Mikhail Aleksandrovich Shubin

December 19, 1944 – May 13, 2020

August 5, 2020

Mikhail Aleksandrovich (Misha) Shubin, an out-standing mathematician, Fellow of AMS and Emeri-tus Distinguished Professor of Northeastern Univer-sity passed away after a long illness on May 13,2020. He was born on December 19, 1944 in Ku-jbyshev (now Samara) in Russia and brought up byhis mother and grandmother. His mother, MariaArkadievna, had graduated from the Departmentof Mechanics and Mathematics (mech-mat) of theMoscow State University (MSU) and became an engi-neer and later, after defending her PhD an AssociateProfessor at a Polytechnic Institute.

Initially Misha, who had an absolute pitch, wasmostly interested and planned his career in music.In high school, however, he got involved with mathe-matics, competed successfully in olympiads, and then

decided to become a mathematician. He was admit-ted to mech-mat of MSU in 1961 and became a pupilof a renowned expert in PDEs M. I. Vishik. He de-fended in 1969 his PhD Thesis, devoted to the the-ory (including index) of matrix-valued Wiener-Hopfoperators. In 1981 he defended his Doctor of Sci-ence degree (an analog of Habilitation), addressingthe theory of operators with almost periodic coeffi-cients, where he is one of the leaders.

Since then, M. Shubin had ventured successfullyinto a variety of areas of mathematics , demonstrat-ing a remarkable width of interests and expertise. Inhis papers and books (totaling to around 140), he hasmade major and influential contributions to a vari-ety of areas, including (but not limited to) operatortheory, spectral theory of differential and pseudodif-ferential operators (especially operators with almostperiodic and random coefficients), theory and appli-cations of pseudodifferential operators and their dis-crete analogs, microlocal analysis, geometric analysisand analysis on manifolds (e.g., spectral theory onnon-compact manifolds and Lie groups, index theory,general Riemann-Roch theorems, and invariants ofmanifolds), integrable systems, non-standard analy-sis, etc. Besides his many solo publications, he has co-authored works with his students, younger colleagues,as well as prominent experts, such as M. Gromov,V. Kondratiev, V. Maz’ya, S. P. Novikov, S. Sunada,and others. For more details, one can consult withthe memorial article [2].

Besides manifold major research papers, Misha hasalso co-authored several important survey articles.His book “Pseudo-differential Operators and SpectralTheory” [4], first published in Russian in 1978, went

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through several English editions and still remains apopular source for studying microlocal analysis. An-other remarkable book “Schrodinger Equation” [1],which he co-authored with F. Berezin, is also a wellknown and in some ways unique source. The AMShas just published his textbook “Invitation to Par-tial Differential Equations” [6], which is a revisedand extended (edited by his colleagues M. Braver-man, R.McOwen, and P. Topalov) version of his lec-tures given at the Moscow State University [5]. Hespearheaded and edited Russian translations of thefundamental books by F. Treves and L. Hormander.

Misha had been participating in the famousI. M. Gel’fand’s seminar from 1964 till his move tothe USA in the beginning of 1990s. His notes for25 years of the talks at the seminar have been, withthe financial support from the Clay Institute, madeavailable on the internet [3].

Misha Shubin was a remarkable lecturer andteacher, who has influenced lives and careers of manyyoung mathematicians (from high school to graduateschool and beyond). He was broadly educated, cheer-ful, and warmhearted person. In the 90s and 80s hehelped several Soviet mathematician “otkazniks.” Healso fought against notorious discriminatory admis-sion practices at the Department of Mathematics ofthe MSU.

The memory of Misha Shubin, a mathematician,teacher, and a wonderful human being, will stay inthe hearts of his colleagues, students, and friends.

Maxim Braverman

I first met Misha Shubin when I was a graduatestudent in Tel-Aviv. Misha visited Israel and I askedhim, as I understand now, a very basic question.Misha answered the question and gave me some refer-ences. To my surprise, the next morning he broughtme several pages of neatly written hand notes witha review of the subject which even contained somecomplete proofs! “I looked through the references Igave you,” - said Misha, - “And realized that theydon’t explain what you need well enough. So I wroteit down.”

A few days later when we went to the old city ofJerusalem I learned that Misha’s thoroughness goes

far beyond his professional life. I often took ourguests to such tours and considered myself quite adecent amateur guide, but it turned out that Mishahad done his homework. The guide-tourist relation-ship was almost completely reversed and my futuretourist friends benefited greatly from the informationI learned on this trip.

Several years later we were working together ona paper about self-adjointness of Schrodinger opera-tors. At some point I wrote a 12 pages long draftand presented it to Misha. He seemed to like it butsuggested working on some additions and generaliza-tions. This happened again and again. Whenever Isuggested to wrap things up and publish the paper,Misha answered: “but it would be such a service tothe mathematical community to analyze this case aswell”. It was not just a phrase, he really cared aboutmaking a difference in the community. Half a yearlater we released a 50 page long manuscript whichbecame my most cited paper. Most people who useit, cite one of these extra topics which we added “toserve the community”.

Misha loved books. He had a huge library. Thewalls of his apartment were covered with bookshelves.Once I asked him, how could he find a book among allthese shelves. He suggested that I choose any book inhis library and ask him to find it. Each time I tried,he immediately went to the right shelf, took out thebook, and told me when and why he bought the book,what is significant about it and why it stands in thisparticular shelf.

