Zhiyu Zhang

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"My principal guide has been the constant search for a perfect coherence, a complete harmony, that I sensed lay behind the turbulent surface of things, and that I endeavoured to release patiently, tirelessly. A keen sense of "beauty", certainly, guided my instincts and was my only compass. My greatest joy was not so much in contemplating it when it had been brought into the full light, as in seeing it gradually emerge from the cloak of shadow and mist where it liked to hide. Of course, I did not stop until I had managed to bring it into the clearest light of day. I knew then, sometimes, the fullness of contemplation, when all heard sounds contribute to a single vast harmony. But more often, what had been brought into the light immediately became the motivation and means for a new plunge into the mists, in pursuit of a new incarnation of That which remained forever mysterious, unknown --- calling me constantly. " --A. Grothendieck, Récoltes et Semailles.

"All problems in mathematics are psychological". --P. Deligne.

"Since Lang passed away, mathematicians often tell me things like “This problem is too hard,” and so on. But then I think of Lang. I immediately hear, “Stop fooling around and get back to work!” As I start working, I also hear Lang saying “It’s possible, let’s do it, let’s do it right now!”, and then I feel I’ve got to try, as Lang would have done if he were around." --W. Goldring.

“Then, in August, 1991, I learned of a new construction of Flach [Fl] and quickly became convinced that an extension of his method was more plausible. Flach’s approach seemed to be the first step towards the construction of an Euler system, an approach which would give the precise upper bound for the size of the Selmer group if it could be completed. By the fall of 1992, I believed I had achieved this and begun then to consider the remaining case where the mod 3 representation was assumed reducible. For several months I tried simply to repeat the methods using deformation rings and Hecke rings. Then unexpectedly in May 1993, on reading of a construction of twisted forms of modular curves in a paper of Mazur [Ma3], I made a crucial and surprising breakthrough: I found the argument using families of elliptic curves with a common ρ5 which is given in Chapter 5. Believing now that the proof was complete, I sketched the whole theory in three lectures in Cambridge, England on June 21-23. However, it became clear to me in the fall of 1993 that the con- struction of the Euler system used to extend Flach’s method was incomplete and possibly flawed.” --A. Wiles.

"A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies." --S. Banach.

"This compatibility result is, in my humble opinion, the most convincing single piece of evidence for the conjecture of Bloch-Kato. The functional equation relating L(V,s) and L(V∗(1),−s) on the one hand, and the duality formula relating H1f(GK,V) and H1f(GK,V∗(1),s) belong to two different paths in the history of mathematics, the first one to the analytic ideas (often based on Poisson’s summation formula) initiated by Riemann in his study of the zeta functions, the second to the world of duality theorems in cohomology. That they give compatible formulas in the context of the Bloch-Kato conjecture seems to me a strong argument in favor of a deep link between L-functions and Selmer groups." -- J. Bellaïche.

I began to read them and to reflect. At that time, it was generally conceded that Harish-Chandra's work was important. At the same time it was considered difficult and seldom, if ever, studied. So I immediately achieved a certain fame: I could read and apparently understand his papers. That meant: (i) I could serve as a referee; (ii) I could serve as a reviewer; (iii) and, as the most demanding obligation, serve as a silent interlocutor, who could listen to him as he described, on his long daily walks, his latest discoveries. This was not problematic as long as I was elsewhere, when he was at Columbia and I at Princeton, or he at Princeton and I in Turkey or at Yale, but when I came as a colleague to the IAS in 1972, I had to persuade him, without offending him, that I had projects of my own to which I hoped to attend. -- R. Langlands, Harish-Chandra---IISER Pune.

“As for the mathematics, I began to reflect on Shimura's results, but on the basis of my own experience and knowledge. My thoughts were informed on the one hand by the principles enunciated in the letter to Weil, on the other, by the newly created theory of the discrete series. This theory, apart from its beginnings in the hands of Bargmann, the work of a single mathematician, Harish-Chandra, is, in my view, one of the great mathematical creations of the second half of the twentieth century, not sufficiently appreciated in its time and not yet today. Although the study of the zeta-functions of Shimura varieties demands inevitably also a great deal from algebraic geometry and number theory, those number theorists or algebraic geometers who attempt to develop it in ignorance of the discrete series and other pertinent aspects of nonabelian harmonic analysis are in danger of condemning themselves, whatever the immediate advantages, to ultimate irrelevance." -- R. Langlands, Comments on Letters to Lang.

