{ "type": "Article", "authors": [ { "type": "Person", "familyNames": [ "Marcolli" ], "givenNames": [ "Matilde" ] } ], "identifiers": [], "references": [ { "type": "Article", "id": "bib-bib1", "authors": [], "title": "\nM. F. Atiyah, N. J. Hitchin, V. G. Drinfel’d and Yu. I. Manin,\nConstruction of\ninstantons. Phys. Lett. A 65, 185–187 (1978)\n", "url": "https://dx.doi.org/10.1016/0375-9601(78)90141-X" }, { "type": "Article", "id": "bib-bib2", "authors": [], "title": "\nV. V. Batyrev and Yu. I. Manin, Sur\nle nombre des points rationnels de hauteur borné des\nvariétés algébriques. Math. Ann. 286,\n27–43 (1990) ", "url": "https://dx.doi.org/10.1007/BF01453564" }, { "type": "Article", "id": "bib-bib3", "authors": [], "title": "\nK. Behrend and Yu. Manin,\nStacks of stable maps\nand Gromov–Witten invariants. Duke Math. J. 85, 1–60\n(1996) ", "url": "https://dx.doi.org/10.1215/S0012-7094-96-08501-4" }, { "type": "Article", "id": "bib-bib4", "authors": [], "title": "\nA. A. Beĭlinson and Yu. I. Manin, The Mumford form and the Polyakov\nmeasure in string theory. Comm. Math. Phys. 107, 359–376\n(1986) ", "url": "https://doi.org/10.1007/bf01220994" }, { "type": "Article", "id": "bib-bib5", "authors": [], "title": "\nJ. Franke, Yu. I. Manin and Yu. Tschinkel,\nRational points of bounded\nheight on Fano varieties. Invent. Math. 95, 421–435\n(1989) ", "url": "https://dx.doi.org/10.1007/BF01393904" }, { "type": "Article", "id": "bib-bib6", "authors": [], "title": "\nV. A. Iskovskih and Yu. I. Manin, Three-dimensional quartics and counterexamples\nto the Lüroth problem. (in Russian) Mat. Sb. (N.S.) 86(128),\n140–166 (1971); English translation: Math. USSR Sb. 15, 141–166 (1971)\n", "url": "https://doi.org/10.1070/sm1971v015n01abeh001536" }, { "type": "Article", "id": "bib-bib7", "authors": [], "title": "\nM. Kontsevich and Yu. Manin, Gromov–Witten classes, quantum cohomology, and\nenumerative geometry. Comm. Math. Phys. 164, 525–562 (1994)\n", "url": "https://doi.org/10.1007/bf02101490" }, { "type": "Article", "id": "bib-bib8", "authors": [], "title": "\nYu. I. Manin, Algebraic curves over fields with differentiation. (in Russian) Izv.\nAkad. Nauk SSSR Ser. Mat. 22, 737–756 (1958); " }, { "type": "Article", "id": "bib-bib9", "authors": [], "title": "\nYu. I. Manin, Rational points on algebraic curves over function fields. (in Russian)\nIzv. Akad. Nauk SSSR Ser. Mat. 27, 1395–1440 (1963)\n" }, { "type": "Article", "id": "bib-bib10", "authors": [], "title": "\nYu. I. Manin, Correspondences, motifs and monoidal transformations. (in Russian) Mat.\nSb. (N.S.) 77 (119), 475–507 (1968); English translation: Math. USSR Sb. 6, 439–470 (1968)\n", "url": "https://doi.org/10.1070/sm1968v006n04abeh001070" }, { "type": "Article", "id": "bib-bib11", "authors": [], "title": "\nYu. I. Manin, Le groupe de Brauer–Grothendieck en géométrie\ndiophantienne. In Actes du Congrès International des\nMathématiciens (Nice, 1970), Tome 1, pp. 401–411, Gauthier-Villars, Paris\n(1971); also in:\nSelected papers of Yu. I. Manin, World Sci. Ser. 20th Century Math. 3, pp. 191–201, World Scientific Publishing, Singapore (1996)\n", "url": "https://doi.org/10.1142/9789812830517_0009" }, { "type": "Article", "id": "bib-bib12", "authors": [], "title": "\nYu. I. Manin, Cubic forms: Algebra, geometry, arithmetic. (in Russian) Izdatel’stvo “Nauka”, Moscow (1972); English translation: North-Holland, Amsterdam (1974) " }, { "type": "Article", "id": "bib-bib13", "authors": [], "title": "\nYu. I. Manin, Parabolic points and zeta functions of modular curves. (in Russian) Izv.\nAkad. Nauk SSSR Ser. Mat. 36, 19–66 (1972); English translation: Math. USSR Izv. 6, 19–64 (1972)\n", "url": "https://doi.org/10.1070/im1972v006n01abeh001867" }, { "type": "Article", "id": "bib-bib14", "authors": [], "title": "\nYu. I. Manin, Computable and noncomputable. (in Russian) Cybernetics, Soviet. Radio, Moscow (1980) " }, { "type": "Article", "id": "bib-bib15", "authors": [], "title": "\nYu. I. Manin, Modular forms and number theory. In Proceedings of the\nInternational Congress of Mathematicians (Helsinki, 1978), pp. 177–186, Academia\nScientiarum Fennica, Helsinki (1980) " }, { "type": "Article", "id": "bib-bib16", "authors": [], "title": "\nYu. I. Manin, What is the maximum number of points on a curve over\n𝑭2? J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28,\n715–720 (1982) ", "url": "https://doi.org/10.15083/00039602" }, { "type": "Article", "id": "bib-bib17", "authors": [], "title": "\nYu. I. Manin, Gauge field theory and complex geometry. (in Russian) Nauka, Moscow (1984);\nEnglish translation: Grundlehren der mathematischen Wissenschaften 289, Springer-Verlag, Berlin (1988)     ", "url": "https://doi.org/10.1007/978-3-662-07386-5" }, { "type": "Article", "id": "bib-bib18", "authors": [], "title": "\nYu. I. Manin, Theta-function representation of the partition function of a\nPolyakov string. JETP Lett. 43, 204–206 (1986)\n" }, { "type": "Article", "id": "bib-bib19", "authors": [], "title": "\nYu. I. Manin, Closed fibers at infinity in Arakelov’s geometry (Christmas\ntale at R. Coleman’s seminar, November 1989). Preprint (1989)\n" }, { "type": "Article", "id": "bib-bib20", "authors": [], "title": "\nYu. I. Manin, Reflections on arithmetical physics. In Conformal invariance\nand string theory (Poiana Braşov, 1987), pp. 293–303, Perspect. Phys., Academic\nPress, Boston, MA (1989) ", "url": "https://doi.org/10.1016/b978-0-12-218100-9.50017-0" }, { "type": "Article", "id": "bib-bib21", "authors": [], "title": "\nYu. I. Manin, Three-dimensional\nhyperbolic geometry as ∞-adic Arakelov geometry. Invent.