 67 59 45 43 39 77 75 73 57 37 55 53 35 25 61 71 51 31 11 7

The Poetry of Analysis

Antonio’s life

http://www.uam.es/antonio.cordoba Antonio’s webpage: A. Calderón (Chicago, 1985); Oberwolfach meeting on (1986); Analysis Harmonic on meeting Oberwolfach 1985); (Chicago, Calderón A. 65 the celebrate to conference the 1983); (Pisa, Conference Analysis conference in honor of Antoni Zygmund's 80 the 1979); (Williamstown, Conference Analysis Harmonic Williamstown mention: He has been plenary speaker at many international meetings, among which we may participate inworkshopsnonlinearPDE’s,organizedbyL.Caffarelli). to 1996, of (spring Austin Texas, of University the at and 1992) of (spring Study mester in , organized by J. Garnett); at the Institute for Advanced by Y. Meyer to take part in the Harmonic Analysis Seminar); at the ETH Zurich (se Analytic Number Theory); at the Universitè de París, Orsay (winter of 1978, invited invitation from L. Carleson to participate in a workshop in Harmonic Analysis and an after 1979, of (spring Institute Mittag-Leffler the at visitor term long a was He been fullprofessorattheUniversidadAutónomadeMadrid. has Antonio 1979, December Since (1994-95). University Princeton at professor visiting and (1988-89); Princeton Study, Advanced for Institute the of member (1983-84); Chicago of University the at professor visiting (1977-79); Madrid CSIC, (1974-79); senior researcher at the University Consejo Superior de Investigaciones Científicas Princeton at professor assistant as positions academic held Antonio “The Kakeya maximalfunctionandthesphericalsummationmultipliers”. thesis: the with Fefferman Charles of supervision the under 1974 in PhD his tained ob he where Chicago, of University the at Program Doctorate the joined Antonio 1971, in Later, Complutense. Universidad the at study to Madrid to Antonio Córdoba was born in Murcia, on January the 12 s h Jit M-SE etn (eil, 03, h Erpa Mt Society Math European the 2003), (Seville, Meeting AMS-RSME such Joint meetings the of international as Committees Scientific the of part been has He Bilbao, Oviedo,Vigo,UIMP, etc. Valencia, Granada, Cádiz, Fe, Santa de Universidad Aires), (Buenos Matemáticas de Argentino Instituto Murcia, Zaragoza, UPC), UB, Federigo (UAB, Barcelona (Inst. Enriques), Milano (Paris), Orsay Austin, UCLA, (Wisconsin), Madison MIT, a been also has Colloquium speaker at many universities: Chicago, Antonio Princeton, Rutgers, Maryland, 2009). Alto, Palo (AIM, equations Navier-Stokes Euler and on Workshop 2008); (Pisa, Mechanics Fluid on Workshop 2007); Madrid, de Autónoma (Universidad Números Teoríade de Jornadas 2006); Barcelona, de (E. Analysis Harmonic on Workshop Schrödinger Institute, Vienna, 2003); Barcelona 2003); Analysis Conference (Universidad Orsay, Paris de (Univ. Coifman R. and Meyer I. of honor in Meeting Analysis Harmonic 2003); (Seville, AMS-RSME Meeting Joint 1996); and 1983 (1979, Analysis Harmonic on Conference Escorial El th birthday (Chicago, 1981); Harmonic th , 1949. In 1966, he moved th birthday of birthday Vita - -

The Poetry of Analysis 7 8 The Poetry of Analysis tion ofafinitenumberperiodic structures, perfect Bragg’s peaks in the X-ray diffraction spectrum, then we have a combina gets one if that fact basic the for proof mathematical the gave Antonio instance, For 1985. around uncovered were quasi-crystals of properties the when graphers crystallo to relevant those in especially problems, inverse several in interested also was He authors. different by extended and used been has theory coeffi a cients, packet wave of rearrangements just were transformations” “canonical where calculus pseudo-differential flexible more a have to order in transform” packets “wave of theory the introduced they Fefferman, C. with collaboration In which neverthelessarestillopen. and developing a plan to treat the difficult higher dimensional Kakeya questions, functions, maximal associated their and curves along transforms Hilbert tegrals, in singular for inequalities weighted lemmas, restriction multipliers, Fourier on L differentiates the integrals of a function dinate axes with one of their dimensions depending monotonically on the others, in parallelepipeds of basis The Chicago: of school analysis harmonic the by desire” of “object an considered was which and 1930) (around Zygmund A. by formulated conjecture a to answer positive a in parallelepipeds by satisfied property covering type” ering lemmas. In a collaboration with R. Fefferman, they proved the “exponential functions associated with different kind of geometries, obtaining some new cov During the early stages of his mathematical career, Antonio investigated maximal T. Tao, amongothers. or Bourgain J. like of work the in continuation to leading results his thesis, have become a standard topic in recent Fourier analysis nowadays, his in introduced were which function, maximal Kakeya name, its and concept The terms of the eccentricity of the involved rectangles, in the two-dimensional case. the and function maximal Kakeya in norms, their of estimates sharp contains and multipliers, summation spherical the about was Dissertation PhD Antonio’s Research Agency)andMinisteriodeEducaciónyCiencia(Spain). National the by Science supported Foundation, the USA-Spain projects Cooperation Committee; NATO, several CYCIT (Spanish in Tenerife researcher and leading the Real (2000) and of (2002), Madrid at held Committee Meetings Scientific Española the Matemática of Sociedad President was He 2009). Fefferman C. (Princeton, Prof. of honor in Conference the or 2000), (Granada, Conference (log +

L ). He pursued in that line of research for several years, with publications with years, several for research of line that in pursued He ). f R so long as 3 , whose sides are parallel to the coor the to parallel are sides whose , i.e. acrystal. f belong locally to the space R n . Later (1978), he gave he (1978), Later . ------When Antonio returned to Spain after a long period of work, mainly at the uni the at mainly work, of period long a after Spain to returned Antonio When of waves. theory water the or cells Hele-Shaw the problem, Muskat the in interfaces fluid of evolution the studying as well as equation quasi-geostrophic the with related models different in singularities for searching Mechanics, Fluid on publications recent his finally, and conjecture, Giorgi´s de Ennio with surfaces connections their minimal and theory, field mean Ginzburg-Landau the around phase of transitions theory a develop C. to with Caffarelli L. with collaboration work his in Seco; L. papers and Fefferman several with contributed he which to atoms, large of energies atomic of question the functions; band-limited of theory the the Analysis; Harmonic and Crystallography of interface the at principle, problem a is uncertainty which the and coefficients the of sice the series account Fourier into of taking summation the to related works other published has Antonio ditive Combinatorics. papers which could now be considered as belonging to the emerging field of Ad several published Antonio Theory), Number in students PhD two (his F.Chamizo and Cilleruelo J. With graphs). their of dimensions fractal properties, entiability are frequencies whose series Fourier of ties proper the and arcs, small in points lattice of distribution the problems”, point “lattice- so-called the with related those especially Theory, Number in problems different in involved get to him led functions maximal Kakeya and lemmas tion restric with connection their and multipliers Bochner-Riesz the about Questions together withJoséLuisFernández. journal, this of Director the still is Antonio 1985. in Iberoamericana Matemática Revista of creation the mention also should we work, this all to addition In drid). Ma Escorial, (El Analysis Harmonic on Conferences International two first the as well as Equations), Differential Partial and Mechanics workshops Fluid (with Theory, Number IAS on the at then Caffarelli, L. with collaboration in Pelayo dez ganized a Mathematical Summer School at the Universidad Internacional Menén or He years. several for Committee Scientific its of president as acting (RSME), Society Mathematical Spanish Royal the of “re-foundation” the in part took also priate policies with the governmental agencies in charge of research in Spain. He appro develop and suggest to commissions as and panels well many in as participating Madrid, de Autónoma Universidad the at Department Mathematics the of “creation” the including Spain, in Mathematics of situation the proving im of task the in involved deeply became he Chicago, and Princeton of versities L p convergence of series in eigenfunctions of the Laplacian in balls of in balls Laplacian of the in eigenfunctions of series convergence k th-powers ( th-powers L p boundedness, differ boundedness, R n and ------

The Poetry of Analysis 9 10 The Poetry of Analysis plants, flowersandasplendidcactuscollection. many her of care taking gardening, enjoys and teacher school high a is she ics; Mathemat in degree a has also who Amelia, to married been has Antonio 1995 Since Thurston. William of supervision the under dissertation PhD her wrote who topologist dimensional low a was 1993, in away passed who Maricarmen, wife, first His age). tender a at still granddaugther a Ainhoa, and Miguel, and Adrián twins (the grandchildren three and laboratory), research medical a at working sons (Diego, who is also a , and Rubén, a PhD in Organic Chemistry two has also He Complutense. Universidad the at Seismology, in specializing ics, found on page has 25. who a Antonio is brother of a younger Geophys professor Mathematics, Crelle, of CPAM, Advances in Mathematics, etc. JournalA detailed list of publications can be American PNAS, Inventiones, Mathematics, of Annals as such journals, prestigious most the in papers 80 than more published has He Crystallography). Seehisfamilytreeonpage31. chanics: stability of matter); and Pablo Fernández-Gallardo (Fourier Analysis and Me (Quantum Balodis Pedro Theory); (Number Chamizo Fernando and Cilleruelo Vega Luis Javier functions); surface); limited (band conical Barceló Antonio a Juan operators); to (Schrödinger transform Fourier the of (restriction Barceló Bartolomé summability); Bochner-Riesz functions, (maximal López-Melero Bernar problems); do (inverse Ruiz Alberto to advisor PhD the been has Antonio - - - - AC: Yes, although he wasn’t the only one. There were some teachers working at working teachers some were There one. only the wasn’t he although Yes, AC: PF: Aspecialkindofpersonthen? AC: I did my Baccalauréat at the at Baccalauréat my did I AC: PF: Whatdoyourememberaboutyourfirstyearsofstudy? CÓRDOBA: ANTONIO PABLO FERNÁNDEZ: stitute, both with the teachers and with my fellow students, with whom I was I whom with students, fellow my with and teachers the with both stitute, In the at home at much very felt War.I Civil the after jobs their lost had who but Republic, the during Universities at teaching been had who Institute that out tobemuchmoredifficult. working on an analogous method for the deaf, although it apparently turned for a study on how to teach mathematics to blind children. At that time he was where he was teaching at the time, and he told me he’d been awarded a prize ever to have been to university. to been have to ever cians inyourfamily? I met him again many years later in Valencia, at the at Valencia, in later years many again him met I process. the complete and blackboard the to go he did angles different from the problems he set us, and only when we’d turned the question over together times; he used to sit down among us, the better to stimulate discussion about those for unusual was that teaching of way a had He career. of choice my on influence an had certainly who teacher, Mathematics my Soto, Francisco like extraordinary, were teachers my of Some times. those of memories fond have Murcia. around villages and towns different in teacher supply a as worked she years they were political, she never passed the competitive state exams, so for many deduced I although about, spoke never she reasons some For again. subjects same the all in exams the sit to herself upon it took she so invalid, declared That was during the Republic. finally she decided it would be more sensible to do a Teacher Training course. but University, the at that studying about thinking was she and Poetry, and Literature liked she that once me told She study. to encouraged they whom youngest, the was who mother, my was it children three their of out but ate, illiter practically were grandparents maternal My Baccalauréat. her do to rather odd, because in Murcia in the late twenties it was unusual for a woman To begin with, Antonio: Have there ever been any mathemati

No, not at all. As far as I know, I’m the first one in my family my in one first the I’m know, I as far As all. at not No, An interviewwithAntonio After the Civil War, all those qualifications were Instituto Alfonso X el Sabio el X Alfonso Instituto My mother was a schoolteacher, which was which schoolteacher, a was mother My Escuela de Magisterio de Escuela in Murcia, and I and Murcia, in - - - ,

The Poetry of Analysis 11 12 The Poetry of Analysis AC: It was particularly involved in political struggle; there were people belong people were there struggle; political in involved particularly was It AC: PF: Notaveryencouragingoutlook. Whatmemoriesdoyouhaveofthatcourse? together and Olympiad, Mathematical the from grant a awarded been I’d AC: PF : Thatwasin1966. Ancochea aswell. tions, such as Father Alberto Dou, a cultured man with a more open mind, and excep honourable some were there although regime, the of members official were lecturers the of poor,many very and were FacultyMathematics the of at standards teaching the speaking, Generally University. that of mediocrity the student mobilization. On the other, I felt very disappointed when I discovered and effervescence political of full highly were years a those was atmosphere; stimulating it hand, one the On headquarters. their had Barba Peces and Oreja Marcelino Lavilla, Landelino as such Transition, the in part important it was a centre of Christian-Democratic activities where people who played an place; interesting very a be to to out turned college That expenses. my all cover grant a with me provided they and accepted, was I conditions. tageous the at places for applied we Bilbao, Mikel Olympiad, the in met I’d friend a with to moveMadrid. decided I exposure, public much too escape to partly and Murcia, in ematics Math of Faculty no was there because partly Nevertheless, colonel. army an even and professors, university journalists, local well-known alongside ince You can imagine what it was like for a 15 year-old kid, representing the prov the team from Murcia was on – not to mention the success I had with the girls! time every down closed about just which city, the in known well very became I and semi-final) the reached (we successful very was Murcia from team the was enormously popular. So much so that for the year and a half that it lasted, programme the and Spain in television of years early the were those but day, [“ Stand” We “United TV: on contest cultural of kind a in Murcia of province the representing team the joined I how that’s and Murcia, of University the from lecturers some with contact into came I Institute, the at teachers my to Thanks calculus. integral fascinated by them, and they got me involved in the world of differential and was I hands; my into came Pastor’sbooks Rey of some that then about was It very popular, partly for the studies themselves and partly because of football. Colegio

Mayor Pío XII Pío Mayor Did youhaveanyhelporcontactsinMadrid? La unión hace la fuerza” la hace unión La , which was offering some very special and advan and special very some offering was which , ]. It may sound incredible to incredible sound may It ]. ------Murcia, 1965 AC: Absolutely. We were lucky in so far as the atmosphere, given that the Franco the that given atmosphere, the as far so in lucky were We Absolutely. AC: might one teachers, university the and course that of description your From PF: from other degree courses took part, and where I met other people such as such people other met I where and part, took courses degree other from he took political leadership of the assemblies and meetings in which students ( CNT the of secretary general become to later was who Buendía, María José remember I example, For persuasion. pro-Chinese more a of others and Party) Communist (Spanish PCE the to ing where else. some future professional a seeking about seriously thinking already was I my life because, given the way things were in Spanish universities at the time, Miguel, via provided me with the chance of going to Chicago. That was a decisive step in who, Calderón was it Then attend. to able was I which erators, op pseudo-differential on course a giving months, of couple a for stayed he where Madrid, to Calderón Alberto invited who Miguel was It contributions. make could one which to and investigation required that questions still were there that complete, from far was edifice mathematical the that discover to shock a was it me for and thought, for food provided that problems him with me that Miguel de Guzmán had just come back from Chicago. Miguel brought told who he was It him. with relationship special a had always I and Faculty, of rest the from aloof little a remained who man elegant an Ancochea, met I on it purpose, as did a way we of showing up course, the system. I Of was a acquired. fourth-year student when we’d knowledge the all with hard what some times, lecturers our gave we that remember I them. with working and books these reading night at hours and hours spent We books. French mainly were They on. so and analysis, functional on Dieudonné analysis, complex were able to read we books by Bourbaki, So Halmos on measure them. theory, Rudin on importing real and were which Santos, de Díaz or Aguilar as such bookstores through available, more become to beginning were abroad from books example, For relaxed. more bit a was end, an to coming was regime guess thatyouallcarriedoutakindofself-learning. paths fortheirlives. different find to forced were who students, best its of some “expel” to aged man University that that feeling the had always I’ve people. brilliant ularly Academically speaking, I think it was an exceptional course with some partic ( UAM the at again up met I whom with and Physics studying were time that at who López, Cayetano Pacoand Bernis Confederación Nacional de Trabajo de Nacional Confederación Universidad Autónoma de Madrid de Autónoma Universidad ). ); - - - -