"In 1980, in a series of lectures in Paris, published as Les débuts d'une formule des traces stable, I sketched the theory as it had developed by then: introduction of the notions of transfer factor and of stabilization and a statement of the fundamental lemma. Even a cursory examination of the text shows that important details were lacking, above all a precise definition of the transfer factor. At the time of the lectures, I expected that the fundamental lemma, an apparently elementary combinatorial statement, would be quickly proved. This was not to be so and it yielded, after initial exploratory efforts by myself, J. Rogawski and others over a full but discouraging decade only slowly to much more sophisticated attacks by Kottwitz, Hales, Waldspurger, Goresky-MacPherson, Laumon and, finally and successfully, by Ngo Bao Chau. The proof of the lemma, at first formulated for \(p\)-adic fields, passes through a proof of equivalence of the \(p\)-adic lemma with a similar lemma for power-series fields over finite fields, an equivalence that has, apparently, some element of mathematical logic in it, but which was proved by hand by Waldspurger in a marvelous tour-de-force and a proof for power-series fields that entails, in the work of Laumon and Ngo, a global argument for curves over finite fields. It is worthwhile to mention in passing that, so far as I understand, a precise definition of the transfer factor is essential to the argument. This precise definition was only possible thanks to the very careful analysis of Harish-Chandra's theory of nonabelian harmonic analysis in Shelstad's treatment of the transfer over archimedean fields." -- R. Langlands, Comments on Letters to Lang.

"The second third is taken up with a first discussion of possible proofs of the conjectured equality. It is not easy to follow and no longer, as far as I can see, of much interest. By 1972, at the time of the Antwerp lecture on the Eichler-Shimura relation and related matters, I had already begun to use a comparison of the trace formula with the Grothendieck-Lefschetz fixed point formula, a method that has been developed in general, at first by me, later, more deeply, with much better results, by Kottwitz. Both of us were handicapped by the lack of the fundamental lemma. Now that it is available, I hope that these methods will be taken up again. I believe that they still offer the best prospects for a complete and systematic treatment of the zeta-functions of Shimura varieties and their relation to automorphic \(L\)-functions, at least of the unramified factors.
The full comparison of the associated Galois representations with the automorphic representations will require, in addition to a full development of endoscopy, more sophisticated algebraic geometry and algebraic geometry. I have never thought about these matters in any serious way." -- R. Langlands, Comments on Letters to Lang.

"The last third of the letter is a discussion of the complex cohomology of Shimura varieties, Matsushima's theory, and Blattner's conjecture for the discrete series and their relation to each other. One major question raised, but only implicitly, by the letter was not discussed: whether indeed, if \(\pi_\infty\otimes\pi_f\) is an automorphic representation with \(\pi_\infty\) in the discrete series and if \(\pi'_\infty\) is a second element of the discrete series, the representation \(\pi'_\infty\otimes\pi_\infty\) is also an automorphic representation and whether it occurs with the same multiplicity? This would now be recognized as a question about endoscopy and global \(L\)-packets, but at the time was formulated more elementarily. As aleady observed, these questions were considered at the very first only for \(\mathrm{SL}(2)\), in part by me in conversation and correspondence with Labesse, and then by Shelstad for real groups, and it was only slowly that the theory reached even the stage of the 1980 lectures. Then, aside from substantial but largely unnoticed progress by Kottwitz, it languished for almost twenty years.
The second letter is somewhat more technical. It establishes, in a somewhat informal manner, that the ideas in the first, including the use of the automorphic \(L\)-functions introduced in my letter to Weil, are compatible with the behavior of the \(\Gamma\)-factors proposed a couple of years earlier by Serre. Its core is a relation, whose proof is preserved in my personal notes and included here but was not sent, so far as I can see, with the letter. Both the letters and the appendix suggest a very industrious young man." -- R. Langlands, Comments on Letters to Lang.

“ Steve’s erudition was astonishing. Rather than looking up a reference in the mathematical litera- ture, it was much easier to call Steve on the phone to find out the facts. Of the papers we wrote together the most interesting ones are also the most tentative. There are papers where we conjecture or simply suggest new relative trace formulas. Steve’s erudition was of paramount importance in formulating these conjectures. ” -- H. Jacquet, Remembering Steve Rallis.

"I was very fortunate to have Steve as my thesis advisor. He was very generous with his time and ideas. I recall one summer when I used to meet with Steve and Cary Rader every day for a couple of hours. This was before I had obtained any results towards my thesis, and it was very frustrating to me when every angle of approach seemed to fail. I still remember Steve’s comment that I was actually learning to be a mathematics researcher—you get to a good idea only after hitting a million dead ends, and the key is to keep working. This advice has stayed with me since then and has indeed helped to shape me into the mathematician that I am today." -- A. Pitale, Remembering Steve Rallis.