\nMath. 104, 223–243 (1991) ", "url": "https://dx.doi.org/10.1007/BF01245074" }, { "type": "Article", "id": "bib-bib22", "authors": [], "title": "\nYu. I. Manin, Frobenius\nManifolds, Quantum Cohomology, and Moduli Spaces. Amer. Math. Soc. Colloq. Publ. 47,\nAmerican Mathematical Society,\nProvidence, RI (1999) ", "url": "https://dx.doi.org/10.1090/coll/047" }, { "type": "Article", "id": "bib-bib23", "authors": [], "title": "\nYu. I. Manin, Real multiplication and noncommutative geometry (ein\nAlterstraum). In The legacy of Niels Henrik Abel, pp. 685–727, Springer,\nBerlin, Heidelberg (2004) ", "url": "https://doi.org/10.1007/978-3-642-18908-1_23" }, { "type": "Article", "id": "bib-bib24", "authors": [], "title": "\nYu. I. Manin, Les mathématiques comme métaphore. Les Belles Lettres, Paris (2021)\n" }, { "type": "Article", "id": "bib-bib25", "authors": [], "title": "\nYu. I. Manin and M. Marcolli,\nHolography principle and\narithmetic of algebraic curves. Adv. Theor. Math. Phys. 5,\n617–650 (2001) ", "url": "https://dx.doi.org/10.4310/ATMP.2001.v5.n3.a6" }, { "type": "Article", "id": "bib-bib26", "authors": [], "title": "\nYu. I. Manin and M. Marcolli,\nContinued fractions,\nmodular symbols, and noncommutative geometry. Selecta Math. (N.S.)\n8, 475–521 (2002) ", "url": "https://dx.doi.org/10.1007/s00029-002-8113-3" }, { "type": "Article", "id": "bib-bib27", "authors": [], "title": "\nYu. I. Manin and M. Marcolli,\nModular shadows and\nthe Lévy–Mellin ∞-adic transform. In Modular forms\non Schiermonnikoog, Cambridge University Press, Cambridge, 189–238 (2008)\n", "url": "https://dx.doi.org/10.1017/CBO9780511543371.012" }, { "type": "Article", "id": "bib-bib28", "authors": [], "title": "\nYu. Manin and M. Marcolli, Kolmogorov complexity and the asymptotic bound for\nerror-correcting codes. J. Differential Geom. 97, 91–108\n(2014) ", "url": "https://doi.org/10.4310/jdg/1404912104" }, { "type": "Article", "id": "bib-bib29", "authors": [], "title": "\nYu. I. Manin and M. Marcolli,\nSemantic spaces.\nMath. Comput. Sci. 10, 459–477 (2016) ", "url": "https://dx.doi.org/10.1007/s11786-016-0278-9" }, { "type": "Article", "id": "bib-bib30", "authors": [], "title": "\nYu. I. Manin and M. Marcolli, Computability questions in the sphere packing\nproblem. (2022) arXiv:2212.05119", "url": "https://arxiv.org/abs/2212.05119" }, { "type": "Article", "id": "bib-bib31", "authors": [], "title": "\nYu. I.\nManin and M. Marcolli, Homotopy spectra and Diophantine equations.\n(2021) arXiv:2101.00197;\nto appear in The Literature and History of Mathematical\nScience, International Press, Boston\n", "url": "https://arxiv.org/abs/2101.00197" } ], "title": "Pierced by a sun ray", "meta": {}, "content": [ { "type": "QuoteBlock", "content": [ { "type": "Paragraph", "content": [ { "type": "Emphasis", "content": [ "Ognuno sta solo sul cuor della terra, \ntrafitto da un raggio di sole: \ned è subito sera." ] }, " \n(Salvatore Quasimodo)", { "type": "Note", "id": "idm12", "noteType": "Footnote", "content": [ { "type": "Paragraph", "id": "footnote1", "content": [ "Everyone stands alone over the earth’s core, / pierced by a sun ray: / and immediately it’s nightfall." ] } ] } ] } ] }, { "type": "Paragraph", "id": "p2", "content": [ "Nostalgia, in Greek, is the pain of returning: the ", { "type": "Emphasis", "content": [ "νόστος" ] }, "\nof the Homeric heroes trying to find their way back in a world forever altered.\nReturn\nand remembrance are a painful illusion, our longing for a past that can speak\nto us in a familiar voice." ] }, { "type": "Paragraph", "id": "p3", "content": [ "I promised him I would come in the spring. By the time I was\ndone with my cancer treatment and able to travel again, I was\nfour months too late. I ended up spending a week\nin his\nMPI office, looking through\nthe writings he left behind: one week for an entire lifetime. That’s when I started writing this piece.\nOn the small board outside the door of his office, along with his photograph and\nthe cover pages of his recent preprints, he had posted a copy of Quasimodo’s poem\n", { "type": "Emphasis", "content": [ "Ed è subito sera" ] }, "\n(Figure ", { "type": "Cite", "target": "S0-F1", "content": [ "1" ] }, "). Indeed, even after a long life full of so many remarkable\nachievements, that final nightfall still comes very suddenly and unexpectedly. It still\ncomes much too soon.\nThere was still so much joy of life, of thought, so much energy and creativity\nleft in him, up until the very end of his life.\n" ] }, { "type": "Figure", "id": "S0-F1", "caption": [ { "type": "Paragraph", "content": [ "The door of Yuri Manin’s office at the Max Planck Institute for Mathematics (MPI) in Bonn" ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "YuMofficedoor.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p4", "content": [ "I used to spend late night hours at the institute, in the eight years when I was\nhis colleague at the MPI. Yuri and Xenia used to tease me when I would\nwalk back to my office, instead of going home, as we were coming\nback from a concert or a movie and a dinner together. Yes, I used to work\nat the MPI late at night, but I had never before\nseen the dawn from the institute windows, like I am doing now every night.