The Poetry of Analysis 13 14 The Poetry of Analysis AC: Well, that was really a late decision. When I was doing my degree I felt more felt I degree my doing was I When decision. late a really was that Well, AC: PF: Butwhendidyoudecidetobecomeananalyst? and Cuerva García although 1971, of autumn the in alone Chicago to went I AC: PF: So then you joined the doctoral programme in Chicago. Did any other of your And since I’d already taken a look at these things, I gave him the answer, the him gave I things, these at look a taken already I’d since And subject. the about anything knew who someone for class) a but set a not are category a of objects (the question elementary an question, rhetorical almost far.so us to explaining been he’d an what really in was inconsistencies It any noticed had we if asked Lane Mac class in day One theory. category to other the and Theory, Galois to devoted was half one Lane: Mac Saunders by given was year first the during doing was an I courses the There’s of one then; time. from anecdote special very a was It there. also was Charlie course of and out, come just had integrals singular on book Stein’s and Zygmund, was so and there, was Calderón Chicago: was there then And integrals. singular and later, when I met Miguel, I changed direction and moved over to Fourier series But theory. category as such fields in home at more felt and books Donald’s drawn towards Algebraic Geometry. I’d read Grothendieck’s and Atiyah-Mac Charlie. by supervised thesis doctoral first the being mine about uncertainty the for except idea, good a was this thought Calderón him. with work to decided already I’d and in fact), younger, months (two me as age same the about just was [Fef who Charlie ferman], with friends become and met already I’d then by but Boston, to me invited MIT. Alberto the at Boston in more six and country home his in months six spending meant which , from offer an accepted derón to at the However, end of I problems year, that which thought. some Cal to began devote some me set Calderón more, What’s me. for helpful very were courses so those even but my degree, doing when subjects to more troduced in been I’d because companions, my of those than better back bit a was my ground Maybe case. my in happened what was which for years, fellowship more a three with course the continuing of chance the had you results, your to according year, first the of end the At geometry. of three al and gebra of three courses, analysis three of consisted Chicago the in of course year doctoral first The later. months few a over came Carmen Mari course, of And mediation. Miguel’s to thanks also Louis, St. to Torre went la de Alberto fellow studentsgotostudyintheUnitedStates? ------Princeton, 1976 AC: Well, it was a really smooth transition. Just after completing my thesis in in thesis my completing after Just transition. smooth really a was it Well, AC: return to decide you did When thesis? your completed you after about What PF: and Kakeya problems, and so on. I was lucky, and by January I’d solved them, Bochner-Riesz the study I suggested who Charlie, with thesis my on working was I that going out to be an analyst instead found of an algebraist. So he in the summer of when 1972 I was later disappointed bit a felt he guess I dent. stu favourite his became I on then from and lot, a quite him surprised which experience. For some months I had to live side by side with those who’d been I who’d had those by to months For with side some side live experience. Complutense Universidad a got and ( them post of one sat I so exams, competitive for down me put to ble trou the took Guzmán de Miguel Madrid in was I while and Stockholm, and Paris Madrid, between I spent which year, a sabbatical taken I’d In 1978 end. the to out things stretch to need no was there But not. or there continue whether I’d about decision a make would Princeton before months few a only was it And Missouri… Louis St. in University Washington with Yale, with tact to of was there to live I hard going take. the was in thought also con Murcia, in born someone back, for that, going admit to my have I lot on a city the keen liked I were although but they too Chicago In California. in living of by chance the tempted I really was attractive: particularly seemed that UCLA from one was There had. I offers job different the to up came weighing was it I end as an and lot, a learned I which during University). period the great really with a was rested It decision the which after contract, 5-year a of consisted (which track tenure to promoted was I work this After Annals. the in paper first my was which Zygmund by a conjecture desire school, Chicago of the of objects the of one for was and that problems horizons, the of my one solved broadened I example I Princeton In place. the of relevance or was importance the of Charlie aware so I wasn’t time, that the At it. with do to a lot had fact there the suppose I Princeton. from offer an got I Chicago, to Spain? would havewanted,naturally. I success the without problem, multi-dimensional the with struggling was I my thesis in the summer of 1974, and during all that time, among other things, read I unsolved. still it’s today well, and dimensions… more for problem the solved have we’d months few a in that toast a made I and Charlie optimism, had to celebrate it. So we went out to dinner, and over dinner, with misplaced and so my thesis was completed. Charlie, of course, was delighted and said we profesor agregado agregado profesor [tenured lecturer], it was called at the time) at the the at time) the at called was it lecturer], [tenured . However, I have to say that it wasn’t a very good good very a wasn’t it that say to have I However, . - - -

The Poetry of Analysis 15 16 The Poetry of Analysis AC: Yes. There was a professor’s post open at the UAM, one that had just been just had that one UAM, the at open post professor’s a was There Yes. AC: PF: To the on theSpanishscene. point reference a become had Department the 1980s the of beginning the by that so fruit, bear to began soon policy that that say to have I Ros... Antonio get to tried also we well; as Francia de Rubio and Fefferman, with Princeton at thesis his done who’d Moriyón, Roberto brought we example, For prestige. of people hire to vacancies these used We leaving. by up ended them of most but theses, their for subjects propose to offered even I Doctorates. their have to have would Department the in lecturers the all course, of And, place. one just choose to were places different in worked who those changed; had tion situa the that know them let and together Department the of members the all got I So time. that at course the for par was which left, and classes gave their just schools) high in jobs teaching with work their combined them of a kind of academy for imparting classes; the people who worked there (many only really was It thesis. their done had who of some only lecturers, 30 hardly for – structures these All research. operations and statistics for other the and topology, and geometry for one algebra; for one equations; functional for one of); charge in be to going was I one the and numerous, most (the tions but five, which was the general rule in that period: one for the theory of func UAM, the at such as Department Maths one just wasn’t there time that At me. to exciting seemed perhaps naïvely that adventure an scratch; from place a ed sharply with the tricks we’d seen others up to. That was a chance to start in contrast trained) been he’d which in school, French the from (coming rigour say that he’d created a good impression; a cultured man whose mathematical to have I and Complutense, the at student a as time my during Viñas nández Fer- known I’d fact, In Murcia. to gone had who Viñas, Fernández by vacated back. come to decided I So 4. Diego and old year 1 was Rubén time, that At suitcase.” a lugging place to place from going not I’m but Madrid… in or USA the in live got either can We up. you’ve mind your make “Antonio, to said: and down foot her put Maricarmen Then year. from whole the consent spent I where with Princeton, to back go to decided I 1979, Complutense, the in so discouraging, very modernization, was at It effort unthinkable. any was structure, the change to attempt any more, What’s least. the say to unpleasant was which years, dark the in teachers my Universidad Autónoma ? - - - Antonio Córdoba. essays and articles further there also See publicaciones-ensayos.shtml http to befoundat UAM", la de Matemáticas de Departamento "El Spanish) (in text the to referred is reader The ://www.uam.es/antonio.cordoba/ by AC: Yes, you could feel the winds of change blowing in a lot of ways. As far as far As ways. of lot a in blowing change of winds the feel could you Yes, AC: PF: I suppose you mean the first Socialist Governments. You were involved with all even that, decided we disappear; just Todidn’t AC: Departments the with, begin PF: Howdidyoumanagetomakeallthosechangessoquickly? though they continued to exist formally, they wouldn’t carry on working as working on carry wouldn’t they formally, exist to continued they though any. That said, I never wanted to be on any teaching commissions! No sir; to sir; No commissions! teaching any on be to wanted never I said, That any. have didn’t practically circles mathematical in prestige and renown certain a have to supposed were who people some that out turned actually it in, came assessments and reports the when because some, of liking the to wasn’t That abroad. from experts of participation the included which system, assessment complete a up draw to managed we and field, that in Spain in all at tradition no was There assessment. project doubt without was programmes of those interesting all most The direction. right the in steps some take to had just I good for me, bearing in mind that I wasn’t much older than 30, but I felt that was than commissions more on was I Maybe time. that at up set being were variable”. Pedro always managed to get me involved in the commissions that complex and “real subtitle the with II”, Analysis Mathematical of “Chair the called was one, mention to just Mine, spectacular. really were which of some chairs, university the of many of names the down!) (cutting changing was tures and the first project assessments. One of the first measures, for example, together the reform programme in the areas of knowledge, Department struc put who one the was He Solana). with then Maravall, with (first Education of Ministry the at factotum the was Pascual Pedro concerned, were universities kinds ofcommissionsatthattime,weren’tyou? pened tobetherighttime. hap just it Maybe changes? many so about bring to much, so do to able we were How analyst. an really was I that find to surprised were later again with up met I students the of some occasion, On Algebra. of professor “official” staff in some areas that for several years, for example, I was the Department’s of short so were We Universities. switched they so intolerable, was calculus eral Algebra professors thought the obligation of giving classes in differential sev that remember I some; for much too was That system. teaching tational ro shared, a up set to was on, insisted absolutely I demand a requirements, turers, which was where all the decisions were made. One of the indispensable lec the all of assembly the instead, Division Mathematics the up set We such. - - - - -

The Poetry of Analysis 17 18 The Poetry of Analysis AC: The forerunner of the RMI was the was RMI the of forerunner The AC: popular.very the However,you change made to affairs those suppose don’t I PF: with the galleys all ready… just when the president of the Council resigned. Council the of president the when just ready… all galleys the with etc., Fefferman, Nieremberg, Cafarelli, from contributions with fact, in one, excellent an publication, for ready issue first the had already We ticipants. Yves Meyer, Caffarelli, along with some members suggested by the other par get together an initial editorial board of some standing: Fefferman, Calderón, to managed We scratch. from again start and commitments outstanding any the for was Mathematics) and that seemed all right to him. One of the conditions we made ies ( vista Hispanoamericana the of viability the study to precisely was me to entrusted they tasks first the of One council. advisory the join to me asked he and references, my had apparently who Elguero, José CSIC, the of president new the from call a got I still, draw up a new one, more adapted to the times. Almost at the same time, again and told me that they were now ready to accept the proposal. Or better me with touch in got Abellanas and García Luis Pedro UAM, the at was I when 1984, in later, years few A accepted. weren’t they and demanding too were nowned mathematicians onto the project. But it seemed that those conditions you can least ask for a the serious journal. I promised It’s to use my on. contacts for getting so re and decisions, editorial for freedom complete with and institutions other of independent board, editorial prestigious a fact: in cial, spe very nothing conditions; of set a together put I and did, I what is which it, recycling for scheme a prepare to me asked Abellanas Pedro particular, In it. revive to attempts several were there 1970s late the In low. very fallen had journal the of quality the about talking we’re time the at and downhill, went it Then decent. was it least at but class, top was it saying not I’m considered. cas mericana the up setting for process the was What subject: of other systems were really qualified to decide which was the best one for us. experience the had who’d those Only me. to natural and it reasonable that seemed say to have also I chores, those of out get to it using admit I though even and excessive, bit a was condition the imagine, can you As abroad. to be had which of one universities, three least at in taught have to had plans get out of it I always set one condition; those who came to me to discuss study , a journal that just before the Civil War, in Rey Pastor’s time, was very well Real

Sociedad Matemática Española (RMI)? Hispanoamericana . So I told him about my negotiations with other bod to publish a final issue, in order to wind up wind to order in issue, final a publish to Revista Hispanoamericana de Matemáti de Hispanoamericana Revista [RSME] – the Spanish Royal Society of Revista Matemática Iberoa Matemática Revista Re ------Los Narejos,1986 AC: At the start, the start, the At AC: PF: Butthatcollaborationdidn’tlastverylong… AC: Yes, and it’s attained a very respectable status. Instead of rankings or cita or rankings of Instead status. respectable very a attained it’s and Yes, AC: PF: The of the RSME started dropping hints about making some changes to the edito the to changes some making about hints dropping started RSME the of the regards As story. another that’s but halt, a to CSIC the of up setting the for project the bring to was decision first his concerned, was mathematics as far As place. his took Ruiz Trillas, with whom I’d previously had some disagreements at the Ministry, tions, I’d rather illustrate the fact by pointing out that, at present, Charlie present, at that, out pointing by fact the illustrate rather I’d tions, And reallythat’showweremained untiljustashorttimeago. institution… any to attached being without while, a for limbo” “in remain to came independent of the Council and the RSME. At that time I thought it better me it was a really worthwhile undertaking. Anyway, that’s how we finally be was a project that by then wasn’t costing them any money and, of course, for it because me, to surprise a as came That it. with liked we what do could we on the Council had anything published in it, and as far as they were concerned the with on go to want didn’t CSIC the that me told they which at attended I meeting a remember I matter. the in interest lost had CSIC the then by But self-supporting. be to journal the for years 4 took only it that say to have I pursuing. worthwhile be wouldn’t tions, it would be a sign that it hadn’t aroused enough interest and the project in at most 5 years: if by then it was still unable to fend for itself with subscrip the initial proposal we pointed out that the handled through the Council). To put it mildly, it didn’t work out very well. In be all would it (although Ministry the by partly and CSIC the by partly vided they coulddowascongratulatemeandencouragetogoahead. all situation, that issue. with first Faced the of galley-proofs the with me, and Guzmán de Miguel Charlie, Yves, there, up turned we So me. corner to ning plan were members the suspected I Council, the of meeting a to called was I When Madrid. in me visiting were Fefferman and Meyer then just that pened hap so It weeks. difficult really were They together. it getting in made we’d effort the all after issue first that lose to want didn’t I However, precedent. a set would it because but – well as that although – up came that names the of because much so not me, to dangerous particularly seemed That board. rial Revista hasjustcelebratedits25thanniversary. Revista needed a subsidy, which was supposed to be pro be to supposed was which subsidy, a needed Revista , among other reasons because nobody because reasons other among , Revista Instituto de Matemáticas de Instituto had to be self-supporting Revista , the president the , at the at ------