Misha always took very careful notes of every talkhe attended and any mathematical discussion he had.His notes from the Gel’fand seminar in Moscow arewell known. But there were many more. All of themwere stored in file-cabinets in his office. Many timeswhen we discussed math, he went to these cabinetsand produced his notes from some talk he attended10 years earlier where some of the questions we werediscussing were addressed. Sometimes those were thenotes from my own talks about which I have com-pletely forgotten!

Over time, I learned that math was not enough tosatisfy his curiosity. He wrote a very interesting essayabout his visit to Egypt and conducted nearly pro-fessional studies of whether Richard III really killed

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his nephews and why the Titanic drowned. It wasalways a pleasure to hear him talking about music,linguistics, history, and cultures of different places hevisited.

Misha was always eager to help his colleagues, es-pecially the younger ones. His role in my own careeris impossible to overestimate. When I already be-came an established researcher I mostly worked onthe problems not directly related to Misha’s mathe-matical interests. Still I often talked to him about myresearch. Only when Misha retired and I could nottalk to him anymore, I realized how important thoseconversations were for my research and how muchhelp he was “invisibly” giving to me. I know manymore mathematicians who owe a lot to Misha. Thehelp he was giving to others throughout his career isa legacy in the field of mathematics that rivals hisown research.

Days with MishaArnold Dikansky

I have known Misha for more than 50 years. Hismathematical knowledge was enormous and dedica-tion to his profession – legendary. Misha’s adviceto me when I was working on my PhD disserta-tion was crucial. Several distinguished mathemati-cians describe here his mathematical achievementsand how he influenced his students and many otherresearchers. So, I will speak here only about the cul-tural dimension of Misha’s multidimensional person-ality and about his family, M&M (Misha and Masha).

We shared with Misha love for classical music andespecially for Richard Wagner’s operas. We had at-tended for many years various Ring operas at theMetropolitan Opera in New York. With the samelevel of thoughtfulness as he showed in his pro-fessional life, Misha approached upcoming perfor-mances, diligently studying librettos. When a newproduction of the Ring came out in 2012, I invitedM&M to come to New York, stay with me, and go tothe performance of Die Walkure. It was his last visitto New York, which he loved and especially appreci-ated its multicultural character.

It is interesting that many mathematicians enjoydeeply the classical music. Maybe the reason is that

both music and mathematics touch something that lievery deep in human nature. Both demand an enor-mous amount of time and effort in order to succeed.In his early life, when choosing the profession, Mishawas torn between mathematics and music. He hasachieved a professional summit in mathematics andenjoyed music and art in general as an amateur. Be-sides music, reading was Misha’s another love. Hehad read the complete Shakespeare (in English, aswell as in Russian translations).

I would like to say a few words about Misha’s wifeMasha. I have visited them on many occasions intheir apartments in Moscow and Boston as well as, inthe last several years, in the nursing home. The carewhich she gave to him was beyond what it seemedto be possible. It has lasted through many ups anddowns in his condition, required tremendous perse-verance, the control over her own emotions and reac-tions, and just physical strength.

Arnold Dikansky Professor Emeritus, St. John’sUniversity New York

Memories of MishaLeonid Friedlander

Probably I met Misha first time in 1975. I was stillan undergraduate, and I was attending Misha’s lec-tures on microlocal analysis. I remember him goingin small details through all the calculus, and then hespent some time on the index theory. At first, hisstyle looked too pedantic to me; only later I appreci-ated how important the details are. That was one ofthe classes, on the basis of which he wrote his book”Pseudodifferential Operators and Spectral Theory”,which is still one of the major textbooks on microlocalanalysis. A bit later, I started going to the SpectralTheory seminar that he was running in the MoscowState University. I spoke there several times. Quiteoften, Misha would ask a question that looked obvi-ous. My first reaction then was: every idiot knowsthat; and then, in a couple of moments: I have noidea.

In 1979 I applied for emigration, and I was finallyallowed to leave the Soviet Union in 1987. That 8years of my life were not easy; most of the time Idid not have a regular job. What made these years

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not only bearable but actually quite good was anenormous moral support that I received from my col-leagues. In Misha’s case, the support was not onlymoral. He arranged for me to translate a book intoRussian; he himself edited the translation. Helpinga political undesirable was not a neutral act. Mishacould have big problems as a result of that, and heknew it well. As I learned later, I was not the onlyperson who got a helping hand from him. The hon-orarium that I received was substantial enough tosolve my financial problems. Actually, when I wasfinally leaving, I paid the compulsory renouncing-of-citizenship fee from that money. Translating thatbook also gave me an opportunity to work closelywith Misha. I was coming to Misha’s apartment reg-ularly, and we would go together through the newportion of the translation (there were no comput-ers, no e-mail, everything was handwritten). Misha’smotto was that the translation must me everywherecorrect, regardless of whether or not the original is.If there is a mistake, it must be corrected, if there agap, it should be filled. After two or three hours ofwork, usually at 10 or 11 p.m., we would go to thekitchen, and Masha, Misha’s wife, would put khacha-puries (Georgian bread filled with cheese) into theoven, and sometimes one or two Misha’s friends wholived nearby would come, and, with tea and khacha-puries, we would be talking till after midnight aboutfiction, politics, linguistics,... Misha had unlimitedcuriosity. That was true in mathematics, that wastrue in everything else.

L. Friedlander and M. Shubin

Several years later, in May of 1991, Misha came tovisit UCLA, where I was a post-doc at that time. Wewent together to San Diego. On the way back, afterhaving visited Anza Borrego, rather late at night, Iwas driving on an almost empty freeway, with mywife sitting next to me, and Misha on the back seat.For half an hour or so, Misha was absolutely silent;we thought that he was sleeping. Then, he said: Ihave been looking at the road and thinking: how dothey manage to maintain it so that all the bulbs areworking. What bulbs? Turned out that he thoughtthat reflectors are electric bulbs (there were no reflec-tors on roads in Russia, so he saw them first time).What he saw did not make sense, and he was tryingto solve the problem.