\nI am seeing it from his office windows: a beautiful sight of the first light of day\non Bonn’s Münsterplatz.\nI use this time during the night, before anyone else comes in, to reflect on\nwhat all this had meant to me over the past twenty years, to read through\nstacks of his handwritten notes,\nto look through his books on the shelves –\nstill where I remember them –, to experience the last remnants of his physical\npresence in the place where he spent the last thirty years of his life." ] }, { "type": "Figure", "id": "S0-F2", "caption": [ { "type": "Paragraph", "content": [ "A page from a 1965 notebook." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "YuMnoteItAlgGeom2.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p5", "content": [ "The oldest handwritten notes I came across in his office are in a small notebook dated 1965.\nThey are notes about results from the Italian school of algebraic geometry: on the\nfirst page a commentary on a 1948 paper by Predonzan, \n", { "type": "Emphasis", "content": [ "Intorno agli ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "Sk", "meta": { "altText": "S_{k}" } }, "\ngiacenti sulla varietà intersezione completa di più forme" ] }, " (Figure ", { "type": "Cite", "target": "S0-F2", "content": [ "2" ] }, "). Just the previous\nevening I had found among the recent papers left on the desk of his home office a\nshort note expressing his enthusiasm for the classical Italian algebraic geometry\n(Figure ", { "type": "Cite", "target": "S0-F3", "content": [ "3" ] }, ")." ] }, { "type": "Paragraph", "id": "p6", "content": [ "Two events in the 1960s\ninfluenced significantly his view of this field of mathematics: a visit to Pisa, during\nwhich he learned Italian, read Dante and an extensive combination of writings by\nItalian algebraic geometers, and the subsequent visit to Grothendieck at the IHES\nin Paris, which lead him to write what became the very first published paper on\nGrothendieck’s theory of motives [", { "type": "Cite", "target": "bib-bib10", "content": [ "10" ] }, "]. During the second half of the\n20th century the field of algebraic geometry witnessed a sharp divide between the abstract\ncategorical language of the Grothendieck approach, on one side, and the hands-on\ngeometric example-driven approach that can be traced back to the older Italian school.\nWe all know algebraic geometers who fall squarely into one or the other of these\ntwo camps, but fewer embrace\nboth views simultaneously, finding ways to combine them with ease in their own work.\nYuri always carried with him this simultaneous dual view of algebraic geometry\nthat came to characterize his approach to the field." ] }, { "type": "Paragraph", "id": "p7", "content": [ "By that time, Yuri was already widely famous for his proof, in 1963, of the\nMordell Conjecture over function fields [", { "type": "Cite", "target": "bib-bib9", "content": [ "9" ] }, "]. The conjecture\nestablishes a relation between topological and arithmetic properties of curves:\nif the genus is ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "g≥2", "meta": { "altText": "g\\geq 2" } }, ", then the number of rational points is finite. The\ncase over function fields can be restated in terms of a pencil of curves\nover a projective algebraic surface. The main tool involved in the proof is\nan abstract formulation of Picard–Fuchs equations in the form of a connection\non a bundle whose fibers are de Rham cohomologies, and whose\nflat sections are the differential equations that govern the structure of period\nintegrals [", { "type": "Cite", "target": "bib-bib8", "content": [ "8" ] }, "].\nIt was Grothendieck who started calling this construction the ", { "type": "Emphasis", "content": [ "Gauss–Manin connection" ] }, ".\nIt has since then become a fundamental and widely applied tool in algebraic and\narithmetic geometry." ] }, { "type": "Paragraph", "id": "p8", "content": [ "Classical algebraic geometry works by Enriques, Segre, and especially Fano formed the\nbackground for his result with Iskovskih [", { "type": "Cite", "target": "bib-bib6", "content": [ "6" ] }, "] on the existence of smooth quartic threefolds\nthat are unirational but not rational, which provided a negative answer to the\nLüroth problem in 1971." ] }, { "type": "Figure", "id": "S0-F3", "caption": [ { "type": "Paragraph", "content": [ "A recent note left on Manin’s home desk." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "YuMnoteItAlgGeom_c.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p9", "content": [ "The beginning of the 1970s is also the time when he published his first book.\n", { "type": "Emphasis", "content": [ "When I felt myself grown up enough to set writing my first book" ] }, ", he\nsays in the short note left\non his home desk, alongside the words ", { "type": "Emphasis", "content": [ "omaggio alla geometria\nalgebrica italiana" ] }, ", homage to Italian algebraic geometry\n(Figure ", { "type": "Cite", "target": "S0-F3", "content": [ "3" ] }, ").\nThe note also contains the Latin translation of Plutarch’s\n", { "type": "Emphasis", "content": [ "πλεῖν ἀνάγκη, ζῆν οὐκ ἀνάγκη" ] }, " (to sail is indispensable, to live is not).