The Poetry of Analysis 19 20 The Poetry of Analysis AC: You have to remember that when I was doing my degree I wanted to be an be to wanted I degree my doing was I when that remember to have You AC: fields: different in involved been you’ve work; research your to back go Let’s PF: nection existed, and I realized right away that I could apply my knowledge my apply could I that away right realized I and existed, nection con the least at but world, different a It’s theory. C-Z the to belonging those is that the kernels that appear in these problems are much more singular than difficulty big the analysis, harmonic of view of point the From sets… special in spectra with series Fourier on sets, Sidon on questions open: still are them of many true, it’s Today, theory. Calderón-Zygmund the beyond went field the in problems some that discovered I where one last this on was It theory. number analytic on Narasimhan by another or relativity, on Chandrasekhar cally, that I could learn everything. And I attended all the courses, like one by seemed to me that the high point had passed. At that time I thought, pedanti I’m not saying that very good things haven’t been done after the 1960s, but it over. was equations) to applications their and operators pseudo-differential its (with school integral singular the of age golden the that feeling the had I then by But club. the of member more one was I if as analyists, of school that in comfortable very felt I and mathematics, for enthusiasm my rediscovered I Chicago to got I When Physics. liked always I’ve But surreal! – air of index practical assignment where we spent several months calculating the adiabatic a remember I example, For myself. by manage to difficult more was labs, the all with what course, degree Physics The own. my on Mathematics study to in the third year. I thought it was a waste of time, and I believed I was cut out Physics dropped I but courses, degree both started I fact in and physicist, a be to wanted I me. for good were made), already language a (with problems tough tackling and language creating training, of types both But side. other algebraic geometer, although later, when I went to Chicago, I went over to the out ontheway? stand milestones What like? been evolution this process, this What’s tions… equa differential theory, number physics, , harmonic year wemanagedtomakeagoofit. little by little, and with the help of Josechu Fernández, after the third or fourth was no money on the line, but a lot of prestige. The first years were tough, but English, of course) was a letter of introduction for Spanish Mathematics. There here (even though it was international and most of the articles were written in published journal a having that us to seemed It map. scientific international vista the and Annals through solely production his all channelling is Fefferman . It started out as an adventure, as part of the plan to put Spain on the on Spain put to plan the of part as adventure, an as out started It . Re - - - - Intitute forAdvancedStudy,1989

In reality, all these continuations were in a certain sense natural. Apart from Apart natural. sense certain a in were continuations these all reality, In fashion, in were variables complex several time that at seminars; the all to titatively, by the number of published papers. At Princeton I continued going quan it, measure to want we if is, that career; my in mistake strategic a was up them following not maybe equation… Schrödinger’s problems, perbolic hy to applications the well, as equations the were There questions. those to ematics withhis dadwasn’tsobadafterall.And ofcourseIwasdelighted. math about speaking that out found he maybe, situation the given States, United the from back came he when but me, with mathematics about talk to wanted never Diego degree his for studying was he when short, story long a cut To me. for encouragement of source a been hasn’t involvement Diego’s that denying not I’m Although techniques. analytic various using we’re ics, the dark. For instance, in the work we’re doing together now, on fluid mechan in leaps all not it’s thread; unbroken an also there’s this all behind Although and that, of course, shows you where you really stand in life. comeacrosssomeone likeCharlie, different someonea of magnitude,order of ple, but you could still say, well, I’m pretty good, too. And then all at once you serve to put you in your right place. Until then, sure, I’d met some brilliant peo They’re two very special friendships, in every sense. Because, for example, they me it was the first time he was able to talk and work with someone his own age. Withhimself. thanolder colleaguesesmuch surroundedby beenalways he’d Charlieisthesame ageasme. When met Ihim hewas barely 20years old, and led to the work we’ve done on Di Giorgi’s theorem, phase transitions, and so on. nated by minimal surfaces, so our collaboration came about quite naturally and alreadydevelopedhadboundariesfreefasci alwayshisbeen I’dtheory, and ested in compressible fluids, shockwave theory, but we didn’t get very far. Luis I took a sabbatical and was in Chicago. Luis was there. At the start we got inter soon become good friends – a friendship that has grown over the years. In 1986 meetingand a coincideat happenedto we 1974; Minnesotain arrivedin Luis pleasure. a been has thing, natural a being from apart them, with laborating col and Caffarelli, Luis and Fefferman Charlie friends, special very two have to driving lucky I’m friendship. been always other has career professional my behind the force field, own my to close were or liked I that subjects being of pseudo-differentialoperators. generalization a on working were example, for I, and Charlie because things, other various Among problems. theory number my and Kakeyasets my to ful faith remained I that. about excited very never was I say to have I although ------

The Poetry of Analysis 21 22 The Poetry of Analysis dif do you’d things some are there again, over life one’s live to had one If AC: look of a kind reflection, final a for you ask to like I’d Antonio, off, Toround PF: the result with Javi Cilleruelo, for example, on the number of points on small on points of number the on example, for Cilleruelo, Javi with result the But solve. to managed finally I’ve those and do to tried I’ve things the tween be gap big a there’s course, of theory, number In nice. pretty is lelepipeds, paral for lemmas covering the with conjecture, Zygmund’s of proof the that think still I one. that like still I result; serious first my operators, maximal of theorem the of proof the achieved: I’ve results the of some like particularly I only inreallyoriginalthings.Ifnot,what’sitallfor? gone downhill, that the mathematician ought to be a kind of artist, interested we’ve gone to the other extreme. I think that the status of now mathematicians has but published, got nobody on, served I committee assessment project first the during that true It’s all. at like don’t I frankly quite which on, so and days, however, what we’re seeing is a rat race of publications, impact ratings difficult problem, and not just to increase their number of publications. These really a solved they’d when exquisite, something was it when paper a lished pub only who mathematicians gentlemen, of Zygmund types as – admired I people or – Calderón like people remember I Chicago, In Mathematics. in In the last 40 years I’ve seen the general attitude in Science change, especially But anyway,atthattimeIfelthadtodoit. here. in involved got I things the all on effort and time much so spend to had certainly have been more successful if I hadn’t come back to Spain; if I hadn’t would mine then published, articles of number the by career professional a judge we If attempt. the in energy much too wasted I maybe a and naïve, was bit It Spain. in Mathematics in similar something do to necessary was it fashion, about having to start the revolution with small groups, and I thought in were that Che by phrases some were there time, that At course. of ferently, back atthethingsyou’veenjoyeddoing,yourfavouriteresults,andsoon. ship andharmonicanalysis. in which Weyl sums appear. Once again, another adventure guided by friend and table, periodic the with for do to calculations have the that developments in asymptotic these help to able was I So team. the of part being me on insisted who thesis, his do to Princeton to sent I’d student a as who Seco, Luis was it Here line. same the in less or more is mechanics quantum on work The - - - - - Soto delReal,1992 This interviewwas conductedinMarch,2009. Antonio Córdoba's(AC)scientific descendents. Pablo Fernández(PFintheinterview)isoneof

Kakeya’s now and problem, famous a is point lattice the know, you Well, AC: ones, classical the problems, “famous” the of any tackle to try ever you Did PF: There were some problems I tackled that I couldn’t solve. To take an obvious To obvious an take solve. I couldn’t that I tackled problems some were There fasci always electrodynamics (quantum integrals Feynman used we where Chamizo, Fernando with got I results the Or arcs. small on accumulate to not points the for precisely necessary was it where transform, Fourier the for lems prob restriction studying were we when up came that question a was That people. by cited increasingly being it’s curiously and like, particularly I arcs, idea, but it wasn’t to be. Now I’m working on Navier-Stokes, so you never never you know… so Navier-Stokes, on working I’m Now be. to wasn’t it but idea, new a hy onto was I Riemann’s thought even I with years it recent in tried times of I couple a that pothesis; deny can’t I said, That too. is problem let’s say? I’d as easy beginning. the at as thought wasn’t it symmetry, radial the breaks estimate Iwaniec best the that true it’s While results. classical the for proofs new find to managed only I that; like out turn didn’t it But way. long a go could I Bochner-Riesz, in done I’d as symmetry, radial the breaking by that thought I cut. to and where how rather but relevant, really wasn’t it that Bochner-Riesz in learned I’d and properties, function Bessel of the basis on the worked They methods. their on improve could I that away right realized I done had Hardy Littlewood and what read I When time. one at me intrigued also lat problem The point tice integrals. singular extremely are appear that kernels the because L to belong squares the in quencies fre with series the whether deciding solved: be to still is favourites theory my number in of One dimensions. higher in problem Kakeya the example, I devotedmyselftothem. They interestedmeatthetime,theydidn’tseemtrivialtome,andthat’swhy results. my all almost by stand still I’d think I speaking, Generally me). nated p for for p less than 4. It’s a tough problem, problem, tough a It’s 4. than less - - - - -

The Poetry of Analysis 23

• • • • • ARTICLES • • • • • • • • • • • • • • • • • • • • • • • • • . ódb, . ódb ad . acd: h Ryeg-alr odto for condition Rayleigh-Taylor The Gancedo: F. and Córdoba D. Córdoba, A. Muskat problems.To appearin A. Córdoba, D. Córdoba and F. Gancedo: Interface evolution: the Hele-Shaw and transform applied to a nonlocal transport equation. M. and Córdoba D. Córdoba, A. Fontelos: A. no. 1,1-39. equations. transport nonlocal and nonlinear some for P. Balodis and A. Córdoba: An inequality for Riesz transforms implying blow-up (2007) In Analysis. A. Córdoba: Encounters at the Interface between Number Theory and Harmonic U.S.A in appear interfaces. To fluid irrotational of evolution the value atrisk. R. Brummelhuis, A. Córdoba, M. Quintanilla and L. A. Seco: of capillaryjets. A. Córdoba, D. Córdoba, C. L. Fefferman and M. A. Fontelos: J. Math.PuresAppl.(9) F. Chamizo and A. Córdoba: A path integral approach to lattice point problems. (2003), no.26,15316-15317. with applications to partial differential equations. derivatives fractionary for estimate pointwise A Córdoba: D. and Córdoba A. Fontelos: M.A. and constraint forcapillaryjetbreakup. Fefferman C.L. Córdoba, D. Córdoba, A. equations. A. Córdoba and D. Córdoba: A maximum principle applied to quasi-geostrophic R. Soc.Mat.Esp. A. Córdoba and D. Córdoba: Charles Louis Fefferman: el poder del análisis. 203-223. equation. quasi-geostrophic the of model 1D a in Fontelos: A. M. and Córdoba D. Córdoba, A. Chae, D. 1377-1389. velocity. nonlocal with equation transport Fontelos: A. M. and Córdoba D. Córdoba, A. termediate layers. Córdoba: A. and Caffarelli A. L. (2006), no.6,529-540. . , 1-19.BibliotecadelaRevistaMatemáticaIberoamericana,2008. Comm. Math.Phys. rceig o te Sgna Jraa d Toí d Números” de Teoría de Jornadas “Segundas the of Proceedings Math. Finance

Proc. Natl.Acad.Sci.USA 7 (2004),no.3,757-765. J. ReineAngew.Math.

81 (2002),no.10,957-966.

12 (2002),no.1,23-43.

249 Ann. ofMath. Phase transitions: uniform regularity of the in the of regularity uniform transitions: Phase ������������������������������������������������ (2004),no.3,511-528. Adv. Math. List ofpublications

593

99 Ann. of Math. (2) Math. of Ann. Integral inequalities for the Hilbert the for inequalities Integral (2006),209-235. (2002), no.17,11006-11007. Formation of singularities for a for singularities of Formation

187 Adv. Math. Adv. Proc. Natl. Acad. Sci. USA (2004),no.1,228-239. J. Math. Pures Appl. (9) Finite time singularities time Finite Adv. Math. Adv. Principal component Proc. Nat. Acad. Sci. Acad. Nat. Proc. Drops: the collapse

162 194 geometrical A (2005), no. 3, no. (2005), (2005), no. 1, no. (2005),

214 (2007), Gac.

100

86 -

The Poetry of Analysis 25 26 The Poetry of Analysis • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 5776-5778. nonrelativistic theory of atoms. of theory nonrelativistic A. Córdoba, C. L. Fefferman and L. A. Seco no. 3,741-750. J. Cilleruelo and A. Córdoba: Lattice points on ellipses. minimal surfaces’’. L. A. Caffarelli and A. Córdoba problem. L. A. Caffarelli and A. Córdoba Seco oscillations. A. L. and Fefferman L. C. Córdoba, A. Riemann’s graphs”. Córdoba A. and Chamizo F. 7-8, 1087-1102. the Thomas-Fermi density. Seco A. L. and Fefferman L. C. Córdoba, A. J. NumberTheory Córdoba F.A. and Chamizo Special Issue,859-870. 1996)] Escorial, (El Guzmán de Miguel Professor Córdoba A. rearranged Fourierseries. Fernández P. and Córdoba A. Soc. Mat.Esp. figuras. y números Riemann: de Fractales Córdoba: A. and F.Chamizo fractal Fourier series. some of dimension and Differentiability Córdoba: A. and Chamizo F. A. Córdoba: Por el giro de una aguja. series. P. Balodis and A. Córdoba: The convergence of multidimensional Fourier-Bessel Nivola, 2000. siglo XX: una mirada en 101 artículos A. Córdoba: Hitos del análisis matemático del siglo XX. In Soc. Mat.Esp. cálculo. del fundamental teorema del geométrica razón La Córdoba: A. 260. numerorum. Disquisitio Córdoba: A. 59-76. Univ.LaRioja,Logroño,2001. In retículo. del Puntos Córdoba: A. and Chamizo F. J. Anal.Math. Comm. PureAppl.Math. : Rev. Mat.Iberoamericana atc pit. Poedns f h cneec ddctd to dedicated conference the of [Proceedings points. Lattice

Adv. Math 1 3 (1998),no.1,37-47. (2000),no.3,435-446.

65 Differential IntegralEquations C. R. Acad. Sci. Paris Sér. I Math. (1997),no.1,101-104.

77 (1999),269-286. . 142 SIAM J.Math.Anal. : Comm. Partial Differential Equations One-dimensional crystals and quadratic residues. quadratic and crystals One-dimensional : rau: Te rca dmnin f fml of family a of dimension fractal “The Erratum: (1999),no.2,335-354. : :

Uniform convergence of a singular perturbation : Correction: “An elementary regularity theory of ovrec ad iegne f decreasing of divergence and Convergence Proc. Nat. Acad. Sci. U.S.A. Sci. Acad. Nat. Proc.

48 , 281-286. Sociedad Canaria Isaac Newton/ Gac. R. Soc. Mat. Esp. Gac. R. Soc. Mat. Esp. Mat. Soc. R. Gac. (1995),no.1,1-12.

11 (1995),no.1,165-226. : A trigonometric sum relevant to the : A number-theoretic estimate for estimate number-theoretic A :

el us n aoi energy atomic and sums Weyl 29 . J. Fourier Anal. Appl. Anal. Fourier J. (1998),no.5,1129-1139.

8

320 (1995),no.1,223. Margarita Mathematica Margarita Duke Math. J. (1995), no. 5, 649-650.

2 Las matemáticas del

4 (1999), no. 1, 58-65. (2001), no. 1, 249- 1, no. (2001),

91 (1994), no. 13, no. (1994),

21 (1996), no.

76

3 (1997), (1994), Gac. R. Gac. Gac. R. Gac. , • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A. Córdoba A. graphs. Córdoba: A. and Chamizo F. 335-338. Córdoba A. J.A. Barceló and A. Córdoba: A. and Barceló J.A. A. Córdoba:Diraccombs. Córdoba, A Proc. Amer.Math.Soc. Córdoba A. and Cilleruelo J. Arith. $B_2[ Córdoba: A. and Cilleruelo J. surfaces. Córdoba A. and Caffarelli A. L. 81-82. no. 3,vii,215-226. A. Córdoba: Geometric Fourier analysis. 305. In multipliers. for Vol. Zygmund, Antoni of honor in analysis inequalities harmonic valued Vector Córdoba: A. Soc. multiplier. disc the for inequality integral An Córdoba: A. 1983) functions, maximal summation, square functions and all that. spherical In lemmas, Restriction Córdoba: A. Math. Soc.(N.S.) and Hilberttransformsalongflatconvexcurvesin$\ functions maximal for estimates $Lˆ^p$ Weinberg: D. and Wainger S. Vance, J. Francia, de Rubio L. J. Duoandikoetxea, J. Córdoba, A. Christ, M. Carlsson, H. along curves. A. Córdoba and J. L. Rubio de Francia: Hilbert transformsalongconvexcurves. Weinberg D. and Wainger Vance,S. J. Nagel, A. Córdoba, A. Amer. Math.Soc.(N.S.) J. A. Barceló and A. Córdoba: 306 A. Córdoba Amer. Math.Soc. (1988),no.8,373-376.