Misha had a strong moral compass. He believedthat certain things are worth fighting for. Back inthe USSR, he was not a dissident, but showing in-dependence was also fighting the system. As a re-sult of that, he had serious career problems. On theother hand, he had tolerance to human weaknesses.I remember well a conversation that we had in hiskitchen in Moscow. We were talking about one spe-cific case, and I was a hardliner, as most young peo-ple are. Misha was trying to convince me that, inthe particular situation, one should show more un-derstanding and be more forgiving. We respectfullydisagreed then. Now, after almost 40 years, I thinkthat he was probably right.

I met Misha many times and in different places.When I was on Sabbatical in MIT in 2004–2005, wetalked a lot. He invited me to give a talk at the BasicNotions Seminar that he was running in the North-eastern University. That was his wonderful invention.I spent much more time preparing that talk than Iwould usually spend for a research talk. Last time Isaw him in 2009 at the conference for his 65th birth-day. I was fortunate to have known Misha and to behis friend.

A few words about Misha ShubinMisha Gromov

What was striking in Misha were his unlimitedfearlessness and his insatiable love for knowledge. His

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eyes were radiating with light when he was comingacross a new idea, especially a mathematical one.

His knowledge and his understanding of mathemat-ics was extraordinary, his sensitivity to everything inthe world was astonishing and he generously sharedhis knowledge and his ideas with everybody.

I cherish the memory of spending time in his com-pany, I wish there were more such people as Mishaamong us.

My friend Misha ShubinVictor Ivrii

I met Misha Shubin for the first time in early 70swhen I, a graduate student of the Novosibirsk StateUniversity, came to the Moscow University to giveseveral talks. Misha had already defended his dis-sertation; we instantly became friends. At that time“Vishik’s seminar” was the main center for Microlo-cal Analysis in USSR, and Misha was a very promi-nent member of this seminar.

I have visited Moscow many times after that,from Novosibirsk and later from Magnitogorsk, of-ten stayed in Misha’s apartment; we met at severalconferences in the Soviet Union and abroad. In par-ticular, in October of 1981 we went together to theconference “Differential Geometry and Global Anal-ysis” (Garwitz, GDR=East Germany). We were partof the Soviet delegation (in the USSR that was theway to send scientists abroad). After many rejec-tions, it was my first visit abroad, and one of Misha’sfirst such trips, after many rejections as well. At thattime, a simple conference trip abroad required a verylong chain of permissions, and majority of academicsnever ventured abroad. And finally we got one! Wefelt like inmates let from our cell to the inner yard.It was a kind of a test: were we “politically mature”and could we be trusted to visit countries which werenot the part of the Eastern Block? We failed thistest miserably and due to a report written by twomembers of the same delegation (one of whom wasan official snitch) that would have been our last tripbeyond borders of the USSR, if not the Perestroika.

That was a large international conference, at-tended by a number of our colleagues from WesternEurope. We had known one another scientifically,

but never met in-person (one should remember thatthere was no email available for us, leave alone Skypeor Zoom). .No wonder that, unlike some more seniormembers of our group who kept their own companyand spoke exclusively Russian, we talked a lot andnot only about mathematics. We spoke English, notunderstood by our “colleague,” who asked “What areyou talking about?” We responded “It is of no inter-est to you, it is mathematics”.

In the early 1990s we were roommates at the Insti-tut des Hautes Etudes Scientifiques. For both of usit was the first visit beyond Eastern Block (which al-ready had collapsed). Everything was new and veryexciting, from streets of Paris to museums. Misha’sbiggest complain was “We will leave before we tasteone tenth the yummy staff in the grocery stores!”Later I visited him and his family several times inBoston and stayed at his place. He also was visitingme and my family in Toronto, and also stayed in ourhouse.

Misha has played a very important role in mymathematical life: it was he and Boris Levitan whosuggested that I try to prove Weyl’s conjecture in1978.

Misha was a remarkable mathematician, verybroad, with a great mathematical taste. He was col-laborating with many mathematicians on the largevariety of topics. His book “Pseudodifferential opera-tors and spectral theory” played very important rolein promoting Microlocal Analysis and related top-ics in Soviet Union and despite availability of otherbooks it has become one of my favorites, as well ashis and Felix Berezin’s book “Schrodinger Equation”.He also was an editor of several volumes of the series“Encyclopaedia of Mathematical Sciences. ModernDirections”.

Due to space limitations, I will mention just a cou-ple of his many results. He and Vladimir Tulovskyhave developed the Method of the almost spectral pro-jector in the theory of spectral asymptotics, whichwas in some sense an intermediate between Weyl’sand Courant’s variational approach and the Taube-rian approach due to T. Carleman. In this method apseudodifferential operator was directly constructedthat was close to the spectral projector.

In his paper with D. Schenk “Asymptotic expan-

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sion of the state density and the spectral functionof a Hill operator” the complete spectral asymp-totics was derived for the integrated density of statesfor one-dimensional Schrodinger operator with peri-odic potential. This result was much later general-ized to multidimensional case by L. Parnovski andR. Shterenberg. The most amazing thing there wasthat the complete spectral asymptotics does not con-tradict to the fact that for one-dimensional operatorswith generic potentials all spectral gaps are open (butdue to complete asymptotics these gaps are very nar-row and asymptotics itself cannot not be differenti-ated).