"Unfortunately, it appears that there is now in your world a race of vampires, called referees, who clamp down mercilessly upon mathematicians unless they know the right passwords. I shall do my best to modernize my language and notations, but I am well aware of my shortcomings in that respect ; I can assure you, at any rate, that my intentions are honourable and my results invariant, probably canonical, perhaps even functorial. But please allow me to assume that the characteristic is not 2" -- A. Weil, a letter to the editor.

"If, unwilling to stumble into metaphysics, one should prefer to remain on the hardly more solid ground of history, the same questions reappear, although in different guise: are we witnessing the beginning of a new eclipse of civilization? Rather than to abandon ourselves to the selfish joys of creative work, is it not our duty to put the essential elements of our culture in order, for the mere purpose of preserving it, so that at the dawn of a new Renaissance, our descendants may one day find them intact?"
" There are examples to show that in mathematics an old person can do useful work, even inspired work; but they are rare, and each case fills us with wonder and admiration. Therefore, if mathematics is to continue to exist in the way in which it has manifested itself to its votaries until now, the technical complications with which more than one of its subjects is now studded, must be superficial or of only temporary character; in the future, as in the past, the great ideas must be simplifying ideas, the creator must always be one who clarifies, for himself and for others, the most complicated tissues of formulas and concepts. "
" It is undoubtedly true that the modern mathematician does not know certain details of the theory of conic sections as well as Apollonius did, or as a candidate for a French competitive examination, but this does not lead any one to think that the theory of conic sections should form an autonomous science. Perhaps the same fate is in store for some of the theories of which we are proudest. " -- A. Weil, The Future of Mathematics (1950).

" For 1942–43 he was an as- sistant professor at Lehigh University, with most of his salary again paid by the Rockefeller Foun- dation. He described his role as “to serve up predi- gested formulae from stupid textbooks and to keep the cogs of this diploma factory turning smoothly.” He was able to continue work on his book Foundations, albeit more slowly; this was also the year in which he interacted with S. S. Chern. The year 1943–44 was the low point. The Rockefeller Foundation money had run out, and Weil’s job at Lehigh was to teach “the elements of algebra and analytic geometry” for fourteen hours a week to army recruits who were being kept busy before being shipped elsewhere. Afterward he and Eveline vowed never to mention the name “Lehigh” again. " -- A. W. Knapp, André Weil: A Prologue.

"It is interesting to observe how many different perspectives and motivations lead mathematicians at the end of the 80s to consider these kind of questions. Understanding the precise link between them, and finding a framework for actually proving some of these conjecture kept the community occupied for several decades!" --G. Baldi, What makes an algebraic curve special?.

"My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful." -- H. Weyl.

"I entered Princeton University as a graduate student in 1959, when the Department of Mathematics was housed in the old Fine Hall. This legendary facility was marvellous in stimulating interaction among the graduate students and between the graduate students and the faculty. The faculty offered few formal courses, and essentially none of them were at the beginning graduate level. Instead the students were expected to learn the necessary background material by reading books and papers and by organising seminars among themselves. It was a stimulating environment but not an easy one for a student like me, who had come with only a spotty background. Fortunately I had an excellent group of classmates, and in retrospect I think the "Princeton method" of that period was quite effective." --P. Griffiths.

"An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well.
I believe that two ecological effects were much more important in putting a damper on the subject than any exhaustion of intellectual resources that occurred." --B. Thurston.

"Polynomials and power series. May they forever rule the world." --S. Abhyankar.

"Abhyankar had a unique perspective of mathematics. He often rebelled against "fancy mathematics", demanding that all theorems should have detailed concrete proofs. He also believed that papers should spell out all the necessary details and he practised this rigorously. As a result, several of his papers are difficult to read, because you have to keep your concentration on every little detail that he has laid down. He would often say that the proofs should be so logical that a computer should be able to verify them! He also had a sense of poetry and rhythm in his papers. He would create multiple subsections with matching words and equal number of subitems, so that the paper had a natural symmetry. Sometimes, he would spend enormous amount of time to create such intricate structures."
“Professor Abhyankar was a charismatic man, and an excellent speaker, who could mesmerise an audience. He had a special way of talking with people about mathematics. He would insist that people explain what they were doing in elementary terms. Often when you began talking with him you realised how poor your understanding was, but at the end of the conversation you had a much deeper knowledge.”
”Perhaps I am wrong, but I think he did not even once use tensor product in his research! One of his motto was never to use a result whose proof he had not read. He broke this rule the first time when he used the classification of finite simple groups. “ -- Remembering S. Abhyankar.