92

, Wadsworth Math.Ser. Wadsworth,Belmont, Ca,1983.

61 57-64. (1984),no.3,407-408.

(1992),no.3,265-270. C. R.Acad.Sci.ParisSér.IMath. Differential IntegralEquations : : : : La formule sommatoire de Poisson. Thediscmultiplier. A counterexample in operator theory. operator in counterexample A The fat needle problem. needle fat The North-HollandMath. Rev. Mat.Iberoamericana

14 313 (1986),no.2,263-267. (1989),no.2,655-669.

115 18 (1988),no.2,163-166. (1992),no.4,899-905. Lett. Math.Phys. Band-limited functions: $Lˆ^p$-convergence. : h fatl ieso o a aiy f Riemann’s of family a of dimension fractal The rgnmti plnmas n ltie points. lattice and polynomials Trigonometric Band-limited functions: L^ functions: Band-limited : Recent progress in Fourier analysis (El Escorial, Duke Math.J.

An elementary regularity theory of minimal of theory regularity elementary An \ Stud. infty]$-sequences of square numbers. square of infty]$-sequences Estimates for Wainger’s singular integrals Bull. London Math. Soc. Math. London Bull. Ann. Inst. Fourier (Grenoble)

Invent. Math.

111 2

6 317 (1986),no.1-2,105-117. (1993),no.1,1-13. .North-Holland, Amsterdam,1985.

17 (1993), no.5,455-460. (1989),no.3,191-196.

58 C. R. Acad. Sci. Paris Sér. (1989),no.1,21-29. Publ. Mat. Publ. \ mathbb{R}ˆ^2$. I, II (Chicago, 1981), (Chicago, II I,

83 p -convergence. (1986),no.1,59-71. Proc. Amer. Math. Amer. Proc. : $L^ˆp$ bounds for bounds $L^ˆp$

37 25 ofrne on Conference (1993), no. 2, no. (1993), (1993), no. 1, no. (1993), Bull. Amer.

32 (1982), I Math. Trans. Bull. 295- Acta

The Poetry of Analysis 27 28 The Poetry of Analysis • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • (1977), no.3,581-588. polygon. the for problem multiplier summation The Córdoba: A. spherical the and function multipliers. maximal Kakeya The Córdoba: A. Fís. Natur.Madrid compleja. variable y Fourier de Análisis Córdoba: A. Sci. U.S.A. integrals. of differentiation On Fefferman: R. and Córdoba A. A. Córdoba:$s Comm. PartialDifferentialEquations operators. integral Fourier and packets Wave Fefferman: L. C. and Córdoba A. In F.I.O. (1977/1978), and packets Wave Córdoba: A. Amer. Math.Soc.(N.S.) Zygmund. A. of conjecture a of proof a functions: Maximal Córdoba: A. 505-511. operators. Bochner-Riesz on note A Córdoba: A. 1979. spaces, Euclidean in analysis Harmonic In multipliers. Fourier and lemmas covering functions, Maximal Córdoba: A. 779. Univ. Sem., A. Córdoba. Multipliers of ${\ 161. LectureNotesinMath. In Zygmund. A. of Univ. Sem., problem (Proc. analysis harmonic a functions: Maximal Córdoba: A. Escorial, 1979), In operators. invariant Translation Córdoba: A. de Guíxois,1980), Feliu (Sant II Part mathematics, on conference Spanish-Portuguese seventh In armónico. análisis de problemas Algunos Córdoba: A. Stein. López-Melero: B. and Córdoba A. suppl. 1,75-80. Seminar on Harmonic Analysis (Pisa, 1980)] the of Littlewood-Paley[Proceedings the theory. on remarks Some Córdoba: A. clásico. Cienc. Exact. armónico análisis de teoremas Cinco Córdoba: A. Springer, Berlin,1980. Ann. Inst.Fourier(Grenoble)

74 1-9. Maryland, College Park, Md., 1979), Md., Park, College Maryland, Amer. J.Math. (1977),no.6,2211-2213. Fís. Natur.Madrid \ 117-176. Asoc.Mat.Española \times t\

Publ. 177-182

71 (1977),no.3-4,515-535. Math. Orsay

\ 1 times \ (1979),no.1,255-257. .

Publ. Sec.Mat.Univ.AutònomaBarcelona 779. 99 \ cal F}(L^ˆp)$. In (1977),no.1,1-22. \ Phi(s,t)$. Springer, Berlin,1980.

15 , 1982. 78

peia smain a rbe o E M. E. of problem a summation: Spherical 31 . Univ.París XI,Orsay, (1981),no.3,x,147-152.

Maryland, College Park, Md., 1979), Md., Park, College Maryland, 3 (1978),no.11,979-1005. Institut Mittag-Leffler 29-50. Proc. Sympos. Pure Math. Pure Sympos. Proc. 29-50. . Rend. Circ. Mat. Euclidean harmonic analysis (Proc. éiar dAaye Harmonique d’Analyse Séminaire 1 , 1980. Fourier analysis (Proc. Sem., El Sem., (Proc. analysis Fourier 162-177. Lecture Notes in Math. in Notes Lecture 162-177. Duke Math. J. Math. Duke Rev. Real Acad. Ci. Exact. Ci. Acad. Real Rev. n. f ah (2) Math. of Ann.

rceig o the of Proceedings 1978. Palermo (2) e. el Acad. Real Mem.

Proc. Nat. Acad. Nat. Proc. 46

9 , 1978. (1979), no. 3, no. (1979), Euclidean 21 , 1980. (1981), Bull.

154- 105 35 ,

• • • BOOKS • • • • • • • • • • ESSAYS AND REVIEWS • • • • • • • • • • • • • • • . ódb ad . . efra: wihe nr ieult fr singular for inequality norm weighted integrals. A Fefferman: L. C. and Córdoba A. Nat. Acad.Sci.U.S.A. analysis. Fourier in operators multiplier and maximal of classes certain A. CórdobaandR.Fefferman:Ontheequivalencebetweenboundednessof Apuntes deCienciayTecnología melancólica. experiencia Una Córdoba: A. realizado paraelColegioLibre deEméritos,2007. A. Córdoba: UPC. FME, Vol. V, Curs Riemann (2007-2008), Trigonométricas. Series las sobre Riemann de tesis La Córdoba: A. la UniversidadComplutensedeMadrid,1976. Córdoba: A. de Extremadura,1987. A. Córdoba: A. Córdoba,J.Cilleruelo: A. Córdoba: Amer. Math.Soc. function. maximal Kakeya related a and multiplier radial A Córdoba: A. Bull. Amer.Math.Soc. A. Córdoba and R. Fefferman: A geometric proof of the strong maximal theorem. Ann. ofMath.(2) A. Cordoba and R. Fefferman: A geometric proof of the strong maximal theorem. 26 A. Córdoba: Bounded variation and differentiability of functions. Math. basis. differentiation a of properties covering Vitali the On Córdoba: A. (1975),no.3,227-238.

57 (1976),no.1,91-95. Studia Math. Lecciones de teoría de los números Libros de texto de Matemáticas en el Bachillerato español. La sagadelosnúmeros Operadores pseudodiferenciales y aplicaciones y pseudodiferenciales Operadores

81 102 (1975),428—430.

(1975),no.1,95-100. 74

81

(1977),no.2,423-425. 57 Lateoríadelosnúmeros. (1975),no.5,941. (1976),no.1,97-101.

25 (2007)(Spanish). . Ed.Crítica,2006. Facultat de Matemàtiques i Estadística, Notícies SCM Notícies . Publicaciones de la Universidad Ed.Mondadori,1992.

23 . Publicaciones de Publicaciones . (2007) (Catalan). (2007) Collect. Math. Conferéncies Informe Studia Proc. Bull.

The Poetry of Analysis 29 30 The Poetry of Analysis • • • • • • • • • • • • • • • • • • • • • • • • A. Córdoba: Las matemáticas en el siglo de las luces. A. Córdoba:Unamentebella. A. Córdoba:Labibliotecadelrector. A. Córdoba:FelipeII,eldiabloylasmatemáticas. A. Córdoba:Poesíaentreteoremas. Ozámiz, matemático y humanista”, Gac. R. Soc. Mat. Esp. Transición.la A matemáticoen UnCórdoba: A. matemáticas. las y conocimiento computador El Córdoba: A. 1986. Córdoba: A. Rev. Mat.Iberoamericana A. Córdoba: José Luis Rubio de Francia (1949-88): semblanza de su vida y obra. el tercermilenio A. Córdoba: Una mirada al análisis matemático. En matemáticas ensusaplicaciones, A. Córdoba: Historias analíticas de tangencia y cuadratura. En A. Córdoba:Portodaslasrazonesprácticas. enó d mtmtcs españoles. matemáticos de Reunión

214

1, (2007),120-130. 93-110. SociedadEstatalEspañaNuevoMilenio,2002.

4 (1988),no.1,1-10. Saber leer 127-158. MECD,2002. Mètode, yearbook El Mundo

155 (2002),8-9. Saber leer (2002). Suplemento “Miguel de Guzmán de “MiguelSuplemento Saber leer ouets e a CAICYT, la de Documentos La ciencia y tecnología ante Saber leer (2004),83-90.

137 trps Hels del Huellas nthropos:

7 (2004), no. 3. 29-47. (2000),10-11.

161 El lenguaje de las

(2003),10-11. 148 (2001), 8-9. Family Tree

The Poetry of Analysis 31 32 The Poetry of Analysis The Congress http://www.uam.es/gruposinv/ntatuam/cordoba/program.html Fotografía de Marcel.lí Bayer June 25 Pilar Bayer’stalkwillbeheld onThursday, th , at16:45pm. jacobian varieties, K. Ribet was able to prove a very particular case of a deep a of case particular very a prove to able was Ribet K. varieties, jacobian their of models Néron the of properties fine using By 1990s. the in provided was for spaces moduli coarse are algebras quaternion non-split to associated curves Shimura Fake ellipticcurvesandtheirmoduli sität Göttingen,Germany. In 2004 she was named Medal Greece, Poland, Russia and Tunisia. In 1998 she was awarded the France, Spain, Austria, Germany, in centres research and universities at seminars and lectures given has She projects. research numerous and theses PhD 10 vised Galois theory, elliptic curves, modular curves and Shimura curves. She has super forms, automorphic functions, zeta on focus publications Her theory. number is field research Her Barcelona. of University the at Algebra of professor full is she 1982, Since (1980-1981). Santander of University and 1977-1980) (Germany, sität Univer Regensburg 1981-1982), (1969-1977; Barcelona of University Autonomous the (1968-1975), Barcelona of University the at lecturer a been has She Barcelona. in 1967, she qualified as a piano teacher at the Municipal Conservatory of Music of obtained her doctorate in mathematics at the same university in 1975. Previously, and 1968 in Barcelona of University the at mathematics in graduated Bayer Pilar CM-points. shall at values we their obtain to as well (2008), as functions these compute to Travesa method a present A. and (2005) Guàrdia J. (2004), Alsina A. with work joint of result a As functions. byautomorphic arithmetic of values fields special of means class of construction theoretical the in role key a play multiplication complex with curves elliptic Fake models. canonical the of theory his to Shimura G. led theory multiplication complex classical the of generalization convenient A proof ofFermat’sLastTheorem. Wiles’ of point starting the was theorem Ribet’s Serre. of conjecture modularity for scientific and technological achievement by the Catalan government. Catalan the by achievement technological and scientific for fake elliptic curves. One of the first applications of Shimura curves Shimura of applications first the of One curves. elliptic Emmy-Noether-Professorin by the Georg-August-Univer Pilar Bayer Narcís Monturiol - - -

The Poetry of Analysis 35 day, June25 Joaquim Bruna’stalkwillbe held onThurs th , at15:30pm. - corresponding to a discrete set of frequencies if The Fuglede conjecture stated that a domain analysis in the resulting Paley-Wiener spaces Paley-Wiener resulting the in analysis All these questions can be restated using Fourier-Laplace transforms and complex a Rieszbasis. In this scenario, the question of the existence of sions. under some further restrictions true on is and dimensions, low in open remains but Tao, by general in false be to Basis of exponentials and related problems he enjoysclimbingandlongroutecycling. Joaquim used to be a very good football player, at least as good as Antonio. Now nine doctoraltheses. supervised has He CRM. Matemàtica, Recerca de Centre the of Director currently UAB, the reform of the Grade in Mathematics and the the promoted he 2001, of Editor UAB. Matematiques the at professor full is he 1984, Madison-Wisconsin Since Paris-Sud,Albany. and at Postdocs Cerdá. Joan of supervision the der un UAB, Barcelona, de Autònoma Universitat the at 1978 in Mathematics in PhD his got and 1975, in Barcelona of University the at Mathematics in graduated He in one dimension a complete description is known of those of known is description complete a dimension one in that condition density critical a and free systems. Very influential work of Beurling and Landau served to establish frames notions: related important other two of point meeting the are basis Riesz intervals does(multibandproblem). of union finite arbitrary an if neither know not do we and exponentials, of basis ball unit the whether know not do we instance, For analysis. Fourier in problems unresolved yet and ones natural most the of one as appears in L L 2 (U) 2 (U) admitsanorthornormalbasisofexponentials(thatis,sines andcosines) , as good as orthonormal bases from the point of view of applications, of view of point the from bases orthonormal as good as , for three years. Chairman of the Math Department from 1998 to 1998 from Department Math the of Chairman years. three for Servei de Consultoria and the and Consultoria de Servei E E (Λ) =  = (Λ) Λ U must satisfy in general, but only for intervals for only but general, in satisfy must , for instance when e e i λ λ t , λ ∈Λ λ , U Λ tiles PW(U) Riesz basis . This conjecture has been shown Joaquim Bruna R n plan for Joint Degrees. He is of signals with frequency with signals of by translations if and only Servei d'Estadistica of the of d'Estadistica Servei U is convex in two dimen

E Λ ( Λ for which for B ) of exponentials admits a Riesz a admits Publicacions E ( Λ ) is ) - -

The Poetry of Analysis 37 38 The Poetry of Analysis (2004), 251-258. dimensions. higher and 5 in false is conjecture Fuglede's T.Tao: [7] Series 33.Amer. Math.Soc.,2004. [6] K. Seip: Dokl. [5] N.K. Nikolskii: Bases of exponentials and the values of reproducing kernels. entire functions. certain of interpolation and sampling for conditions density Necessary Landau: H.J. [4] in theplane. [3] A. Iosevich, N. Katz and T. Tao: The Fuglede spectral conjecture holds for convex bodies translates. $L in Completeness Ulanovskii: A. and Olevskii A. Bruna, J. [2] berger. Birkhauser, 1989. Beurling: A. [1] REFERENCES content in above tosphericalharmonicsandRiemanianmanifolds. the all of generalizations natural consider also will We problems. open several to attention the call and topics these on results known survey will we talk the In so called generatorfunctionsfor the of description the instance for unsolved, remain questions classical, natural very very Although function. fixed a of translates of the capability with spanning deals Wiener to least at back going research of area related closely A basis foritslinearspan). every frame in an abstract Hilbert space is a finite union of Riesz sequences (Riesz Related to the above, the following conjecture is lately receiving much attention: the deepunderstandingofwhyremainingunclear. know, not do we some and not, do some have, Some kernels. reproducing with spaces Hilbert abstract for generally more and functions, analytic of spaces eral gen for considered be can interpolating and sampling simultaneously are that to correspond systems free

21 (1980). Rev. Matem.Iberoamericana Interpolation and Sampling in spaces of analytic functions U Math. Res.Letters . Frames The collected works of A. Beurling A. of works collected The Acta Math. E ( Λ

) correspond to 117 interpolating sequences interpolating

(1967),37-52. 10 L (2003),559-570. p ( R ).