Yuri Kordyukov

I first met Misha Shubin back in 1980, when Iwas a sophomore at the Department of Mechanicsand Mathematics of the Moscow State University(MSU). At that time he was, of course, not Misha forme, but Mikhail Aleksandrovich. He became Mishamuch later, sometime in the late 1990s. Shubin leadour recitations on the ordinary differential equationscourse. It was the time to choose a research advisor.I had been thinking for a long time, attended lecturesof various mathematicians, and in the end my choicesettled on Shubin.

As far as I understand now, at that time Shubinhad just defended his Doctor of Sciences dissertationand was looking for new directions of research. In1981, he wrote a paper on the spectrum distributionfunction for transversally elliptic operators on a com-pact manifold equipped with a Lie group action. In1983, Shubin wrote a paper with his graduate stu-dent Meladze devoted to pseudodifferential calculuson Lie groups. At that time he had a group of stu-dents working on analysis on noncompact manifoldsand index theory. For instance, Alexander Efremovstudied index theory on coverings of compact mani-folds, Dmitry Efremov - spectral theory on hyperbolicplane. Brenner worked on analysis on manifolds withboundary. The weekly seminar led by Shubin meton Fridays. We studied the classic works by Atiyah-Bott-Patodi, the newly emerging works of Bismut,the book by Guillemin and Sternberg on geometricasymptotics, etc. I remember Shubin giving a course

on the Atiyah-Singer index theorem. But his researchinterests were much broader. At the same time, inthe first half of the 1980s, he published his work withBondareva about solutions of the Cauchy problem fornonlinear integrable equations, a survey with Zvonkinon nonstandard analysis, papers with Schenk on com-plete asymptotic expansions of the density of statesfor a periodic Hill operator, and many others. He alsoedited the Russian translations of the two-volumeTreves monograph, the four-volume monograph byHormander, and the book by Rempel and Schulzeon the index theory for elliptic boundary value prob-lems. I have heard that after writing his book onpseudodifferential operators, he began writing a bookon Fourier integral operators, and had already writ-ten 150 pages. But when he learned about the pub-lication of Treves book, he stopped his own writingon the subject and with great enthusiasm switchedto translating the Treves book. Shubin had alwaysbeen very busy and worked hard. It was his naturalstate. He expected the same from me.

As a subject for my research, Shubin suggested tostudy analogs of his results on transversally ellipticoperators for the simplest example of a noncompactLie group — the group of real numbers. After a while,during our discussions, the idea came up to look atthe 1979 paper on noncommutative integration the-ory by Connes, and I switched to studying it. Later,manifolds of bounded geometry came into my view.The results on uniformly elliptic differential operatorson manifolds of bounded geometry formed the basisof the first chapter of my Ph.D. thesis, defended un-der Shubin’s supervision in 1988. The second chapterwas devoted to tangentially elliptic operators on foli-ated manifolds. In 1988, after completing my gradu-ate studies, I left for Ufa, where I found a position atthe Aviation Institute. Shubin was not sure whetherI would continue to work with him, and thus rec-ommended to me some mathematicians with whomI could work in Ufa. In the end, I decided to con-tinue the research that I had started under Misha’sdirection. It was already the beginning of the 1990s,the period of big changes in Russia. In 1991, Shubinwent to MIT for a year and then found a permanentposition at the Northeastern University.

After leaving for the United States, Shubin would

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rarely come to Moscow. I had met him several timesin his Moscow apartment in the 1990s. In 1998, ourcollaboration resumed to continue his work with Gro-mov and Henkin on L2-holomorphic functions on cov-erings. Regretfully, this project has never been fin-ished. In 2001, Shubin found an opportunity to in-vite me to Boston for a month. This was my firstacquaintance with the United States, and Misha washappy to introduce me to various nuances and pe-culiarities of American life. In Boston, he invitedme to join his project with Mathai to study theasymptotic behavior of the spectrum of a periodicmagnetic Schrodinger operator in semi-classical limit.During my stay at Boston, within the framework ofthis project we managed to solve a problem that hadapplications to the quantum Hall effect. These re-sults were published in 2005 in the Crelles Journal.It was our only joint paper with Misha. This re-search triggered my continuing interest in the mag-netic Schrodinger operators. Then we parted ourways again. Shubin was very involved in his collab-oration with Kondratiev and Maz’ya on the spectraltheory of the magnetic Schrodinger operator. When Iwas back to Boston in 2001, he showed me some partsof these works that involved very subtle analytical ar-guments. I last met Shubin in 2009 in Boston at theConference in honor of his 65th birthday. At thattime he was already seriously ill, and we did not talkmuch about mathematics.

Shubin had a great influence on my life, taught mea lot. He has largely determined my path in mathe-matics. He shared with me his vision of mathematics,value criteria of mathematical results, his principlesof joint scientific research and interaction with othermathematicians. I remember that in my second yearat MSU, when I was just choosing my path in mathe-matics, he explained to me the unity of mathematics.He said that mathematics is one big interconnectedarea. Therefore, it does not matter in which part ofmathematics to begin your scientific activity. In thefuture it will be possible, if necessary, to move fromone part to another, with the exception of, perhaps,only a few isolated islands, and the main goal is notto get stuck on one of them. He had a huge libraryof mathematics books (about 5 thousand, as far asI remember), which he had collected back in the So-

viet times. He has managed to move it to America.When I was in Boston in 2001, I stayed at Misha’splace and had accommodation in his library (whichwas also his study). I lived for a month in the midstof all these wonderful books. He told me; “Yura!Why do you need to go to the university? You haveeverything here for work - a computer, books, the In-ternet.” He was in the habit of working at home, de-veloped through long years of work at Moscow StateUniversity. MSU professors did not have individualoffices, and thus they usually worked from home. Hetaught me the skill of making presentations (“youshould write on the board, starting in the upper leftcorner” and so on), explained the cut and paste tech-niques for writing mathematical texts in that distantpre-computer time. He also helped me with variouslife tasks (be it job search after graduation or choos-ing a typewriter). In his turn, he had mentionedmany times that he had learned a lot in mathematicsfrom me. He has never raised his voice, always triedto be extremely tactful and delicate, not to imposehis opinion. He was principled, honest, demanding ofhimself, very humble and sincerely devoted to math-ematics. So he will remain in the memory of manypeople who knew him.