" When Weil arrived in Tokyo in 1955, planning to speak about his ideas on the extension to abelian varieties of the classical theory of complex multiplication, he was surprised to learn that two young Japanese mathematicians had also made decisive progress on this topic.91 They were Shimura and Taniyama. While Weil wrote nothing on complex multiplication except for the report on his talk, Shimura and Taniyama published their results in a book in Japanese, which, after the premature death of Taniyama, was revised and published in English by Shimura. " -- Review of Shimura’s Collected Papers, J. S. Milne.

'' For the lack of the knowledge of real harmonic analysis on real semisimple Lie groups, I have no much progress for long time. But around early 90’s, I noticed the series of papers by Yamashita ([52] and related papers). This gives me the method to compute various spherical functions on Sp(2, R) and on SU (2, 2). And fortunately I had many bright students to work out.''
'' At somewhere Jean-Pierre Serre said that Representation Theory and Alge- braic Geometry are the most difficult fields in mathematics. In the arithmetic theory of automorphic forms the One from which everything comes out is the discrete subgroup Γ, which is the unit group of a non-commutaive arithmetic algebra. The emanated objects are written in these two sophisticated languages. But if one wants to have arithmetic results, the direct application of the known results in these two fields is not enough. At least one has to have coin effective computable results beyond the general framework. This is really a demanding and time-consuming job. The only wise way to cope with this difficulty seems for one to have one’s own style or philosophy; not to follow the fashion of the time to waste time in vain. '' -- T. Oda.

" The fact that this question in representation theory – whether the character χ−1 v of the torus occurs in the restriction of πv or π∗v – was settled by the symplectic epsilon factor ϵv(M ⊗N) was of great interest to me. " " We learned about these results from Joseph Bernstein, at the time I wrote a general paper on Gelfand pairs [25]. Eventually, the full result on multiplicities was established by his students Aizenbud and Gourevitch, and by Rallis and Schiffmann in the p-adic case [1]. The real and complex cases were settled by Sun and Zhu [56]. " -- B. H. Gross, road to GGP.

" Peter’s first paper [21], written when he was a schoolboy at Eton, was about the arithmetic of diagonal quartic surfaces, as were several subsequent papers, for instance [18,24,30]; despite his being better known for the Birch–Swinnerton- Dyer Conjecture, it is fair to say that the arithmetic of al- gebraic surfaces was his lifetime mathematical interest. "
" I returned to Cambridge, where Peter was now working in the computer laboratory, designing the operating sys- tem for TITAN, the machine planned to succeed EDSAC II. We wondered whether there was a similar phenomenon to the work of Siegel and Tamagawa for elliptic curves 𝐸 over ℚ. Specifically, was there a correlation between their local behaviour as described by their 𝐿-function 𝐿(𝐸, 𝑠), and their global behaviour, meaning their group 𝐸(Q) of rational points? "
" I think it was Davenport who told us that Hecke had dealt with 𝐿(𝐸, 𝑠) when 𝐸 had complex multiplication. For the curves 𝑦2 = 𝑥3 − 𝐷𝑥, the critical value 𝐿(𝐸, 1) is a finite sum of values of elliptic functions, easily machine computable if 𝐷 is not too large, and computable by hand using some algebraic number theory when 𝐷 is really small. " -- Bryan Birch, Memories of Peter Swinnerton-Dyer.

" Later, in a deliberate snub to the French and Russian schools of arithmetic geometry, largely centred on the legacy of Grothendieck, Peter insisted that he did not know what cohomology was, and was familiar with only the pre-1950 mathematics. The timing is important: 1954 was the year of his discipleship with Andr´e Weil in Chicago, and it is exciting to speculate if it led Peter and Bryan Birch to their famous conjecture. "
" .. around his eightieth birthday is his work [34]. Like many of his papers, it uses the aforementioned linear algebra ma- chinery of descent. A striking feature of this paper is that in it Peter used the main theorem of Markov chains to obtain an asymptotic distribution of the 2-Selmer rank in a uni- versal family of quadratic twists of an elliptic curve. This was never done before, but turned out to be a very useful tool. Peter was quite happy with this unorthodox inven- tion. “I am a computer scientist at heart,” he commented. " -- Sir Peter Swinnerton-Dyer, Mathematician and Friend, by Alexei Skorobogatov.