\nYuri must have likely written it just weeks before his death. In Greek\n", { "type": "Emphasis", "content": [ "ἀνάγκη" ] }, " is necessity by way of fate. Yuri used to say that\nmathematics chooses us, not the other way around: to him it was fate, destiny, and\nindispensable vital need.\n" ] }, { "type": "Paragraph", "id": "p10", "content": [ "At the end of the introduction to the\nbook ", { "type": "Emphasis", "content": [ "Cubic forms" ] }, " [", { "type": "Cite", "target": "bib-bib12", "content": [ "12" ] }, "], Yuri quotes works of Grothendieck, Segre, and Châtelet\nas main sources of inspiration. The book assembles a significant amount of Yuri’s work in\nalgebraic geometry over the preceding years, combined into a beautiful narrative. What\nhappens to the group law of elliptic curves when one moves up to rational points of a cubic surface?\nThe surprising answer is non-associative commutative Moufang loops. The classical geometry of the\n", { "type": "MathFragment", "mathLanguage": "mathml", "text": "27", "meta": { "altText": "27" } }, " lines on a cubic surface leads to a modern interpretation in terms of a birational invariant of\nthe surface given by the Galois cohomology group ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "H1⁢(k,Pic⁢(Xk¯))", "meta": { "altText": "H^{1}(k,\\mathrm{Pic}(X_{\\bar{k}}))" } }, "\n(that can also be described in terms of Brauer groups), which can be computed\nfrom the partition of the ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "27", "meta": { "altText": "27" } }, " lines into Galois orbits. The book is a treasure trove: minimal cubic surfaces\nand birational transformations, Brauer–Grothendieck group and Brauer equivalence via Azumaya algebras,\nand other intriguing connections between arithmetic and geometry." ] }, { "type": "Paragraph", "id": "p11", "content": [ { "type": "Emphasis", "content": [ "Cubic forms" ] }, " was for a long time the only one of Yuri’s books I did not have.\nIt is hard to find, and only last fall I was able to order a good copy from a bookseller.\nIt did not come for a long time, and I assumed it had been lost in the mail. It was\ndelivered to my home in the evening of January 7, the same day when Yuri died.\nCarl Gustav Jung (another interest Yuri and I shared) called these ", { "type": "Emphasis", "content": [ "meaningful\ncoincidences" ] }, ": random acausal events that happen to take place at crucial times in\nour lives and that our psyche therefore invests with meaning." ] }, { "type": "Paragraph", "id": "p12", "content": [ "There are no more lights in the institute windows at night: there used\nto be plenty, not just mine, but the MPI was a different place at that time.\nI left in 2008. Yes, I left. It seemed like a good decision at the time – the MPI was\nchanging and my job in California was attractive, but the years\nkept passing and I never quite felt like I had rooted my life elsewhere.\nMy center of coordinates had stayed in this room, which is now\nabout to disappear. I kept coming back, year after year." ] }, { "type": "Paragraph", "id": "p13", "content": [ "We started working together almost immediately after I joined the MPI faculty in\nthe summer of 2000. Yuri suggested that we revisit his theory of modular symbols [", { "type": "Cite", "target": "bib-bib13", "content": [ "13" ] }, "] from\na different perspective, related to noncommutative geometry and the “invisible boundary”\nof modular curves determined by the irrational points in the boundary of the upper half plane [", { "type": "Cite", "target": "bib-bib26", "content": [ "26" ] }, "].\nWhile the rational points, the cusps, have an algebro-geometric interpretation in terms\nof degenerations of elliptic curves to the multiplicative group, the irrational points can be\nregarded as degenerations that no longer exist algebro-geometrically but only as noncommutative\ntori. Limiting modular symbols exist at such invisible points, determined\nby an ergodic average of classical modular symbols, and the arithmetic contributions of this part of\nthe geometry can be seen by expressing\nMellin transforms of cusp forms in terms of quantities defined entirely on this\nboundary [", { "type": "Cite", "target": "bib-bib26", "content": [ "26" ] }, "], leading to a more general formalism of modular shadows,\nLévy functions, and Lévy–Mellin transforms [", { "type": "Cite", "target": "bib-bib27", "content": [ "27" ] }, "]. The role of noncommutative\ntori as non-algebro-geometric degenerations of elliptic curves and of the noncommutative boundary\nof modular curves as their moduli space was the source of Yuri’s ", { "type": "Emphasis", "content": [ "real multiplication program" ] }, " [", { "type": "Cite", "target": "bib-bib23", "content": [ "23" ] }, "],\nbased on the beautiful idea that noncommutative tori with real multiplication (non-trivial Morita\nself-equivalences) should play the role of the missing geometry behind the explicit class field\ntheory problem for real quadratic fields, like elliptic curves with complex multiplication for the\nimaginary quadratic case. This promising program still remains unfulfilled.