Misha Shubin, dear friend and colleague

Peter Kuchment

It is hard to believe that Misha Shubin, such a vi-brant, brilliant, and friendly person, is not with usanymore.

We have met probably somewhat more than a half-century ago, when I was still an undergraduate, andhe - a graduate student. We met at one of the firstfamous in the former Soviet Union Voronezh Win-ter Mathematical Schools.These were unimaginablefeasts of mathematics, involving hundreds mathe-maticians (from undergraduate students to the lead-ing experts) from the whole Soviet union. The discus-sions among younger participants like us were veryintensive, every lecture being taken apart. This iswhere many friendships like ours with Misha andcollaborations started. Ours has lasted throughouthalf of a century and different cities and countries.Back in Russia, I visited Moscow and Misha visited

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Voronezh with seminar talks and discussions. Wehave never done any joint work, but our mathemat-ical paths were often closely parallel. For instance,at about same time when Misha worked on almost-periodic PDEs, I did the periodic ones; then thisrepeated with us both looking at operators on ho-mogeneous spaces, discrete problems, etc. His workhas influenced most of mine. His graduate schooldays paper on holomorphic projections on holomor-phic sub-bundles of a trivial Banach bundle, whichused intricate and recent at that time results of SCVgoing back to the work of H. Grauert, has been usedby me and many other operator theorists since. Ihave learned a lot from his work on almost-periodicPDEs. In 1975, being on sabbatical I attended hiswonderful lectures on microlocal analysis at MoscowState University. Later on, they turned into his fa-mous book [4]. The lectures were so meticulouslyprepared that it is no wonder they could be turnedinto a book rather easily. This was my practicallyfirst significant exposure to microlocal analysis, pseu-dodifferential operators and such, which turned outto be crucial for the work I started doing a coupleof years later. I still rip the benefits of his lecturesin most areas of my interest, especially in medicalimaging and PDEs. Very recently I worked with mystudent on extending further Gromov’s and Shubin’sRiemann-Roch type results. Moreover, Shubin’s in-fluence on my growth as mathematician and my re-search extended much further than direct links that Ihave just indicated. Our discussions, his lectures andhis publications have taught me a lot (and I do notthink I was his best student).

When I and my family were leaving the fSU in 1989(see the photo of the farewell party),

S. Orevkov, P. Kuchment, V. Lomonosov,M. Shubin, V. Lin, E. Landis, L. Posicelskii

we all thought that we were parting forever, withno chance to meet ever again. Fortunately, the timeshave changed, and Misha also moved to the USA. Wekept communicating a lot and visiting each other, mecoming to Boston, and him visiting Wichita, KS andCollege Station, TX, where I have worked.

Misha has played a very significant role in my life inmany other regards. He was an extremely broadly ed-ucated person and was happy to share his knowledge.I have read quite a few fiction and non-fiction booksat his recommendation. I had not studied English be-fore 1985 at all (my foreign language was German).In 1985, during my another sabbatical in Moscow,Misha recommended me a wonderful immersion En-glish class, which he had taken previously. He alsomade sure I could get into it (it required a recommen-dation). This has played a major role in my futurelife, especially after my emigration to the USA.

In difficult and corrupt Soviet times, he was oneof the staples of moral fortitude, which often did notmake his life any easier, to say the least. He wasalways ready to help people in difficult circumstances.I remember once, when I lived in Kansas and he wason a business trip to Mexico, he called me from thereto give an advice about universities choice for mydaughter, who was graduating from the high schoolat that time.

Misha Shubin was a wonderful person in all re-gards: as a mathematician, teacher, writer, and mostimportantly, as human being. We all will miss himthoroughly. We are grateful to the fortune for letting

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us know him.

Misha Shubin, An Exceptional EruditeVladimir Maz’ya

It was by pure chance that Misha Shubin and Ihappened to share a compartment in a night trainfrom Moscow to Ruse (Bulgaria) in 1975. We bothwere participants at a conference and for both of usit was the first trip abroad. As novices in travelingoutside the USSR, we could only be allowed a trip toa country within the Soviet bloc.

The train trip lasted almost three days and we hadplenty of time to speak about mathematics as well asabout our life in two different mathematical commu-nities. He was telling me about the Division of Dif-ferential Equations at Moscow University, where heworked, and I told him about my research position atthe Research Institute within the Math Departmentof Leningrad University, at the laboratory headed bySolomon G. Mikhlin. I reminded Misha that I hadbeen present at two of his talks at Vishik’s seminarin Moscow. Seven years younger than me, Misha sur-prised me by his broad knowledge in mathematics.

As it often happens, our acquaintance was warmedby shared food. There was no restaurant on the trainand we offered each other ”delicacies” like cookedeggs, boiled chicken, tomatoes and cucumbers. Theonly thing provided by the train hostess was hot teain mornings and evenings. It is interesting that I re-member this train trip with Misha more vividly thanthe conference we attended.