" Deformation theory had been on Spencer’s mind for some time prior to his work with Kodaira (cf. [18], [17], [20]). As he explained it to me, the issue was that they did not know what should play the role in higher dimension of quadratic differentials in the Teichmüller theory on Riemann surfaces. The breakthrough came with (2). With that major “hint” everything began to fall into place, leading to the papers [11], [5], [10], [12] (and relatedly [13], [14], [23]), which brought deformation theory into the core of complex algebraic geometry. "
"Before describing their theory, I want to say that deformation theory seems to some extent to have “been in the air”. In particular, the important work by Frölicher-Nijenhuis independently established the rigidity theorem stated below, and their papers on the calculus of vector-valued differential forms influenced the work of Kodaira-Spencer as well as that of many others working in related areas. " -- Phillip A. Griffiths, Donald C. Spencer (1912–2001)

" Both authors would like to express their gratitude to Roman Bezrukavnikov. The second author would like to say that he learned the main idea of this work from him: in particular, he explained that the ring of differential operators in characteristic p is an Azumaya algebra over the cotangent bundle and suggested that it might split over a suitable infinitesimal neighborhood of the zero section. The first author was blocked from realizing his vision (based on [29]) of a nonabelian Hodge theory in positive characteristics until he learned of this insight. Numerous conversations with Roman also helped us to overcome many of the technical and conceptual difficulties we en- countered in the course of the work. " --Nonabelian Hodge theory in characteristic p.

" Lucien understood early and very clearly the need to be strategic and efficient to adapt to this very difficult envi- ronment. He contacted a few of us (in particular, Mar- guerite Mangeny, Daniel Ferrand, and myself) to orga- nize a working group, a learning seminar, which would meet every week to discuss and study a subject. The main sources were Bourbaki, EGA, SGA, Cartan-Eilenberg, and Zariski-Samuel. Each of us would use as many hours as necessary to explain a theme to the others. "
" Lucien and I understood quickly that meeting Maurice and discussing math regularly with him was a unique piece of luck. Interacting with him, ex- changing mathematical ideas, and joking at the same time with him was not only possible but marvelously easy. "
" We began attending the IHES algebraic geometry seminar. The ex- traordinary stature of A. Grothendieck was very impres- sive. During one of the rare personal discussions we had with him, in his office if I remember well, he gave us an enormous pile of notes (several hundred pages… ) and sug- gested that we should put all this in clear form as a se- rious start in our mathematical life. A magnificent and generous offer which left us rather excited. "
We began to understand that our tastes were diverging. Lucien was more and more attracted by arithmetic. He would regularly describe (with a smile) a project to prove Fermat’s theorem (often by using Frobenius). I had de- cided to classify space curves. He wanted to be in Paris, I wanted to leave Paris. This was the end of a collaboration that both of us had enjoyed deeply, the end of our years of training.
--Christian Peskine, Lucien Szpiro (1941–2020).

“ Masser and Oesterl´e were led to the 𝑎𝑏𝑐 conjecture because Oesterl´e was interested in a new conjecture of Szpiro about ellip- tic curves (smooth cubic curves, such as y2 = x3 + 8) which has applications to number-theoretic properties of elliptic curves. Masser heard Oesterl´e’s lecture on Szpiro’s conjecture and wanted to formulate it without using elliptic curves. Eventually it turned out that the abc Conjecture and Szpiro’s Conjecture are equivalent. ” --B. Conrad, a public lecture at Stony Brook University in September 2013.

“In mathematics, there are many things and some parts are not really understood. I think there is some hidden part, and I want to find those hidden parts.” --Masaki Kashiwara, 2025 Abel Prize.

" This new evolution is not a sudden occurrence but the result of a mental disposition towards the beautiful math that comes to us by reading, thinking, working out problems, writing, and, most impor- tantly, talking to people. "
" I was simply drawn to work on something that fascinated me, and to do that, I needed to learn and discover new math. And it was fun, pure and simple. At some point, suddenly, I found myself already moving in a new direction. " --Changing Focus, Mark Andrea de Cataldo.

" Doing math is akin to unfolding a melody; its first sounds are usually a gift from someone else. My first paper was written in the footsteps of the ADHM classification of instantons. I caught the idea of higher regulators while preparing a talk on S Bloch’s work; this led to conjectures on the values of L-functions (still widely open) and to speculations about mixed motives (largely realized by V Voevodsky and A Suslin). Conversations with R MacPherson and P Deligne brought forth our work with J Bernstein on the Kazhdan–Lusztig conjecture; we played happily with D-modules and perverse sheaves until Bernstein left Russia at the beginning of 1981. " --Autobiography of Alexander Beilinson. " Gee secured the team a room in the basement of the Hausdorff Research Institute, where they were unlikely to be disturbed by itinerant mathematicians. There, they spent an entire week working on Pan’s theorem, one 12-hour day after the next, only coming up to ground level occasionally for caffeine. " --The Core of Fermat’s Last Theorem Just Got Superpowered.