\n" ] }, { "type": "Paragraph", "id": "p14", "content": [ "Modular symbols, periods of modular forms, and ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "L", "meta": { "altText": "L" } }, "-functions formed the central theme of\na series of papers Yuri wrote in the 1970s, relating cusp forms and their period integrals and\nHecke series. Results include explicit formulae for Hecke eigenvalues, algebraicity results\nfor periods of cusp forms, ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p", "meta": { "altText": "p" } }, "-adic Hecke series through ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p", "meta": { "altText": "p" } }, "-adic measures associated to\ncusp forms and the ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p", "meta": { "altText": "p" } }, "-adic Mellin transform,\nan effective algorithm for the computation of the Tate–Shafarevich group of elliptic curves\nbased on Mellin transforms of ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "L", "meta": { "altText": "L" } }, "-functions and modular symbols.\nIn the Helsinki ICM address [", { "type": "Cite", "target": "bib-bib15", "content": [ "15" ] }, "], Yuri summarizes several of these results,\npresenting as the central\ntheme of number theory the interaction between the two classes of ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "L", "meta": { "altText": "L" } }, "-functions, associated\nto varieties through cohomologies, and associated to modular and automorphic forms." ] }, { "type": "Paragraph", "id": "p15", "content": [ "There are two extensive sets of handwritten notes I found in his office, one on Lévy functions\nand what was supposed to be our continuation of [", { "type": "Cite", "target": "bib-bib26", "content": [ "26" ] }, "]\nand [", { "type": "Cite", "target": "bib-bib27", "content": [ "27" ] }, "], and one on the real multiplication project.\nBoth were likely written in the mid-2000s,\nprobably 2005 or 2006. We had frequent conversations at the time on all of these topics, but\nduring those years I got diverted into other projects (perhaps unwisely handled in retrospect)\nand much remained undone: here is where the pain of returning becomes especially hard to bear." ] }, { "type": "Paragraph", "id": "p16", "content": [ "But I am here again, night and day, going through these written remnants\nof Yuri’s work: many more sets of carefully organized handwritten notes, references he was\nreading along with them, drafts of papers, printouts of email exchanges, lecture notes\nfor courses he taught in Moscow, at MIT, at the MPI, at Northwestern.\nI am reading through all of this, in whatever order the folders come into my\nhands: everything has already been sorted out, during these\nmonths after his death, when I still could not be here." ] }, { "type": "Paragraph", "id": "p17", "content": [ "His work: it makes me think of the ocean, with\nbig waves and deep currents. That’s what I see, as I go over folder\nafter folder of manuscripts and notes, those huge waves of work\nstretching over the years. When I moved to\nBonn from Boston in 2000, I arrived in time to catch the last years\nof the long wave of quantum cohomology and Frobenius manifolds,\nthat started shortly after he had himself moved to the MPI from MIT\nin 1993, and that dominated his interests for a decade. His work with\nKontsevich, starting in 1994 [", { "type": "Cite", "target": "bib-bib7", "content": [ "7" ] }, "], set the foundation for the algebro-geometric\nform of quantum cohomology and Gromov–Witten invariants,\na mathematical construction that originated in the physical theory of\nquantum strings propagating on a spacetime manifold. This\nstring theory setting leads to a deformation of the usual product on the\ncohomology of algebraic varieties, where instead of counting the intersection\npoints between cycles, one uses an appropriate counting of algebraic curves\nconnecting them. This algebro-geometric counting requires the geometry\nof moduli spaces of stable maps and the delicate notion of virtual fundamental classes.\nThe axiomatic treatment of Gromov–Witten invariants and quantum\ncohomology leads naturally to the study of Frobenius manifolds\nas the underlying structure\n(a notion originally introduced by Dubrovin), with diverse\napplications in singularity theory and integrable systems.\nA motivic viewpoint on quantum cohomology was developed in\nhis work with Behrend [", { "type": "Cite", "target": "bib-bib3", "content": [ "3" ] }, "], where the moduli spaces of\nstable maps are shown to determine correspondences between the\nmotive of a variety and the motives of moduli spaces of curves,\ngiving rise to an action of the modular operad on motives.\nThis large body of work on Frobenius manifolds, moduli spaces,\nand quantum cohomology culminated in his very extensive research\nmonograph on the subject, published in 1999 [", { "type": "Cite", "target": "bib-bib22", "content": [ "22" ] }, "]. My gift for Yuri’s\n65th birthday was a Frobenius manifolds conference. " ] }, { "type": "Paragraph", "id": "p18", "content": [ "The relation between algebraic geometry and physics is\nthe deeper current underlying this long wave of work on\nquantum cohomology and Frobenius manifold structures.\nYuri was the first to clearly demonstrate the importance of algebraic geometry\nfor string theory with the 1986 results (in [", { "type": "Cite", "target": "bib-bib18", "content": [ "18" ] }, "], and with\nBeilinson in [", { "type": "Cite", "target": "bib-bib4", "content": [ "4" ] }, "]) expressing the Polyakov measure of\nthe bosonic string path integral in terms of moduli spaces of curves,\ntheta-functions, Green functions considered by Faltings in the\nsetting of Arakelov geometry, and holomorphic quadratic differentials.