Later on, we would meet regularly at the Petro-vsky winter conferences in Moscow, at which I regu-larly participated. However, a close contact becamepossible only much later, when we both emigrated.

In 2003, Bob McOwen invited me to spend sixmonths as a visiting professor at the NortheasternUniversity in Boston. My wife Tanya had a lot ofteaching duties, while I was privileged to give justa special topic course for graduate students for twohours a week and participate in the seminar, where Igave several talks.

Besides doing research with Bob McOwen, whowas the Chairman at that time, very soon I becameinvolved in the following interesting mathematical

problem. Let −∆+V be the Schrodinger operator inL2(Rn), where V ∈ L1

loc(Rn) is bounded from below;n ≥ 2. A.M. Molchanov discovered in 1953 that forsome constant γ > 0 the following condition is equiv-alent to the discreteness of the spectrum: for everyd > 0

infF

∫Qd\F

V (x)dx→∞, Qd →∞, (1)

where Qd are open cubes with edge length d, andthe compact sets F ⊂ Qd are negligible in the sensethat cap(F ) ≤ γ cap(Qd). I.M. Gel’fand had raised aquestion about the best possible constant γ. Mishaand I answered this question by proving that γ canbe taken arbitrarily in (0, 1). Moreover, we showedthat the negligibility condition can be weakened to

cap(F ) ≤ γ(d) cap(Qd) (2)

and described all admissible functions γ : (0,∞) →(0, 1). They must satisfy

lim supd→0

d−2γ(d) =∞. (3)

All conditions (1) with functions γ satisfying (2)-(3) are necessary and sufficient for the discreteness ofthe spectrum of the Schrodinger operator. Our solu-tion of this fascinating problem appeared in ”Annalsof Mathematics” in 2005.

Left to right: V. Maz’ya, I. Verbitskii, R.McOwen, M. Shubin

Another question we succeeded in answering becamethe topic of our paper “Can one see the fundamen-tal frequency of a drum?” (Letters in MathematicalPhysics, 2005). The main theorem here states thatthe bottom of the Dirichlet Laplace spectrum λ(Ω)

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for an open set Ω ⊂ (Rn) is equivalent to r−2Ω , where

rΩ is the so-called interior capacity radius of Ω. Thereason for the resemblance of the article’s title to thefamous Marc Kac’s problem about isospectral drumscan be explained as follows. If an eye could filter outsets of “small” capacity, then one could recover “bylooking” the lowest Dirichlet Laplacian’s eigenvalue,or at least get its reasonably good estimate.

When Volodya Kondratiev, who worked at thesame division at Moscow State University whereMisha used to work, came for a short visit to Mishain Boston, three of us became involved in a problemon spectral properties of the magnetic Schrodingeroperator. The first publication by the three of us ap-peared a year later in “Communications in PDEs.”The second one was finished later and published in2009.

We worked with Misha intensively for six months,meeting both at the Department of Mathematics andat his place on weekends. My wife Tanya and I rentedan apartment within five minutes’ walk from Misha’shome. The proximity to Misha’s place was the onlyadvantage of our rather modest apartment, whichwas made tolerable with the generous help fromMasha Shubina (Misha’s wife), who offered kitchenutensils and other necessities.

The six months in Boston were fruitful. Togetherwith Misha we wrote four articles. However, whenwe left Boston, the distance and the lack of technicalmeans of communication like Skype, made our fur-ther collaboration more difficult. We spent a lot oftime on the phone and exchanging emails, but ourcontacts somehow faded away. When we discussedpossible directions of further collaboration with thestarting point being our “Annals” paper of 2005, itbecame clear that we were rather different in our pref-erences. Misha, being an exceptional erudite, wantedto widen the scope of our efforts, while I usually pre-fer going after necessary and sufficient conditions ina problem. This difference in tastes was difficult toovercome. It is a pity that we have not had an op-portunity to work together since then.

Memories of MishaRobert McOwen

Department of Mathematics

I first met Misha Shubin in Spring of 1987 at amathematics conference in Oberwolfach, Germany.Both of us were giving talks at the conference. Idont remember what topic Misha talked about, butI remember him most clearly from one of the eveningsocials towards the end of the conference. It was hisfirst time out of the Eastern Bloc, and he had theopportunity to meet in person many mathematicianswhom he had only known by their work. I rememberhim getting very emotional and, with a big smile onhis face, saying I am so happy!

It was five years later that Misha joined the MathDepartment at Northeastern University in Boston.I remember his arrival having a big impact on ourresearch profile and our graduate program. In Fall1993, he started the Basic Notions Seminar. Topicsof seminar talks varied, but the rule that Misha en-forced relentlessly was that all stated mathematicalresults must be proved from elementary principles.Speakers who gave a series of talks were subject toanother rule that Misha also enforced: subsequenttalks must review all results from previous talks be-fore proceeding to new material. The third rule thatMisha imposed was that questions were strongly en-couraged and must be answered in full detail. Mishausually set the pace himself by asking lots of ques-tions that slowed the speaker down. Needless to say,not a lot of material was covered in any given talk,but the audience always came away from the BasicNotions Seminar with a firm understanding of whatthey had heard.

In 1995, my colleague Christopher King and I co-organized with Misha a Special Session on “PDEs inGeometry and Mathematical Physics” for an AMSConference that was held at NU. Misha had a hugelist of names to suggest as potential speakers, and wehad to select carefully: they were all eager to cometo NU to give a talk in a Special Session that he hadco-organized. Misha served as Graduate Director ofthe Math Department from 1995 to 1997. During thisperiod of time, he introduced many new courses toour graduate curriculum, like a course on the Atiyah-Singer Index Theorem. I think that the course wasonly taught once, by Misha himself, but it shows theambitious standard that he aspired to for our gradu-ate program.