22 (2006),no.1,1-16. sampling sequences , by L. Carleson, P. Malliavin, J. Neu J. P.Malliavin, Carleson, L. by , . The existence of sequences of existence The . ^^^1( \ mathbb{R})$ of discrete of mathbb{R})$

Λ Math. Res. Letters Res. Math. . University Lecture for PW(U) Sov. Math. , while

11 - -

Wednesday, June 24 Xavier Cabré’stalkwillbeheld on th , at16:45pm. The fractional powers of the Laplacian are the infinitesimal generators of the sym population dynamics. in and flames, and fronts reaction certain in equation), quasi-geostrophic (the (American options modelled with jump processes), in geophysical fluid dynamics finance in crystals, in dislocations plasma, in diffusions in appear They Finance. powers (- Long-range or “anomalous” diffusions, such as diffusions given by the fractional for fractionaldiffusionequations transitions phase and propagation Front main fieldofresearchare His Partial DifferentialEquations. 2008). (since Catalunya de Politècnica Universitat the at Professor Full and 2003) (since Catalunya now de Politècnica Universitat the at Professor is Research ICREA He 2002-03). Austin, at Texas of (University Professor Associate Tenure and 2001-02) Austin, at Texas of (University Fellow Faculty Harrington 1994-95, Princeton, Study, Advanced for Institute the of Member 1995). University, York 1998. in VI Curie-Paris also got the University. York New Institute, Courant the at the Nirenberg, Louis of under supervision 1994, in PhD his got He 1966. in Barcelona, in born was Cabré Xavier should propagate at exponential speed -in contrast with the classical case with In [2]westudythepropagationoffrontsforfractionalKPP equation tional ellipticAllen-Cahnequation. frac the for transitions phase as well as equation, heat fractional nonlinear the In this talk, I will mainly describe recent results concerning front propagation for been mostlydevelopedinthelastyears. has diffusions fractional for analysis nonlinear view, of point mathematical the sion processes that combine Brownian motion together with a jump process. From diffu are flights- Lévy called -also These processes. diffusion Lévy stable metric which there is propagation at a constant KPP speed. In particular, no travelling no particular, In speed. KPP constant a at propagation is there which α

∈(0,1) ∆) Δ α Habilitation à diriger des recherches of the Laplacian, attract lately great interest in Physics, Biology, and . In [8,9], by heuristic considerations it was predicted that fronts that predicted was it considerations heuristic by [8,9], In . He has been awarded with the Kurt Friedrichs Prize (New Prize Friedrichs Kurt the with awarded been has He at the Université Pierre et Marie Xavier Cabré α =1 for He - - -

The Poetry of Analysis 39 40 The Poetry of Analysis symmetry resultofGidas-Ni-Nirenberg type. a and space, half a in theorems type Liouville powers, subcritical for type Spruck Gidas- of estimates priori a as well as solutions positive of results regularity and cylinder the in problem the of realization local domain bounded (- where Finally, wewilldescriberesultsfrom[6]ontheproblem De Giorgitypeforthenonlocalequation. the improve to allow estimates These solutions). shaped” [1] we find sharp energy estimates for these and other solutions (such as “saddle- in ness direction. We establish several properties of these solutions, such as their unique there exists a “layer” solution -meaning, essentially, a solution increasing in one on condition ary in posed equation elliptic erate ∈(0,1) with In [1,3,4]weareconcernedwiththeequation -instead oftheexponentialtailGaussiancorrespondingto infinity at tail decaying power a has equation heat fractional the of solution tal nonlinearity. Such exponential speed originates from the fact that the fundamen µ every level set of For instance, given an initial condition with compact support in when exist should wave *=f’(0)/(n+2 α

is a result of [7] which allows to realize this nonlocal equation as a degen ∈(0,1) R ∆ , minimality, symmetry in certain dimensions, and decay at infinity. In infinity. at decay and dimensions, certain in symmetry minimality, , Dir Δ ) 1/2 α . The case The . stands for the unique positive square root of the Laplacian the of root square positive unique the for stands ) and u R Ω is located at time n =

∂ \c f(u) { R R + n+1 n α α with zero Dirichlet boundary conditions. Using also a also Using conditions. boundary Dirichlet zero with is equal to equal is . In [3,4] we characterize the nonlinearities the characterize we [3,4] In .

=1/2 was studied in [5]. Crucial to our analysis for for analysis our to Crucial [5]. in studied was =1/2 < 1 < . In [2] we establish mathematically these results. these mathematically establish we [2] In . R + n+1 together with a nonlinear Neumann bound Neumann nonlinear a with together t , up to an error, near { I| u(1-u) or to another concave monostable concave another to or Ω × [0, × ∞ ), we establish existence establish we ), 1 D symmetry results of results symmetry D x I| =exp( R n α , we prove that = µ 1 . *t) f for which for }, where in a in α - - - -

anomalous diffusion. [9] Mancinelli, R., Vergni, D., and Vulpiani, A.: Front propagation in reactive systems with (2003), 018302. diffusion systems with Levy flights: a fractional diffusion approach. diffusion fractional a flights: Levy with systems diffusion reaction- in Lynch,dynamics V.and Front A., E.: B. Carreras, D., Del-Castillo-Negrete, [8] Comm. PartialDifferentialEquations [7] Caffarelli, L. and Silvestre, L.: An extension problem related to the fractional Laplacian. root oftheLaplacian,preprint. [6] Cabré,X.andTan, J.:Positivesolutionsofnonlinear problemsinvolving thesquare Comm. PureAppl.Math. reactions. boundary for half-space a in solutions Layer J.: Solà-Morales, and X. Cabré, [5] existence, II: Laplacians fractional for uniqueness, andqualitativepropertiesofsolutions,preprint. equations Nonlinear Y.: Sire, and X. Cabré, [4] mum principles,andHamiltonianestimates,preprint. [3] Cabré, X. and Sire, Y.: Nonlinear equations for fractional Laplacians I: regularity, maxi tional diffusion,preprint. [2] Cabré, X. and Roquejoffre, J.-M.: Front propagation in Fisher-KPP equations with frac nonlinear fractional of equations, preprint. solutions for estimates energy Sharp E.: Cinti, and X. Cabré, [1] REFERENCES Physica D.NonlinearPhenomena

58 (2005),1678-1732.

32 (2007),1245-1260.

185 (2003),175-195. Phys. Rev. Lett. Rev. Phys.

91 - -

The Poetry of Analysis 41

portation theoryandtheMonge-Ampereequation. trans optimal between connection the is work his of direction current Another media. random or periodic in equations nonlinear fully and type”, divergence “non- of equations media, periodic in boundaries free and surfaces minimal in collaborators his with work his continues He diffusions. local non for linear problems non of and homogenization of theory the in results recently, more and, theory transportation optimal to equations, solutions differential and partial problems elliptic nonlinear boundary free of regularity the are contributions significant most his of Some combustion. like applications nonlinear to equations fully elliptic to solutions of regularity the about questions theoretical from ear partial differential equations and their applications. His research has reached nonlin elliptic of area the in been has research Caffarelli’s Professor of focus The Prize forLifetimeAchievement(2009). P.Leroy Steele AMS the and 2005) Sciences, of Academy Swedish (Royal ematics Argentina. Plata, la de Universidad and Madrid, de Autónoma Universidad Paris; perieure, Su Normale l'Ecole from Causa Honoris Doctor awarded been has He Sciences. of for Advanced Study in Princeton. In 1991, he was elected to the National Academy Institute the of member permanent a was he 1996 to 1986 From University. York , and the Courant Institute of Mathematical Sciences at New the of Minnesota, of University University the at professor the a been has also at He Austin. Texasat Mathematics in Chair Richardson Sid the held has he 1996 Since Aires. Buenos of University the at (1972) PhD and (1968) Science of Master 8 December born was Caffarelli A. Luis mal surfaces". mini "integral corresponding the of properties geometric the and phenomena, proportionally to an "integral version" of mean curvature. We will describe these moves surface transition limiting corresponding the interactions, range long of phenomena, for instance as a limit of phase field models In the case of slow decay transition phase of modelling the in appears curvature, mean its to proportional curvature, mean by Movement non localoperators for surfaces minimal and transition Phase

He has received the Bôcher Prize (1984), the Rolf Schock Prize in Math i.e we a ufc eovs ih oml speed normal with evolves surface a when , th , 1948, in . He obtained his obtained He Aires. Buenos in 1948, , Luis Caffarelli - - - - -

The Poetry of Analysis 43

Friday, June26 Fernando Chamizo’stalkwill be heldon th , at12:30pm. Intertwining Analytic Number Theory and Harmonic Analysis expertise touch. collaboration with his big family combining Arithmetic and his harmonic analytic in Antonio by produced results the of some present to is talk this in purpose My influential forme. “Lessons in Number Theory” [10] in the University of Extremadura which were very course graduate-level a of notes the published had he previously and [8] bers” num of theory “The book the Cilleruelo) J. with (jointly co-authored has also He sons areinproject. E. Cristóbal (2009), and around half a dozen of new grandsons and great-grand and (2006) Ubis A. (1998), Trujillo C. (1995), Jiménez-Urroz J. grandsons, 4 ated F.and (1990) Cilleruelo J. subject, this gener in have theses who (1994), Chamizo PhD two supervised has He descendants: theorist number of number a quite in materializes devotion his and Arithmetic of fond analyst harmonic a is Antonio Antonio CórdobaandNumberTheory theory andnon-holomorphicmodularforms. since 1997. His research area is Number Theory. He has worked specially in lattice point Madrid de Autónoma Universidad at professor is currently and Córdoba Antonio of supervision the under 1994 in degree PhD his got Chamizo Fernando has been drastically empowered), awakening actively his interest in this topic [6]. relation between Fourier Analysis and Additive Theory is very old (and nowadays an equivalently the planar led him to study lattice points in short arcs [8], [9] or the boundary, etc. keep a perfect analogy (see [4] in problems the kernel, the of chopping the duality, The contribution). standing topic inAnalytic aclassical Number Theory, through Bochner-Riesz problems, means (a problem in point which he did an to lattice out arrived firstly Antonio lytic NumberTheoryfocusmainlyonspecificsimpleexamples. Ana while functions) wild (containing spaces function general very with deals Analysis Harmonic hand other the on that add should One method). circle sical clas the in boundary natural the to associated instance (for singularities with wild kernels employs latter the and kernels) Calderón-Zygmund instance (for lytic Number Theory is that the former considers kernels with simple singularities Ana and Analysis Harmonic classical between difference noticeable a that is ion L 4 , L 2 theorem for some short trigonometric polynomials [7]. The [7]. polynomials trigonometric short some for theorem Fernando Chamizo § 3.2). Restriction theorems for . Antonio’s opin ------

The Poetry of Analysis 45 46 The Poetry of Analysis [1] F. Chamizo and A. Córdoba. The fractal dimension of a family of Riemann’s graphs. Antonio’s workstouchingNumberTheory for formula closed the of proof easy and to papers expository in times several back come has He independently. proof Beukers’ got Antonio that able ple and short proof that largely killed the attention for the new ideas. It is notice stir caused by Apéry’s proof of the irrationality of thoughts rationals and Irrational a bigchallengeandhopeforAntonio. any new main term. Could one explain the periodic table with Arithmetic? This is avoids that energy the of asymptotics the in term “quasiperiodic” unexpected an is result The Theory. Number Analytic in methods common with [16] in treated is that sum exponential an to leads model mechanic quantum the from coming number atomic the when energy the for case) and some authors have formula faced the problem asymptotic an of giving hydrogenian the of exception classical the (with solution explicit any expect to involved too is It Hamiltonian. quantum certain a of spectrum the of bottom the Theory! Number Atomic duce Fourier series counterexamples for hypothetical rearrangementpro theoremsprimes)(squares, [17].arithmeticalsequences some and [3]behaviour fractal a Theory:Number to linked under some conditions polynomial frequencies produce strongly contributions his of couple a here mention We always Antonio). captivated has (that conjecture Rudin instance for flavour, theoretical number a questions have natural the of Some series. trigonometric on volumes Zygmund’s in chapter missing the is it that sometimes claimed has Antonio case. sublacunary speak, to so the, about much too know not do we but good, quite are gressions) pro geometric as growing frequencies with series Fourier mean (we series rier Strange Fourier series. Esp [2] F. Chamizo andA.Córdoba.FractalesdeRiemann:númerosyfiguras. Acad. Sci.ParisSér.IMath. . 1 (1998),no.1,37-47. From the harmonic analytical point of view lacunary Fou

The energy of the ground state of an atom is given by given is atom an of state ground the of energy The 317 (1993),no.5,455-460. ζ (2) and (2) . The senior researchers surely recall the great the recall surely researchers senior The . ζ (3) and not so far he found [12] a new new a [12] found he far so not and (3) ζ Z Z (2). goes to goes ζ (3). Later Beukers gave a sim

∞ . The variational problem variational The . Gac. R.Soc.Mat. C. R. - - - - - [12] A.Córdoba.Disquisitionumerorum. Miguel deGuzmán(ElEscorial,1996) In points. Lattice Córdoba. A. [11] Matemáticas. Badajoz,1987. Matemáticas, vol. 20. Universidad de Extremadura. Facultad de Ciencias. Departamento de Córdoba. A. [10] 741–750. ellipses. on points Lattice Córdoba. A. and Cilleruelo J. [9] [6] J.CillerueloandA.Córdoba.B Fourier series. rearranged decreasing of P.divergence and and Córdoba Convergence A. Fernández. [17] [15] A. Córdoba, C.L. Fefferman, and L. A. Seco. A number-theoretic estimate for the Tho the for estimate number-theoretic A Seco. A. L. and Fefferman, C.L. Córdoba, A. [15] [14] A.Córdoba. istic theoryofatoms. [13] A. Córdoba, C. Fefferman, and L. Seco. A trigonometric sum relevant to the nonrelativ [8] J.CillerueloandA.Córdoba. Math. Soc. [7] J. Cilleruelo and A. Córdoba. Trigonometric polynomials and lattice points. no. 3,265–270. Pures Appl. [5] F. Chamizo and A. Córdoba. A path integral approach to lattice point problems. Univ. LaRioja,Logroño,2001. In points. Lattice Córdoba. A. and F.Chamizo [4] Fourier fractal some series. of dimension and Differentiability Córdoba. A. and Chamizo F. [3] Rev. Mat.Iberoamericana oscillations. energy atomic and sums Weyl Seco. L.A. and Fefferman, C.L. Córdoba, A. [16] mas-Fermi density. Adv. Math

115 (9) SIAM J.Math.Anal. (1992), no.4,899–905. 81 La sagadelosnúmeros Lecciones de teoría de los números los de teoría de Lecciones . (2002),no.10,957–966. 142 Comm. PartialDifferentialEquations Proc. Nat.Acad.Sci.USA. (1999),no.2,335–354.