\nThe interaction between algebraic geometry and string theory has\ndeeply transformed, in the decades that followed, both the field\nof high energy physics and algebraic geometry itself." ] }, { "type": "Paragraph", "id": "p19", "content": [ "At various times, Yuri proposed the idea that not only algebraic geometry,\nbut also arithmetic geometry and number theory should play a fundamental\nrole in physics [", { "type": "Cite", "target": "bib-bib20", "content": [ "20" ] }, "]. One of our first joint papers\nproposed the use of ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p", "meta": { "altText": "p" } }, "-adic geometry in the AdS/CFT holographic\ncorrespondence of string theory [", { "type": "Cite", "target": "bib-bib25", "content": [ "25" ] }, "], reinterpreting\nin holographic terms an earlier result of Yuri on the fiber at infinity of\nArakelov geometry [", { "type": "Cite", "target": "bib-bib21", "content": [ "21" ] }, "]. This idea of a ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "p", "meta": { "altText": "p" } }, "-adic form of the\nAdS/CFT correspondence became popular with physicists fifteen years later.\nI have to admit that Yuri’s paper on the fiber at infinity of Arakelov geometry [", { "type": "Cite", "target": "bib-bib21", "content": [ "21" ] }, "]\nis probably the single paper that was most influential in the development\nof my own view of mathematics, not only in terms of work directly\ninspired by it, but more generally in terms of exemplifying what I find beautiful and\nvaluable in mathematical research. Along with the published version of\nthat paper, I kept a copy of the unpublished preprint that preceded it [", { "type": "Cite", "target": "bib-bib19", "content": [ "19" ] }, "], full\nof his heuristic explanations, drawings, and analogies that formed the\nbackground to the final polished result\n(Figure ", { "type": "Cite", "target": "S0-F4", "content": [ "4" ] }, ")." ] }, { "type": "Figure", "id": "S0-F4", "caption": [ { "type": "Paragraph", "content": [ "Two drawings from [", { "type": "Cite", "target": "bib-bib19", "content": [ "19" ] }, "]." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "Paragraph", "content": [ { "type": "ImageObject", "contentUrl": "InftyArakelov2.png", "mediaType": "image/png", "meta": { "inline": false } }, { "type": "ImageObject", "contentUrl": "InftyArakelov3.png", "mediaType": "image/png", "meta": { "inline": false } } ] } ] }, { "type": "Paragraph", "id": "p20", "content": [ "Within his overall geometrization program, envisioning\nalgebraic geometry as a unifying\nlanguage in number theory, physics, and information theory,\nanother large wave of work one comes across is the\none that happened\nin the 1980s and early 1990s, encompassing the\ngeometry of Yang–Mills instantons, supergeometry, and symmetries\nof quantum spaces. The ADHM (Atiyah–Drinfeld–Hitchin–Manin)\nconstruction and classification of Yang–Mills instantons on the\n", { "type": "MathFragment", "mathLanguage": "mathml", "text": "4", "meta": { "altText": "4" } }, "-sphere is certainly the most famous result of this period [", { "type": "Cite", "target": "bib-bib1", "content": [ "1" ] }, "].\nSolutions of Yang–Mills equations on ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "4", "meta": { "altText": "4" } }, "-dimensional manifolds\nbecame, some years later, a major tool in low-dimensional topology,\nin the form of Donaldson invariants. Yuri’s extensive development of\nan algebro-geometric formulation of supergeometry was motivated by\nhis interest in the physical theories of supergravity and supersymmetric\nYang–Mills, for which he gave a precise mathematical formulation.\nA broad overview of the developments in mathematical\nphysics, around the topics of Yang–Mills instantons, Penrose’s twistors,\nand supergeometry, obtained by him and his students, is presented in\nhis book ", { "type": "Emphasis", "content": [ "Gauge field theory and complex geometry" ] }, " [", { "type": "Cite", "target": "bib-bib17", "content": [ "17" ] }, "].\nWork with Kuperschmidt and Lebedev on integrable systems focused on\nthe hierarchy of higher hydrodynamic equations of Benney type.\nIn this context he also introduced the noncommutative residue of\npseudodifferential operators, generalized by his student Wodzicki.\nAfter Drinfeld introduced quantum groups, Yuri showed that they can\nbe realized as symmetries of quantum spaces,\na natural point of view from the perspective of both physics\nand noncommutative geometry." ] }, { "type": "Paragraph", "id": "p21", "content": [ "Yuri was the first person who suggested the idea of quantum computing,\nin his 1980 book ", { "type": "Emphasis", "content": [ "Computable and uncomputable" ] }, " [", { "type": "Cite", "target": "bib-bib14", "content": [ "14" ] }, "]\n(Figure ", { "type": "Cite", "target": "S0-F5", "content": [ "5" ] }, "), a good two years before Feynman\nmade the same suggestion along similar lines of thought." ] }, { "type": "Figure", "id": "S0-F5", "caption": [ { "type": "Paragraph", "content": [ "Excerpt from the book [", { "type": "Cite", "target": "bib-bib14", "content": [ "14" ] }, ", p. 15] about quantum computing." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "Qcompute_c.png", "mediaType": "image/png", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p22", "content": [ "In the same period he promoted in his\nseminar the importance of algebro-geometric constructions in the theory of classical\nerror-correcting codes (first used by Goppa in 1981), a topic that his students Tsfasman and Vlăduţ\nthen widely developed. He introduced in [", { "type": "Cite", "target": "bib-bib16", "content": [ "16" ] }, "] the asymptotic bound in the geography of error-correcting codes,\nrelating the asymptotic theory of codes to the problem of constructing algebraic curves\nwith sufficiently many algebraic points over finite fields.\nHis impact on the theory of information epitomizes how topics that have\nby now become huge fields of research had germinated very early inside his mind:\nnowadays algebro-geometric codes lie at the heart of the edifice of cryptography and\nquantum information has become an enormous landscape, stretching across\nmathematics, physics, and computer science." ] }, { "type": "Paragraph", "id": "p23", "content": [ "The themes of computability, of classical and quantum information, and\nof error-correcting codes resurfaced frequently in his later work, and became\na very significant interest in the last years of his life, including a considerable\npart of our own joint\nwork, for instance [", { "type": "Cite", "target": "bib-bib28", "content": [ "28" ] }, ", ", { "type": "Cite", "target": "bib-bib30", "content": [ "30" ] }, "]. This is another example\nof what I see as the deep currents in Yuri’s work, those long visions that\nstretched through the decades, acting as a background structure, and emerging\nperiodically to the forefront of his research activities." ] }, { "type": "Figure", "id": "S0-F6", "caption": [ { "type": "Paragraph", "content": [ "Manuscript fragments about linguistics." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "Paragraph", "content": [ { "type": "ImageObject", "contentUrl": "Ling1.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } }, { "type": "ImageObject", "contentUrl": "Ling2.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] } ] }, { "type": "Paragraph", "id": "p24", "content": [ "Another long-term current in Yuri’s work is Diophantine equations and the\nproperties of rational points of varieties. The Hasse principle asserts that\nthe existence of rational points of a variety over a number field can be deduced\nfrom the existence of points over all the non-Archimedean and Archimedean\ncompletions (that is, one can go from local to global solutions). Yuri first\nshowed that the Brauer–Grothendieck group determines an obstruction\n(the Brauer–Manin obstruction) to the Hasse principle [", { "type": "Cite", "target": "bib-bib11", "content": [ "11" ] }, "].\nThe existence of this general obstruction changed the perspective on\nDiophantine problems, and the understanding of local-to-global principles.\nHis extensive investigation of rational points of bounded height on cubic surfaces\nbrought him to develop a broad research program relating geometry and\ntopology of varieties to Diophantine properties. Varieties with ample canonical\nclass (intuitively, hyperbolic or negatively curved, like the algebraic\ncurves of genus ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "g≥2", "meta": { "altText": "g\\geq 2" } }, ") are expected to have fewer rational points,\nlying on lower-dimensional submanifolds. For large anticanonical class\n(elliptic or positively curved, like Fano varieties) one expects many rational points.\nWork with Batyrev [", { "type": "Cite", "target": "bib-bib2", "content": [ "2" ] }, "] showed two important aspects of the\nstructure of rational points: the presence of accumulating subvarieties\nand the linear growth of the number of points with bounded anticanonical height\non the complement of accumulating subvarieties (Manin’s linear growth conjecture).\nA series of papers in the late 1980s and early 1990s, starting with [", { "type": "Cite", "target": "bib-bib5", "content": [ "5" ] }, "]\nwith Franke and Tschinkel, provided evidence for the explicit\nform of the asymptotic behavior ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "c⁢H⁢(log⁡H)t", "meta": { "altText": "cH(\\log H)^{t}" } }, " for the number of points\nof bounded height for varieties ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "V", "meta": { "altText": "V" } }, " over number fields, in the ample\nanticanonical case, with ", { "type": "MathFragment", "mathLanguage": "mathml", "text": "t=rank⁢Pic⁢(V)−1", "meta": { "altText": "t=\\mathrm{rank}\\,\\mathrm{Pic}(V)-1" } }, ".\nDuring the last two years of his life, Yuri became interested in the possibility\nof categorifying the height zeta function, so as to encode, in the form of\nscissor-congruence type relations, the presence of accumulating subvarieties,\nand suggested the relevance of homotopy-theoretic methods to this goal [", { "type": "Cite", "target": "bib-bib31", "content": [ "31" ] }, "].\nHe very much considered this a line of thought that he meant to continue developing." ] }, { "type": "Figure", "id": "S0-F7", "caption": [ { "type": "Paragraph", "content": [ "Yuri Manin’s office at the Max Plank Institute for Mathematics, Bonn." ] } ], "licenses": [ { "type": "CreativeWork", "url": "https://creativecommons.org/licenses/by/4.0/legalcode", "content": [ { "type": "Paragraph", "content": [] } ] } ], "content": [ { "type": "ImageObject", "contentUrl": "YuMofficeMPI.jpg", "mediaType": "image/jpeg", "meta": { "inline": false } } ] }, { "type": "Paragraph", "id": "p25", "content": [ "There are so many other things left in these folders, in his office. It’s a beautiful\nlarge office, with a lot of light, windows overlooking the central Münsterplatz,\na beautiful place from where to watch the ending of a world (Figure ", { "type": "Cite", "target": "S0-F7", "content": [ "7" ] }, ").