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Misha became a Matthews Distinguished Univer-sity Professor at Northeastern in 2001. He was thefirst Distinguished Professor appointed in the Math-ematics Department and remained the only one untilAndre Zelevinsky was given the honor after his deathin 2013. Misha earned the honor by being a highly re-spected researcher and author of many published ar-ticles and books. I have always been impressed by thenumber of mathematicians who cited his 1978 book,Pseudo-differential operators and spectral theory, ashaving a big impact on their research. Consequently,I was honored when the AMS asked me to help bringout the textbook on partial differential equations thatMisha had been working on for the last years of hisproductive life. I had not seen the book before, butas I looked through it, I was impressed not just withthe content but with the style of writing: he pro-vided rigorous proofs but written in a friendly waythat made them easily understood. I felt it was im-portant that the book be published, so enlisted theassistance of my colleagues Maxim Braverman andPeter Topalov. I was glad that we were able to com-plete the editing process and have Invitation to Par-tial Differential Equations appear in the AMS list ofpublications slightly before Mishas death this year. Ihope it will serve as a lasting reminder of the gentleman who was an inspirational figure in mathematics.

Coincidence or DestinyIn memory of Misha

Toshikazu Sunada

For Fortuna, the goddess of destiny in Roman reli-gion, there exists no difference between “coincidence”and “destiny,” but for human being who cannot judgethe appearing spots of a die in advance, the encounterwith people, for instance, is neither more nor lessthan a “chance”.

It was in 1993 when I met Misha for the first time,on the occasion of the Japan-U.S. Mathematics Insti-tute (JAMI) Conference “Zeta functions in numbertheory and geometric analysis” held at Johns Hop-kins University. I already knew him by name as aleading figure in the field of analysis, especially, of el-liptic PDEs pertinent to physics and geometry. Oneday, in the tea break of the conference, he approached

me, and introduced himself with a gentle and cordialway of talking. Although I cannot recall exactly thecontent of our conversation then, it is certain, how-ever, that we discussed some mathematics and thathis interest in part turned out to have something incommon with mine. According to my poor mem-ory, our common interest was the spectra of “mag-netic Schrodinger operators” in both continuous anddiscrete cases. His approach was analytical in na-ture, while mine was geometrical, if anything. In anyevent, I was pleased to know that Misha had interestin my research, for I thought of what I was studyingat that time as being isolated in the mathematicalcommunity in Japan. Indeed, our conversation gaveme a great deal of motivations and encouragement.But, that is not all; I was attracted by his personalityduring the course of the conversation. He really is acaring and warm-hearted person.

The next chance to meet him came in 1996. Weboth participated in the symposium on “Partial Dif-ferential Equations and Spectral Theory” held atDurham University, UK. Taking this opportunity, wediscussed not only on mathematics, but also on non-mathematical subjects. I was very much impressedwith his broad knowledge of humanities in general.Above all, he showed his keen interest in Japaneseculture. He has read some Japanese contemporaryliterature, through which he got well acquainted withJapanese mentality and behaviour.

In 2005, I invited him to Japan to do a joint re-search. At that time, I was trying to make the classi-cal T 3-law of specific heat rigorous by means of dis-crete geometric analysis. During his stay in Tokyo,where he enjoyed longed-for Japanese life, we couldaccomplish a joint work entitled “Mathematical the-ory of lattice vibrations and specific heat” that waspublished in Pure and Appl. Math. Quaterly, anewly established journal. It is my great honour tobe a collaborator of Misha.

The last chance to have met him was in 2009. Iwas invited to the Conference in Honour of the 65thBirthday of Mikhail Shubin, “Spectral Theory of Ge-ometric Analysis” held at Northeastern University,Boston. I could meet his beautiful family in the birth-day party at his house. He looked happy and con-tented, surrounded by his family and many friends.

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Eleven years since then, I still remember his warmsmiling face. I wanted to see him again and to have apleasant conversation about a wide variety of things,but his illness precluded me to do it. Last May, I wasdeeply saddened when I heard that he passed away.I really bear a grudge against Fortuna for not givingme the chance to fulfill my desire.

To conclude this eulogy, I would like to add a fewstatements, in relation to his personality. On 11th

March, 2011, Japan experienced Great East JapanEarthquake. The tsunami triggered by the earth-quake killed more than 15000 people along the eastcoast of Tohoku region. Misha was worried about myfamily (in my memory, he gave me an internationalcall from Boston). Fortunately, we were not affectedby this disaster since Tokyo is far away from Tohoku.What I wish to say is that he not only confirmedmy family’s safety, but also made a donation to theMathematics Institute of Tohoku University to assista few teaching staff affected by this disaster (I hadbeen a member of the institute for ten years). Themembers of the institute very much appreciated hiskind offer and heart-warming message. I, too, amthankful to Misha for his solicitude.

Well Misha, my friend and collaborator, your sin-cerity, kindness, warmness, generosity, caring for oth-ers, smiles, humours will stay with me forever. Mayyou rest in eternal peace. I extend my most sincerecondolences to Misha’s family.

Misha Shubin, a few personal memoriesAlexander Zvonkin

Misha Shubin was the most important person inmy life after my family – my wife and children.