11 (1995), no.1,165-226. La Teoría delosNúmeros 2 , [ Proceedings of the conference dedicated to Professor to dedicated conference the of Proceedings 29 ∞ ]-sequences ofsquarenumbers. , volume3,pages859–870,1997. (1998),no.5,1129–1139. . Crítica,Drakontos. Barcelona,2006. Gac. R.Soc.Mat.Esp. , 91 Margarita mathematica Margarita (1994), no.13,5776–5778. . Publicaciones del Departamento de Departamento del Publicaciones . . Mondadori,Madrid,1992. , 21 Duke Math. J. Math. Duke (1996), no.7-8,1087–1102.

4 (2001),no.1,249–260. Acta Arith

76 , pages 59–76. pages , (1994), no. 3, no. (1994), Proc. Amer. . 61 J. Math. (1992) - -

The Poetry of Analysis 47

day, June25 Alice Chang’stalk willbeheldonThurs th , at11:00am. - tion betweencurvatureandtopology. rela direct a provides formula, Gauss-Bonnet the in integrand the Pfaffian, the ization of the Gauss curvature to even dimensional manifolds. Its close relation to general natural the is Q-curvature geometry. conformal in equation curvature Q- order top the with concerned results analytic some survey will I talk, this In Q-curvature inConformalGeometry formal invariants. con via four dimension of manifolds of class a classify to methods PDE applying project a in involved is She geometry. conformal and analysis geometric in are interests research her years, recent on In analysis. concentrated harmonic research classical her in career, problems her of stages early the and thesis her In Beijing, 2002. in Mathematics of Congress International the at speaker plenary a was and low, Fel Guggenheim and Sloan a is She 1999. in University Princeton at Mathematics of Professor a became She (1989-1991). Berkeley U.C. (1981-1998), UCLA including US, the in universities various at taught has she then, Since 1974. in Berkeley nia, Califor of University the at mathematics in degree PhD her receiving States, ed bachelor degree at the National her Taiwan received University in 1970. and She moved Taiwanto the Unit in up grew She China. Xi’an, in born was Chang Alice tions toproblems inconformalgeometry. In thistalk,Iwill giveabriefsurveyofthesubject withemphasizeonapplica these equations intherecentliterature. ress on questions of existence, regularity and classification of entire solutions for four may touch zero. In spite of these difficulties, there has been significant prog than greater dimensions of manifolds on equation Q-curvature order fourth the of solution the whether known not is it example, for principle: maximum of lack at the critical exponent, for which the Sobolev imbedding is not compact; (ii) the mention the following: (i) the lack of compactness: the nonlinearity always occur we these Among features. common of number a share equations These equation. equation analogous to the Yamabe equations which we shall call the Q-curvature elliptic semi-linear a to rise gives operator such Each operators. covariant mally confor of general in construction systematic a provides Graham, and Fefferman zian geometry in dimension four. The ambient metric construction, introduced by Lorent of Paneitzcontext covariant the conformally in order operator fourth the of conformally covariant operators. In 1983, Paneitz gave the first construction of context the in geometry conformal in naturally arises Q-curvature of notion The Alice Chang ------

The Poetry of Analysis 49

Wednesday, June 24 Diego Córdoba’stalkwillbe held on th , at15:30pm. fluids which provides weak solutions to the incompressible porous media and Euler equations. media porous incompressible the to solutions weak provides which fluids dimensional two by given equations dynamics contour of family a discuss will I as wellfortheirapplicationsinphysics. talk two physical models that are of interest from this mathematical point of view the singularities in the formation of patterns. For this purpose I will describe in my by played role the understand to crucial becomes it scenarios these In gularities. on solving problems that involve the possible formation and propagation of sin focused is mechanics fluid of analysis mathematical the in topic research main A and Eulerequations Interface dynamics:theMuskatproblem includes harmonic analysis,partialdifferentialequationsandfluidmechanics. interest His research Spain. in Científicas Investigaciones de Superior Consejo 2002 the of Matemáticas in Ciencias de Instituto Spain the at works he to then since and returned he University, Princeton at Professor Assistant and Chicago of University the at Instructor Dickson Princeton, at Studies Advanced of Institute the doctoral of Member a his being After received 1998. in University and Princeton from degree Fefferman Charles of student a was Córdoba Diego Antonio Córdoba (mydad)andFranciscoGancedo. with work joint a is This data. initial al the upon cases, both condition, in depending, though Rayleigh-Taylor the does as time, short a for prevails positivity such that carefully check to problem evolution the of part is It positive. strictly initially be quotient arc-chord the that imposing by property that quantify We self-intersect. not does interface initial the that assumed is it problems both in surface tension which leads to equality of the pressure on the free boundary, and without case the treat We evolution. the its modifying in without component direction tangential any subtract may we which from amplitude the of integral by an amplitude. The dynamics of that interface is governed by the Birkhoff-Rott multiplied interface the at distribution delta a then is vorticity the that follows It (Hele- Shaw and Muskat) or law Bernoulli’s law (irrotational incompressible Darcy’s Euler equation). by given field velocity free divergence a with scalar, active an as considered density, the for equations transport as models these regard We Diego Córdoba - -

The Poetry of Analysis 51 52 The Poetry of Analysis medium withdifferent densities. [4] D. Córdoba and F. Gancedo. Contour dynamics of incompressible 3-D fluids in a porous in 2D [3] A. Córdoba, D. Córdoba and F. Gancedo. problems. To appearin [2] A. Córdoba, D. Córdoba and F. Gancedo. Interface evolution: the Hele-Shaw and Muskat problem A F.Gancedo. and Córdoba D. Castro, A. [1] REFERENCES non-analytic initial data.ThisisajointworkwithÁngelCastroandFranciscoGancedo. for ill-posedness show we strength the of equation the For data. initial analytic for well-posedness prove we which for sheet vortex the of motion mean strength. In this context we choose a term in the tangential direction for the with with boundary densities ismodeledinsuchawaythatthevorticityconcentratedonfree where theevolutionofaninterfacebetweentwoimmisciblefluidssame of thetwodimensionalEulerequation The classicalVortex Sheetequationprovidesoutstanding kindofweaksolutions h . Preprint2008,arXiv:0810.5340. g ( ( x . PreprintarXiv:0810.0731 a ) a test function. We study solutions with finite energy which implies zero ; t z ) thevortex-sheetstrength,i.e. ( a ; t ), andisgivenbyaDiracdistributionasfollows: AnnalsofMath. Comm. Math.Phys.

2008. Preprint arXiv:0806.2258. Interface evolution: the water wave problem naive parametrization for the vortex-sheet the for parametrization naive w

is ameasuredefinedby 273, 2,445-471(2007). June 23 Guy David’stalkwillbeheld on Tuesday, rd , at12:30pm. is Reifenberg-flat as above, and for instance we control the sum of the squares the of sum the control we instance for and above, as Reifenberg-flat is of parameterizations bilipschitz give to used be also can algorithm this of control T.better with a Torowork where present to try shall We the endofargument. of description good very a get to scale by scale it refine and scale, unit the of morphisms theorem About Reifenberg’stopologicaldisk sets. tions, geometricmeasuretheory,andinparticularminimalalmost His researchinterestsincludesingularintegraloperators,thecalculusofvaria (since 1999). Sciences and Arts of Academy American the of member honorary foreign is and 1986) (Berkeley, Mathematicians of Congress International the at speaker main de Servant prix l’Académie Grand des Sciences (2004) and the the Médaille d’argent du CNRS (2001). He functional”), was Mumford-Shah the for minimizers of sets “Singular book the (for 2004 prize Balaguer i Sunyer Ferrán the Semmes), S. 1990), thePrixInstitutHenriPoincaréGauthier-Villars(Analysenonlinéaire,with (Mathematics, IBM-France Prix the (1987), Salem Prix the awarded been has He rently ProfessorattheUniversitédeParis 11(Orsay),France cur is He (1996-2001). France de Universitaire Institut the of member junior and 97-98) and (87-87 CA Berkeley, Institute, Research Sciences Mathematical the at sity (1983-85), visiting associate professor at UCLA (Spring 89), visiting professor Univer Yale at Instructor Gibbs 1982-89), France, Polytechnique, Ecole (at CNRS recherche, de Chargé then Attaché, been has He Meyer. Yves of supervision the Guy David was born June 1 top-down geometric algorithm where we start from the rough description of rough description top-down geometricalgorithmwherewestartfromthe The proof, which we shall try to allude to in the lecture, is a very nice example of embedded. centered on affine an to distance) Hausdorff normalized (in enough E set a if that says roughly theorem disk topological Reifenberg’s E , then it is locally equivalent to a R n . In particular, it has fairly good parameterisations and is nicely is and parameterisations good fairly has particular,it In . st , 1957, in St Omer, France. He got his PhD in 1986, under d -plane through bihölder homeo E , or parts of parts or , d Guy David pae n vr ball every in -plane ⊂

R E n , when , is close is E E - at at E - - -

The Poetry of Analysis 53 54 The Poetry of Analysis amples ofthis. ex many give to time much too have not probably shall we but sets, minimal Taylor’sJ. in used is similar something instance for and sets, and almost on work surfaces minimal control to used been has theorem Reifenberg’s Traditionally, and someproofssimpler. operators on integral singular of properties boundedness or parameterizations, good of tence mation by planes) are connected to more analytical properties (such as the exis and many others where geometric properties of Thus this work connects with results of C. Bishop, P. Jones, G. Lerman, S. Semmes, contained inarectifiablecurve(withfinitelength). in sets compact the characterize to Okikiolu and Jones by used also were They curves. Ahlfors-regular and Lipschitz on operator integral Cauchy the proximation of P.the of numbers Jones E ). Our additional uniform flatness assumption makes the situation, E by d -planes; they were initially introduced, when b q (x,r) . Recall that these numbers measure the good ap good the measure numbers these that Recall . E (such as good average approxi d =1, to control R n that are that - - - - Tuesday, June 23 Charles Fefferman’stalkwill be heldon rd , at9:30am. A tributetoAntonio'swork. Selected TheoremsofAntonioCórdoba emy ofSciences,andtheAmericanPhilosophicalSociety. Acad National the Sciences, and Arts of Academy American the of member a is as served the chairs has He Berkeley. in Institute currently Research Sciences Mathematical the of trustees of He board and department Madrid). mathematics de Princeton the Autonoma of Universidad chairman Hon the (Doctor from doctorates Causa honorary several oris and prize, Bôcher Prize, Bergman the Medal, Fields the Award, Waterman the Prize, Salem the include honors His computational geometry. and mechanics, fluid mechanics, quantum geometry, conformal variables, plex com several equations, differential partial analysis, Fourier classical in worked has Fefferman 1974. since Princeton at again and 1974, to 1970 from Chicago of pervision of E. M. Stein. He taught at Princeton from 1969 to 1970, at the University the University of Maryland in 1966 and his Ph.D. at Princeton in 1969 under the su at B.S. his received He 1949. in D.C., Washington, in born was Fefferman Charles Charles Fefferman - - - -

The Poetry of Analysis 55

Friday, June26 Andrew Granville’stalkwillbe heldon th , at11:00am. values of multiplicative functions in terms of analysis as well as arithmetic. We arithmetic. as well as analysis of terms in functions multiplicative of values mean the about questions explore we which in Balog, and Breteche la De as well as Soundararajan, author,with the by research recent present will we talk this In method andarithmetic Analysis innumbertheory:Thecircle their doctoralstudies. during months few a for America North to came who students Spanish three with worked has and Ubis, Adrián and Jiménez-Urroz Jorge Cilleruelo, Javier with pers school in wonderful Santander organized by Antonio Córdoba. He has written pa 1991 the to back going theory number Spanish with links well-established has He these areas. between links and science computer theoretical combinatorics, enumerative ry, theo graph combinatorics, additive geometry, arithmetic theory, number braic alge in articles with also though theory, number analytic in primarily works He Research ChairattheUniversitédeMontréal. Canadian a is Granville Now Georgia. of University the at professor a becoming before Princeton, in Bombieri Enrico and Toronto in Friedlander John with tions posi postdoctoral held he Then PauloRibenboim. of direction the under 1987 in PhD his completed he Canada, in University Queens' and Cambridge at Educated in termsfamiliarfromthecirclemethod. interpreted be can values mean these about results classical certain how see will Andrew Granville - - - -

The Poetry of Analysis 57

Wednesday, June 24 Miguel ÁngelHerrero’stalkwill beheldon th , at11:00am. Mathematical problemsinblood disciplinar (IMI)atUCM. Inter Matemática de Instituto of Director currently is he (2004-8), UCM Madrid, de Complutense Universidad at Aplicada Matemática de Departamento of Head been having After 2003). (since Technology and Engineering Science, in lation Simu and Modeling on Series Birkhäuser and 1997) (since Mathematics Applied of Journal European of Boards the to belongs currently he and 2003, to 1997 from ogy (ESMTB). He has been also an Editor for SIAM Journal of Mathematical Analysis serves at the Board of the European Society for Mathematical and Theoretical Biol the Scientific Committee of the Real Sociedad Matemática Española, and currently cia Nacional de Evaluación (ANEP) and ANECA in Spain. He has been a member of Agen and (USA) Foundation Science National Union), (European Council search Re European and Foundation Science European the including agencies, several with collaborated has He Biology. mathematical in models and systems fusion reaction–dif equations, differential of theory the include which interests, tific scien his of subjects the on papers research 80 over published has Ángel Miguel at UniversidadComplutenseinMadridsince1988. Mathematics Applied of Professor is He abroad. and Spain in institutions, several in Lecturer Invited been has he that after and Oxford, Parisand in VI visitor toral postdoc been has He 1951. on (Spain) Madrid in born was Herrero Ángel Miguel stages of thrombi formation. Such phenomena is characterized by the onset of onset the by characterized is phenomena Such formation. thrombi of stages early the into insight gain to us allows which model mathematical simplified a cade, BC), cas of which many details biochemical are known by now. so-called In this lecture we (the shall derive reactions biochemical tuned, finely and dent, interdepen of array complex a of use makes coagulation blood of process The of number a to thrombotic pathologies. consequently and formation, thrombi abnormal to lead may system coagulation blood the of activation the in increase inordinate an hand, other the On disorder. hereditary serious a haemophilia, of cause is coagulate to blood of ability impaired an instance, For consequences. significant have may system such in disruption Any occur. to injuries minor from bleeding prevents which organisms, human of mechanism security robust a is coagulation Blood coagulation Miguel ÁngelHerrero ------