\nAll of this will be gone soon, though I am trying to help making it\ndigitally available at some point in the future. Among the things I keep finding: a typewritten manuscript\nof the Strugatsky brothers (which appears to be unpublished to my inexperienced eyes),\nwritings on a variety of subjects (including the original manuscripts of several\nessays of [", { "type": "Cite", "target": "bib-bib24", "content": [ "24" ] }, "]), and especially poetry, his own as well as his translations\nof other poets. Poetry is everywhere: it forms a pervasive ubiquitous subtext both in and\naround his mathematical thoughts." ] }, { "type": "Paragraph", "id": "p26", "content": [ "I came across some of his old writings in linguistics\n(Figure ", { "type": "Cite", "target": "S0-F6", "content": [ "6" ] }, "). I often go by as a linguist these\ndays, but our paths to the subject have been very different and in many ways complementary.\nHe wrote about glottogenesis and protolanguages (like the hypothetical Nostratic) and about\npsycho-linguistics. He was interested in mathematics as language. I have been interested\nin language as mathematics, but he often voiced criticism of generative linguistics, as\nan excess of abstraction away from the more intricate specific functioning of languages.\nOn Christmas day of 2015, as Yuri and I met to work on our own joint paper in\nlinguistics [", { "type": "Cite", "target": "bib-bib29", "content": [ "29" ] }, "], I received an email from Chomsky about my\nwork on\nsyntactic parameters. ", { "type": "Emphasis", "content": [ "Speak of the devil…" ] }, " – I told Yuri as I showed it to him. It remained an\ninside joke between us, whenever the topic of linguistics came up in our discussions. In another\nof those uncanny Jungian coincidences, Noam reappeared in my life on the same\nday when the MPI held the memorial service for Yuri, in January,\nto start our currently ongoing linguistic\ncollaboration. Months after Yuri’s death, it is still impossible for me to return to thinking about mathematics.\nHe was in every part of mathematics I ever happened to think about (gauge theory,\nnoncommutative geometry, motives, mathematical physics…), and now everything hurts.\nI always remained an outsider to the world of mathematicians and\nthe one who was able to make me feel at home there is now forever gone.\nI will return to it one day, probably, but right now I just cannot. By a strange twist of fate,\nthe formal mathematical abstraction of generative linguistics that Yuri\nused to criticize turned out to be now the only thing that made it possible for me to\ncontinue working during the past few months." ] }, { "type": "Paragraph", "id": "p27", "content": [ "I’ll take the liberty of ending this piece on a personal note.\nThis is the third time I am asked to write some form of obituary, reminiscence, tribute,\ncommentary on Yuri’s death: making a public spectacle\nof the pain of losing the closest friend I ever had, navigating the acceptable boundary of words.\nIt hurts, every time more. Because I miss him\nmore and more as the months are passing. Because I am here trying to invent\nimplausible motivations for why the show must go on, for why I ought to keep doing\nthe things I used to do, without a main reason for why I did them. I am not requesting that\nyou would all stop asking me: I understand it that, as a long term collaborator and close personal\nfriend, it is my duty to do this kind of writing in his memory.\nI would just like you to reflect on how\nduty becomes a proxy for pain\nwhen the latter has no proper venue of expression, and on how much we lose as a\ncommunity if we try to separate our mathematical achievements from the\ndeeper core we stand on, the value of our humanity." ] }, { "type": "QuoteBlock", "content": [ { "type": "Paragraph", "content": [ { "type": "Emphasis", "content": [ "ἀλλά με κακκῆαι σὺν τεύχεσιν, ἅσσα μοι ἔστιν" ] }, ", \n", { "type": "Emphasis", "content": [ "σῆμά τέ μοι χεῦαι πολιῆς ἐπὶ θινὶ θαλάσσης" ] }, " \n(Homer, Odyssey, XI, 74–75)", { "type": "Note", "id": "idm408", "noteType": "Footnote", "content": [ { "type": "Paragraph", "id": "footnote2", "content": [ "But burn me with my armor and all my\nbelongings, and drop a sign of me on the shore of the grey sea." ] } ] } ] } ] }, { "type": "Paragraph", "id": "authorinfo", "content": [ "\nMatilde Marcolli is currently the Robert F. Christy Professor of Mathematics and Computing and Mathematical Sciences at the California Institute of Technology.\nShe studied theoretical physics at the University of Milano (1988–1993) and mathematics at the University of Chicago (1994–1997).\nShe worked at the Massachusetts Institute of Technology (1997–2000), the Max Planck Institute for Mathematics (2000–2008), the California Institute of Technology\n(2008–2017 and again from 2020),\nthe University of Toronto (2018–2020), and the Perimeter Institute for Theoretical Physics (2018–2020).\n", { "type": "Link", "target": "mailto:matilde@caltech.edu", "content": [ "matilde@caltech.edu" ] } ] } ] }