At a certain point I was going through a difficultperiod: I had not been doing math for almost tenyears. As a rule, it is next to impossible to restartafter such a long interruption — unless you get helpfrom someone. It was a good fortune for me thatMisha and I became neighbors. It started with smallthings: while taking the same bus quite by chance,we discussed various mathematical subjects. ThenMisha proposed that I translate a French mathemat-ical paper. Then he lent me a book on nonstan-dard analysis, the subject that interested him at that

time. This way, step by step, we finished by co-authoring quite a substantial paper. But that wasnot all: Misha convinced me to start participating inthe weekly seminar headed by I. M. Gelfand, and mynext paper was greatly influenced by the latter.

Later, while visiting the Institut des HautesEtudes Scientifiques in Bures-sur-Yvette, Misha rec-ommended me as one of the possible guests, and thatvisit was absolutely crucial for my life and hugelypredetermined it.

I could easily continue with this list but then itwould be about myself, not about Misha.

Misha was a very good person. I really do not knowhow to say it in a more elegant way. He was justa very good person. I remember once talking witha dissident friend of mine who claimed that in ourcountry of omnipresent denunciations and the over-whelming power of the KGB you could not trust asingle person: anyone could turn out to be a snitch.As a counterexample, I told him that Misha was aperson whom one could totally trust. He thought fora while and then, despite of his habit of always in-sisting on his point of view, said: “I think you areright this time; it is impossible to imagine Misha asan informant.”

Misha saw that my way of writing mathematicalpapers was unprofessional, and he found a way to letme know it without hurting my feelings. He askedme whether I knew Asimov’s stories about robotsand then, proceeding from the famous Three Lawsof Robotics, formulated similar laws of good mathpaper writing:

Three basic principles for writing amathematical paper

1. The text of the paper should be math-ematically correct.

2. The text should be written as rapidlyas possible except when this wouldconflict with the First Principle.

3. One should try to make the text as per-fect as possible (comprehensible, acces-sible, self-contained and complete) ex-cept when this would conflict with thefirst two Principles.

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It was almost forty years ago, but I still keep thesheet of paper with these principles written by him.

Our paths have diverged. Misha moved to Boston,while I moved to Bordeaux. Misha has visited us inFrance five times. Those were purely personal vis-its, though we could not avoid talking about math-ematics. His casual remarks, his questions, or hissurprise at certain phenomena helped me to betterunderstand my own results.

He has never made an impression of being in ahurry, but worked amazingly fast. But no matterhow fast he worked, it was never at the expense ofmeticulousness (remember the first principle: “Thetext should be mathematically correct”).

When we discussed the translation of the above-mentioned French article, he asked me how long Ithought it would take. At that time I had a full-timejob at an applied research institute, but hoped to beable to do the translation in a month. “A month!” –Misha was clearly disappointed. “In my experience,one can translate three pages even on a very busyday”, he said. And he immediately started to apol-ogize saying that he did not want to apply pressure,let it be a month, it’s OK! But I perceived it as achallenge. I told to myself: “24 pages means 8 days.”I did work hard, and eight days later the paper wastranslated. “Good job!” said Misha, and that wasone of the most cherished praises in my entire life.

I will give just one more example. Misha was thechief editor of the Russian translation of the famousfour-volume Linear Partial Differential Operators byLars Hormander – more than two thousand pages ofa dense mathematical text. This is what Hormanderwrote in his Preface to the Russian edition (my trans-lation back from Russian):

The Russian edition contains severalchanges, as compared to the English one. Icorrected the errors that I had discoveredby myself, while very attentive translators,headed by Professor M. A. Shubin mademany important clarifications. I was greatlyimpressed by the meticulous work done bythe translating team and I am grateful fortheir collaboration – thanks to it we willbe able to introduce improvements into

further English editions.

So, I was not the only one being impressed by Shu-bin’s style and quality of work, it also impressed LarsHormander! What more can I say?

Maxim Braverman, Arnold Dikansky, LeonidFriedlander, Misha Gromov, Victor Ivrii, Yuri

Kordyukov, Peter Kuchment, Vladimir Maz’ya,Robert Mc Owed, Toshikazu Sunada, Aleander

Zvonkin

References

[1] Berezin, F. A. ; Shubin, M. A. The Schrdinger equation.Translated from the 1983 Russian edition by Yu. Rajabov,D. A. Letes and N. A. Sakharova and revised by Shubin.With contributions by G. L. Litvinov and Letes. Math-ematics and its Applications (Soviet Series), 66. KluwerAcademic Publishers Group, Dordrecht, 1991. xviii+555pp. ISBN: 0-7923-1218-X

[2] M. Braverman, B. M. Buchshtaber, M. Gromov,V. Ivrii, Yu. A. Kordyukov, P. Kuchment, V. Maz’ya,S. P. Novikov, T. Sunada, L. Friedlander, A. G. Khovan-skii Michail Aleksandrovich Shubin, Russian Math. Us-pehi, to appear.

[3] Gelfand Seminar Notes (taken by M. A. Shubin),https://www.claymath.org/publications/notes-talks-imgelfand-seminar

[4] Shubin, M. A. Pseudodifferential operators and spectraltheory. Translated from the Russian by Stig I. Andersson.Springer Series in Soviet Mathematics. Springer-Verlag,Berlin, 1987. x+278 pp. ISBN: 3-540-13621-5 Second edi-tion. Springer-Verlag, Berlin, 2001. xii+288 pp. ISBN: 3-540-41195-X

[5] M. A. Shubin, Lectures on equations of mathematicalphysics (Russian), Moscow, Mosc. Center. Cont. Math.Educsation 2001, 2003.

[6] M. Shubin, Invitation to Partial Differential Equations,Edited by M. Braverman, R. McOwen, and P. Toplalov,AMS 2020.

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