The Poetry of Analysis 59 60 The Poetry of Analysis E. Zlobina,fromtheNationalCenterforHematologyatMoscow,Russia. K. and Guria Th. G. with collaboration in made been has reported be to work The area willbepresented. the on directions future possible and problems of number a Finally, proposed. be will source activation the of that and a MC the of location sources, the between relation pathological external, by induced is coagulation where case the In lysis. and activation describing parameters (thrombotic) biochemical of terms strong in coagulation of triggering the as issues on discuss to model our of use make shall We devices. ultrasound of means by detected be can that (MC) cloud microthrombi a of appearance the to leading process, polymerization strong a day, June26 Henryk Iwaniec’stalkwillbe heldonFri th , at9:30am. - Applications ofHarmonicAnalysis toSieve Honorary CitizenofElblag,whichistheclosesttohisheartlifetimerecognition. Bordeaux University. In 2006 the Town Council of his native city made Iwaniec an ous countries. In 2005 he was honored by receiving the doctorate honoris causa of vari from researchers with collaborates and students graduate teaches Iwaniec in NumberTheory(USA). Prize Cole (Switzerland), Prize Ostrowski (Warsaw), Medal Sierpinski York), (New Jurzykowskiare: received Iwaniec which prizes several Among AwardFoundation the “PolskaAkademiaUmiejetnosci”. and Sciences of Academy National the Sciences, and Arts of Academy American es of Mathematicians. Iwaniec is a member of the Polish Academy of Sciences, the by numerous invitations to give talks at conferences and International Congress acknowledged were accomplishments Iwaniec’s passion. his are numbers Prime The main interest of Iwaniec is in analytic number theory and automorphic forms. which positionheenjoystothisday. Rutgers, at Professor State Jersey New of chair a offered was he 1987 January in the USA (including a long term appointment at the Institute for Advanced Study) ences before leaving for the USA in 1983. After taking several visiting positions in Sci of Academy Polish the of Mathematics of Institute the at professor became Henryk Iwaniec graduated in 1971 from Warsaw University, got PhD next year and exponential integral the is this of example easy One rescue. the to rides analysis, harmonic especially analysis, where is This success. of chances our better the distinction this express one from zero. Moreover, the greater the number of different ways we can find to In counting things, perhaps the most basic ingredient is the ability to distinguish (without evenconstructingany ofthem). objects beautiful some of existence the announce may we positive, if is count but, the pleasure us bring this does only Not on. so and equations, to solutions Introduction: 1) (in collaborationwithJ.B.Friedlander) Theory n ubr hoy e ie o on tig, rm numbers, prime things, count to like we theory number In Henryk Iwaniec - - -

The Poetry of Analysis 61 62 The Poetry of Analysis free fromsmallprimes,hence almost-primes. Weget of allsmallprimessothat sons. Wecall for anynumberofsequences This detectorisparticularlyconvenientforstudyingthesum and Recall thattheMöbiusfunctionisdefinedas In sievetheorywerelyonadifferentdetector, theMöbiusformula for these, saythoseupto of coefficients few first the up scoop to way good a and series power than rather For multiplicative problems the generating functions are given by Dirichlet series with of representations This allows the study of additive problems like that of Goldbach where the number σ > µ ( d 0. ) =0 otherwise. S ( A, P µ ) thesiftingfunctionbecausein ourminds ( d p ) x 1 +p = , iswiththeCauchyintegral (−1) S 2 ( =N A, P r A = if isgivenby ) registersthesupportof d =p ( a n ) whichareinterestingforarithmetic rea 1 …p r , p j distinctprimes, A onthoseintegers P istheproduct - n

integers intheevenshorterinterval where Consider can even succeed when with succeed Summation Poisson 2) analysis. Hereweshallexhibitanumberofsuchinstances. harmonic from tools various by treatment to amenable congruare latter The sums. ence same the precisely to reduced still is problem the nevertheless tages, advan technical some offer but similar are which functions other by replaced counting of to multiples almost-primes counting of study the reduced have we essence In where rier expansion: to introducesmoothingdevices andapplyPoissonsummationratherthanFou In moredelicate(forexample higher-dimensional) casesitismoreconvenient allows ustodososomeextentonaverageover the Fourierexpansion oms howeverwantustothinkthatitis Clearly, if A y d is small compared to compared small is ( A x d tobethecharacteristicfunctionofintegersinshortinterval ) isthecongruencesum > d y d on average over average on as large as possible. Sometimes, thanks to harmonic analysis, we analysis, harmonic to thanks Sometimes, possible. as large as thentheanswerisusuallyzeroorsometimesone.Thesieve axi : In evaluating the congruence sums the main issue is to is issue main the sums congruence the evaluating In : d is so large that the number of elements is less than one! x d , say , . In modern sieve theory the Möbius function is function Möbius the theory sieve modern In . y = x = y y / d . Harmonicanalysis,inthepersonof 1/3 . Now the congruence sum counts sum congruence the Now . d . - - - -

The Poetry of Analysis 63 64 The Poetry of Analysis devices. (think finite field) and the general case can be reduced to these by combinatorial primes are moduli interesting most arithmetically The get. to able be shall we all need results which hold on average only over d we and that problems is a sieve good thing our because that for is that Recall challenging. still yet classical, more objects to leads polynomials quadratic for just sums Weyl the of Understanding as treated be can they that means separate. them of few a only are there that fact the of roots the Here brings bounds. Deligne of powerful theory profound the and fields finite over varieties on sums of exponential products become sums the Weyl situations In sophisticated more F is theWeylsum.Hereoneneedsnotonlyagoodboundfor Fouriertransform where son summationineach,getting we firstsplitintoclasses When evaluatingthecongruencesum for seen be quadratic polynomials. already can This lattice. a over summing simply sequence than subtle the more of characteristics intrinsic with combined is summation Poisson Polynomials Quadratic and Sums Weyl 3) non-zero frequenciesaresmall,atleastonaverageover the from integrals Fourier the that prove to us allows gap spectral This success. the sieve presumption. The fact that the next integer is far from zero is crucial for frequency the from comes term main expected The ˆ butmoreimportantlyforWeylsums,againonaverageover n

n (mod d ) with L -functions constitute the spectrum and and spectrum the constitute -functions : Fascinating features appear when appear features Fascinating : n 2 + 1 0(mod h

=

d 0 . which coincides with coincides which d ) andapplyPois d . - a spectral interpretation, not with respect to the Euclidean Laplacian, but rather but Laplacian, Euclidean the to respect with not interpretation, spectral a over summation the Here, harmonics Weyl the of changes sign the to Due Kloosterman sums. It is almost as if we could use statements about primes to say to primes about statements use could we if as almost is It sums. Kloosterman of sums in cancellation deduce to backwards go resolution, spectral the through the individual Kloosterman sums to deduce bounds for the eigenvalues and then, to appeal we that fascinating is It Weil. André by sums Kloosterman to applied and proved field, finite a over curves for case this in Hypothesis, Riemann the on depends also principle gap successful this that out pointing resist cannot One congruence groupsthearithmeticpreventsthisandSelberg showed that of By analogy it is no surprise that the smallest eigenvalue Riemann surface by Selberg. Because of the congruence architected forms automorphic of theory spectral the to due one hyperbolic the this timeoneofadifferenttype: sum, congruence a evaluating of one becomes issue the problems, sieve initial the sieve offer chances for success in handling some sums over primes. As with the of procedure) (inclusion-exclusion ideas combinatorial elementary the absence, interpretation of the zeros of primes along the lines of the Riemann-Hilbert-Pólya ideas by means of a spectral Groups Congruence over Sums Congruence 4) (mod roots the that criterion, equidistribution Weyl the to due means, this that Note Theorem 1. succeeded inprovingthefollowing: over summing when cancellation substantial a S h ( x p ; ) areequidistributed. q ). In general In ). (Duke-Friedlander-Iwaniec) G 0 ( q ) \ l 1 could be close to zero which would be useless, but for but useless, be would which zero to close be could

H c of the highly arithmetic objects (Weyl sums) admits sums) (Weyl objects arithmetic highly the of . L -functions remains alas an incomplete dream. In its c

0 (mod e p ( . Using modern technology we technology modern Using . h : To perform summations over summations perform To : q n ) the action takes place on the / p ) we expect, for each for expect, we ) l 1 ( q ) determines the size h ≠ 0, n

The Poetry of Analysis 65 66 The Poetry of Analysis cated tostatebutfarmoreuseful,with Here with Theorem 3. were abletosurpassthesquarerootbarrier. we eigenvalues of analysis harmonic non-abelian the with characters of analysis tions as a substitute for the Riemann Hypothesis. Enhancing the abelian harmonic to up almost Q take can one Since with Theorem 2. primes: for bound Bombieri-Vinogradov the gets one inequality) sieve Dirichlet (large characters classical of analysis harmonic the Using us. tells Hypothesis Riemann the what beyond go that primes about things say to information automorphic Hypothesis: Riemann the Beyond 5) explicit formulatosaymoreabouttheprimes. Riemann the by return then and zeta-function the of zeros the about something Q Q = is only slightly larger than larger slightly only is x 1/2+ Let (Bombieri-Friedlander-Iwaniec) d ( x ) forany a ≠ 0 beaninteger. Then d ( x ) ≥ 0theimpliedconstantdependingon M x but we have similar results, more compli more results, similar have we but x M x this result has served in many applica many in served has result this x Actually, there is a way we can use this use can we way a is there Actually, Q aslarge Let a ≠ ≠ 0beaninteger. Then x 4/7 . a . - - Tuesday, June 23 Rafael delaLlave’stalkwillbe heldon rd , at11:00am. away How togetfarwithlittleeffort:skipping tial differentialequationsandotherareasofanalysis. par in occasionally works also he but methods) numerical methods, variational His main area of research is dynamical systems (hyperbolic systems, KAM theory, active inseveralcommitteesrelatedtopublication. matical Physics preprint archive -the first electronic preprint server and has been mathe the and Journal Electronic Physics Mathematical of founders the among Catalunya for over 20 years. He is de in the editorial Politecnica board of Universitat several journals. He visiting was been has He people. 50 over with papers ten writ has He semesters. special and conferences several organize to contributed has He postdocs. of number a and Physics) in (4 thesis PhD 10 supervised has He has beenatUniversityofTexas. he 1989, Since IHES. Minnesota, Univ. IMA, in postdoc a as affiliated been has He Princeton. in Mathematics in professor assistant an as years 5 spent also He 1983. in Mathematics in PhD the received He Princeton. in student graduate a became he grateful- quite is he whom to others and Guzmán de M. Casal, A. Córdoba, A Universidad Complutensein1979andimmediately-thankstothegoodeffortsof Rafael de la Llave was born in Madrid. He became licenciado en Ciencias Físicas in Nekhorosev theorem statesthatallpointsare stable forlongtimes. The time. all for stable are points most that asserts theorem KAM The structions. some appropriate non-degeneracy and regularity conditions), there are two ob mial growth and if not, the solutions remain bounded. In nonlinear systems (with polyno is there then system, the of frequencies natural the of one as frequency For linear systems, the answer is easy, We all know that if the forces have the same age out. whether there are solutions for which the forces accumulate or whether they aver of question the ask can One out. average which forces periodic small to subject systems Hamiltonian of study the is problem the of formulation mathematical A careful planning. a requires it and easy not is this knows, tried has who anybody As effort. small 1 . Introduction. . One of the deepest aspirations of human kind is to get far using far get to is kind human of aspirations deepest the of One Rafael delaLlave ------

The Poetry of Analysis 67 68 The Poetry of Analysis that itwasbetter toleteachpapermakeitsownprecise definition. consensus strong was There diffusion. Arnol’d of definition mathematical canonical a be should there whether of question the posed moderators The others. many and Sinai tadt, 1 touch andskipawayso,thereisnoneedtouseKAMmethods. just manifold, the in time long a around hang to need not do we estimates, ous give estimates on the time for the phenomenon to happen. In contrast and with previ 13] 10, [9, from assumptions some eliminate methods new The before. than By combining these tools we can exhibit large scale effects in much simpler ways scattering map. Wewillalsodiscussthemethodofcorrectlyalignedwindows. so-called the is excursions homoclinic these study to tool main The when themotionisunfavorable. it to close staying favorable, is forcing the when manifold invariant hyperbolic normally the from away jumping as described be can escape of mechanisms The vantageous toconsidernormallyhyperbolicmanifoldswitharichdynamics. ad more is it tori, whiskered just considering than rather that, is idea main The author. the and Seara Gidea, Delshams, of work the following C) approach to pertaining developments. recent Some 3. enon (oftentheseconditionsaresatisfiedforgenericsystems). phenom the experiences system a that show that computations explicit Give C) tions inonepartofphasespace(e.g.positivedefinitesystems)[6,7,4]. condi topology?) (which genericity some under phenomenon the Establish B) A) Constructexamplesofthephenomenon.[3,17,18,11,8,12,14,15]. Among therigorousmathematicalworkwecandistinguishseveralmodalities: tensive numericalwork,someofwhichhasyieldedinvaluableinsights[5,20]. conjectures these of versions precise many are There systems. many in abundant very be should instability. of problem The 2. mechanics papers[1]. celestial the in discovered were phenomena related Some time. all and points all for stability get cannot you that 16] 18, 17, [3, in out pointed was it Nevertheless,

In [19] there was a round table that included Arnol’d, Gallavotti, Herman, Moser, Neish Moser, Herman, Gallavotti, Arnol’d, included that table round a was there [19] In 1 . Of course, besides the mathematical work, there has been ex been has there work, mathematical the besides course, Of . In this lecture, we will cover some developments some cover will we lecture, this In In [2], one can find conjectures that instability that conjectures find can one [2], In ------Berlin, 1992. In potentials. superquadratic in motions unbounded of counterexample Littlewood’s On Levi: M. [16] 702-720. diffusion. Arnold of Geometry Levi. M. and Kaloshin V. [15] ans. [14] V. Kaloshin and M. Levi: An example of Arnold diffusion for nearintegrable Hamiltoni Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verifica rigorous and heuristics problem: gap large the overcoming systems Hamiltonian in diffusion for mechanism geometric A Seara: T.M. and Llave la de R. Delshams, A. [10] tron. Res.Announc.Amer.Math.Soc. results. of announcement problem: gap in large the diffusion overcoming systems for Hamiltonian mechanism geometric A Seara: T.M. and Llave la de R. Delshams, A. [9] topologiquement non elliptique stable endimension4. fixe point de Exemple Calvez: Le P. and Douady R. [8] systems. Hamiltonian unstable priori a in orbits diffusion of Existence Yan: J. and Cheng C.-Q. [7] case tonian systems. Hamil of problem instability Topologicalthe Llave: in la methods de R. and Gidea M. [13] folds ofsmallcodimension. submani densely covering orbits with systems P.Hamiltonian and Martín: Fontich E. [12] Sup. (4) elliptiques. fixes points des instabilité ou Stabilité Douady: R. [11] tion onamodel. [6] C.-Q. Cheng and J. Yan: J. and Cheng C.-Q. [6] Rep systems. oscillator many-dimensional of instability universal A Chirikov: B.V. [5] Math. Soc. systems. Hamiltonian convex in pseudographs of dynamics The Bernard: P. [4] freedom. of Math. Doklady degrees several with systems dynamical of Instability Arnold: V.I. [3] celestial mechanics. and classical in motion of stability of problems and denominators Small V.I.Arnol’d: [2] Gauthier-Villars, Paris, 1971. 1970) (Nice, Mathématiciens des International Congrès du Actes In corps. trois des problème le dans mouvement du finale l’allure Sur V.M.Alexeyev: [1] REFERENCES . . Preprint04-265,[email protected],2004. Bull. Amer.Math.Soc.(N.S.) 52 (1979),no.5,264-379.

21 J. DifferentialGeom. (1968),no.1,1-46.

21 (2008),no.3,615-669. yais eotd epstos n yaia systems dynamical in expositions reported: Dynamics

5 Discrete Contin.Dyn.Syst. (1964),581-585. Mem. Amer.Math.Soc. Russian Math.Surveys C.R. Acad.Sci.ParisSér.IMath. Nonlinear Anal. Arnold diffusion in Hamiltonian systems: 1 a priori unstable priori a 1 systems: Hamiltonian in diffusion Arnold , 67

(2004),no.3,457-517. 45 (2008),no.3,409-427.

9 (2003), 125-134.

179

14 (844):viii+141,2006. 52(2003),no.1,315-327. 18 (2006),no.2,295-328. (1963), no.6,85-191. 296 (1983),no.21,895-898. SIAM Rev. SIAM , Tome, 893-907. pages 2, Ann. Sci. École Norm. École Sci. Ann. , 113-124. Springer, 113-124. ,

50 (2008), no. 4, no. (2008), J. Amer. J. Phys. Elec Sov. - - - - -

The Poetry of Analysis 69 70 The Poetry of Analysis [17] J.E. Littlewood: Unbounded solutions of an equation $\ equation an of solutions Unbounded Littlewood: J.E. [17] Physics, NewYork,1980. 1979) York, New Lab., Nat. Brookhaven (Sympos. interaction beam In freedom. of degrees three with systems iltonian [20] J.L. Tennyson, M.A. Lieberman, and A.J. Lichtenberg: Diffusion in nearintegrable Ham 533 1995)] Spain, (S’Agaró, Institute Study Advance NATO the of ings [19] C. Simó (Ed.): (1966), 491-496. $ of solutions Unbounded Littlewood: J.E. [18] (1966), 497-507. $g(y)/y and bounded, and periodic . Springer, 1999. Hamiltonian Systems with Three or More Degrees of Freedom. ^[Proceed

\ to \ \infty$ as $y as \infty$ \ ddot y +g(y)=p(t)$. y ddot \ to Nonlinear dynamics and the beam- the and dynamics Nonlinear \ pm \ \ ddot y +g(y)=p(t)$, with $p(t)$ with +g(y)=p(t)$, y ddot infty$ . . , NATO Science Series C Series Science NATO , J. London Math. Soc. Math. London J. J. London Math. Soc. Math. London J. , 272-301. Amer. Inst. Amer. 272-301. ,

41 41 - -

day, June25 Antonio Ros’stalkwillbeheld onThurs th , at12:30pm. - works by Collin [1] and López and Ros [2] and recently Meeks, Pérez and Ros [4] Ros and Pérez Meeks, recently and [2] Ros and López and [1] Collin by works in zero genus of surfaces minimal embedded properly of classification the consider can we generally, More laminations. minimal about and disks minimal embedded The years. last the in proof depends on results by Colding theory and Minicozzi about curvature estimates for the of achievements mayor the of one is [5] berg in surfaces minimal nected con simply embedded properly classifying of consists extensions these of One active fieldofresearch. graph on minimal entire unique the as plane the characterizes which Theorem, Bernstein We first consider classical theory of minimal surfaces. As an example we have the particular, of geometry. In our talk we will present some recent results in this field. under a natural constraint, has influenced the development of mathematics and, in Thestudy ofsurfaces ofleast area, ormore generally surfaces with stationary area Area minimizingsurfaces themathematicsof Partialas Differential Equations, Topology Complexor Analysis. Geometric Analysis. This area arises from the interaction ofprofessorAntonio GeometryaUniversity theisRosof Granada ofresearch hisandwith focuses otherin parts spaces with a distance and a measure and, even without leaving the Euclide the leaving without even and, measure a and distance a with spaces manifolds, Riemannian contexts: other many in naturally appears problem the in sits surface the When volume. scribed pre a enclosing those among surfaces minimizing area of study the of consists which Problem, Isoperimetric the is consider to want we topic main second The classic following the of one surfaces: is S Then plane. the in subset open an of pology Theorem. nite topology.Joiningtheseresultswehavethefollowing infi with family this in surfaces the classifying by program this completed have iv) iii) ii) i) ARiemannminimalexample TheCatenoid. TheHelicoid. Theplane. R 3 . The case of nontrivial finite topology was solved as a combination of combination a as solved was topology finite nontrivial of case The . R 2 , whose extensions and generalizations continue nowadays to be and Let S Let ⊂

R 3 be a properly embedded minimal surface with the to the with surface minimal embedded properly a be R 3 . The solution of this problem by Meeks and Rosen and Meeks by problem this of solution The . . R 3 , the solution is the round sphere; but sphere; round the is solution the , Antonio Ros ------

The Poetry of Analysis 71 72 The Poetry of Analysis ary Ros A. [9] ratation. sepa phase mesoscopic and surfaces curvature mean constant periodic Stable Ros: A. [8] 2, 373-388. crystallography. in inequalities Isoperimetric Ros: A. [7] in threespaceforms. [6] M. Ritoré and A. Ros (2005), no.2,727-758. [5] W.H. Meeks III and H. Rosenberg print. Ros A. and Pérez J. W.H.III, [4] Meeks vent. Math. Ros A. and Pérez J. III, W.H.Meeks [3] ferential Geom. Ros A. and F.López [2] J. of Math [1] P. Collin: Topologie et courbure des surfaces minimales proprement plongées de REFERENCES of A. Schoen which is a triply periodic minimal surface of genus 3 in the interestingexample of alocal minimum of the isoperimetric problem is the Gyroid sophisticatedinterfaces appearing inmesoscale phase separation phenomena. An in lem by Barbosa and do Carmo, as the unique local minimum of the isoperimetric prob characterizationperiodicthesphere,casetheobtained theextends to of thors.It Thisresultsharpcombinesandis contributions Rıtoré,by[8,9]Ros otherandau local minimumoftheisoperimetricproblem.ThenSisonefollowing: Theorem. an As open. following the remain discuss will we example situations above the in appearing questions basic Many (per unitcell),where G Isoperimetric Periodic Problem, which consists of describing, among surfaces dividing the space in two the consider also can We boundary). free with surfaces curvature mean constant to connects problem (this region certain a in volume a enclosing surfaces for problem isoperimetric the study can we geometry, an -invariant regionswithprescribedvolumefraction,thosewhichhaveleastarea iv) iii) ii) i) , preprint. A (quotientofa)triplyperiodicsurfacewithg=3. A (quotientofa)doublyperiodicsurfacewithg=2. The (quotientofa)circularcylinder,g=1. The roundsphere,g=0. R . Interfaces FreeBound. (2) 3 : and provides a theoretical support to explain the geometry of certain of geometry the explain to supporttheoretical a provides and

Stability of minimal and constant mean curvature surfaces with free bound free with surfaces curvature mean constant and minimal of Stability Let S

133 145 (1998),no.1, 107-132.

(1997),no.1,1-31. 33 ⊂ (1991),no.1,293-300.

T Comment. Math.Helvet. 3 beaclosedsurfaceofgenusginflat3-toruswhichis : Stable constant mean curvature tori and the isoperimetric problem G : On embedded complete minimal surfaces of genus zero. genus of surfaces minimal complete embedded On is a symmetry group of of isasymmetrygroup

9 (2007),no.3,355-365. : The uniqueness of the helicoids. : :

Uniqueness of the Riemann minimal examples. minimal Riemann the of Uniqueness Properly embedded minimal planar domains planar minimal embedded Properly

67 (1992),no.2,293-305. R 3 , see for instance [7,9]. instance for see , J. Amer. Math. Soc. Math. Amer. J. Ann. of Math. (2) 17 (2004), no. (2004), bcc torus. R 3 J. Dif J. , pre , . Ann. In 161 ------Wednesday, June24 Ovidiu Savin’stalkwillbeheld on th , at12:30pm. ∂N interior of domain polygonal a on ov, Sheffield and others. The functions combinatorics on random surfaces and random and tilings mechanics by statistical Cohn, Kenyon, in Okounk papers recent of series a by motivated is work Our and discussthe in two dimensions for certain classes of convex, possibly non-smooth functions We considertheclassicalproblemincalculusofvariationsminimizing random surfaces Minimizers of convex functionals arising in ferential equations,withemphasisontheregularityofsolutionstoellipticPDEs. dif partial include interests research His University. Columbia at Professor ate Miller Postdoctoral Fellow at UC Berkeley (2003-2006) and from 2006 he is Associ a been has He Caffarelli. Luis of supervision the under 2003, Austin, at Texas of in Mathematics at the University of Pittsburgh, 1999, and his PhD at the University Ovidiu Savin was born in Piatra Neamt (Romania), January 1st, 1977. He got his MS two large classes of non-uniformly elliptic functionals functionals elliptic non-uniformly of classes large two obtain We dimensions. two in Moser Giorgi-Nash- De of results regularity classical the improve also techniques Our dients mustapproach gra corresponding their then point a such to converges points of sequence a if non-differentiability: of points the at result continuity of type a discuss also We obstacle. N which have an end point on are minimizers that says result main Our ational problemcanbethoughtalsoasadegenerateobstacle problem. of degeneracy of of behavior the about assumptions any without ties, . On these segments the minimizer coincides with either the lower or the upper the or lower the either with coincides minimizer the segments these On . . We investigate the regularity of minimizers for functionals with these proper N except at a finite number of points, and they are piecewise linear on C 1 D regularityofminimizers. 2 F can be thought as the union of a finite set with ∂N N . - (called the Newton polygon), they are smooth in the in smooth are they polygon), Newton the (called ∂ Ω Ω and have directions perpendicular to the sides of ∫ Ω

F F ( C which appear in these papers are defined ∇ 1 C regularity of Lipschitz minimizers for for minimizers Lipschitz of regularity u 1 in ) dx Ω Ω except on a number of segments of number a on except Ω F on Ovidiu Savin F defined in in defined ∂N . In this case the set the case this In . ∂N R 2 . One class class . One . This vari F ------,

The Poetry of Analysis 73 74 The Poetry of Analysis consists of functionals for which there exist two open sets O sets open two exist there which for functionals of consists This isajointworkwithD.deSilva. points. of number finite a at except below from only derivatives second bounded have V Λ c { D 2 F

<

Λ I } that cover cover } that R 2 . The second class consists of those functionals that that functionals of those consists class . second The λ c { { c D 2 F

>

λ I } and and } June 23 Luis Seco’stalkwillbeheldon Tuesday, rd , at16:45pm. financial landscape forever, from a financial sector that was linear to another one the changed had seventies the in develop to started had which markets rivatives de the that realized regulators banking good: was second The jobs. and money im lost people bankrupt, went companies two bad: was first The consequences. had portant event This 25%. plunged market stock the 1987, 19th, October On ideas ofAntoniodid. dimension when viewed through a Antonio’s mathematics lens. This is what some It is remarkable that something as prosaic as the banking sector develops a poetic Managing FinancialRiskwithLaplace on financial mainly finance, research risk managementandassetmanagement. mathematical his of decade area the last in the been for have and interests Mechanics Quantum on worked has He Institute ofTechnology. California the at Instructor Bateman a was He 1989. his in University Princeton from obtained He RiskLab-Toronto. Ph.D. his and 1985, of in Madrid de Autonoma Universidad from director degree Bachelor’s the atthe and Program Toronto Finance of the Mathematical University of director the firm, management portfolio Canadian a Management, & Analysis Sigma of also CEO is and President He Toronto.the of University the at Mathematics of Professor a is Seco Luis means convolution of their distribution densities, or taking products on Fourier on products taking or densities, distribution their of convolution means variables random adding because gaussians, are gaussians of combinations ear Lin distributions. gaussian- -or normal by described are factors most risk simple; financial fairly is world financial linear a viewpoint, mathematical a From 1987. in time first the for obvious became this and non-linear, to linear from finance of world the changed seventies The 2007. of bubble sub-prime the of burst the course of and 1998, of crisis default Russian the Bank), Barings ended that Kobe in earthquake the (after 1995 in again then and 1987, in happened what is This consequences. terrifying have and amplified be could somewhere event small up in value more or less the same. Financial derivatives changed this: a seemingly and andliabilities, hadassets go assets 1%, say up, go markets if those: on impact linear had prices in changes banks simple: was finance of world linear The where convexitymatters,alot. Luis ÁngelSeco - - -

The Poetry of Analysis 75 76 The Poetry of Analysis poetry. financial mathematical manifolds: certain of curvatures involving expression cal non- for concept linear portfolios, VaR which expresses the the main sources of of market interpretation risk as a geometric mathemati a is this all of result The tive whichbecameherPhDthesisattheUniversityofToronto. perspec asymptotic this from risks portfolio minimize to methodology a signed de Maite Eventually, Brummelhuis. Raymond include to team collaborative the the original problem of VaR calculation is something that we did after expanding this became Maite’s Master’s thesis; upgrading this mathematical idea to address of statement mathematical The expression. explicit simple a to lead would and specialties- Antonio’s of -one expansion phase stationary a with done be could that something is infinity around tail its of expansion asymptotic the messy, is and one idea came up: whereas the distribution of a quadratic form of gaussians problem, this with deal to how about Antonio to talk to started myself, and to- Toron of University the at student graduate a -then Quintanilla Maite 1997, In gaussians, quadraticformsofgaussiansarenoteasilytractableobjects. something as simple as a quadratic correction is that, while sums of gaussians are with problem The inaccuracies. large of expense the at but explicitly, can solved be which one, linear a into back problem non-linear the turns this course, of simple terms, is a first order in Taylor expansion of the P which, VaR delta-normal the called is simplification known best The this wasadauntingtask.Thereforesimplificationswereneeded. their P&L as a function of market values, but generating a random variable out of time, computing of hours after only but accurately less or more computers, with calculate to ability the had banks expression; mathematical a by given function the of quantile P or function, profit-and-loss a but nothing is terms mathematical in which VaR, or at-Risk, Value- concept: new a on based was regulation of piece new Their Committee. P.J. Bank the by developed voluntarily Basel the by adopted and 1994 in Morgan been had which methodology, RiskMetrics the was regulation of piece new this linearity will lead to mathematically sophisticated regulation. One component of work in the early nineties, they realized -in non-mathematical terms- that loss of to set regulators Banking When gaussian. are gaussians of products and space; & L, of a bank. The problem is that the P the that is problem The bank. a of L, & L as a function of the risk factors; & L is not a not is L - - - - June 23 Luis Vega’s talkwillbeheldonTuesday, rd , at15:30pm. and logconvexityproperties. turbations of zero order. Our approach is based on the use of Carleman estimates a have gaussian that decay at two equation different times. Then Schrödinger we extend the free result considering per the of solutions for result uniqueness a as result Hardy’s some restate present we where I’ll Ponce and talk Kenig Escauriaza, the with work In joint vanish. must they then Gaussian given a than fast er decay transform Fourier its and function a if that states theorem Hardy’s uncertainty principle andapplications Hardy’s to approach new A of thescatteringoperatorslinearandnon-linearwavepropagators. study the to and dynamics, vortex and equations Schrödinger non-linear tween work is related to Hardy’s version of the uncertainty principle, the connection be recent His propagation. wave in phenomena dispersive of analysis the in cretely con more and Equations, Differential Partial and Analysis Harmonic on works He (Spain). Bilbao in Vasco/EHUPaís del Universidad at professor a is Vega Luis Luis Vega - - - -

The Poetry of Analysis 77

Preparación de la edición: Pablo Fernández Gallardo Imprime: Zürich color, S.L. C/ Méjico, 37 - 19004 Guadalajara