1.1. The Alfred Jurzykowski Foundation.

Alfred Jurzykowski was born in 1899 in Cieszyn Silesia. As an officer of the Polish army, he took part in the Polish-Bolshevik War in 1920. In the interwar period, he founded his own department store in Warsaw and ran an export company in Gdańsk. He took part in the September Campaign. After its defeat, he left for Italy via Romania and from there to the USA, where he ran his own company. During his stay in Brazil in 1950-1960, in 1956 he launched the first Mercedes-Benz bus and truck factory in that country, in which he initially had 50% and later 75% of shares. When he returned to the USA in 1960, at the urging of the Polish consul in Chicago, Rudolf Rathaus, he created a foundation bearing his name.

The Alfred Jurzykowski Foundation was established by Alfred Jurzykowski in 1960 in New York to support (including financing) scientific and cultural institutions located outside Poland. Beginning in 1964, the Foundation awarded prizes to artists of Polish origin (working in Poland or abroad) for outstanding achievements in the broadly understood field of culture: science and humanities, medicine, literature (creative work, translations, literary criticism), fine arts, music (composition, performance, musicology), theatre, film, etc. The awards were presented at the Foundation's headquarters in New York and amounted to six thousand dollars in the years 1990-1997.

1.2. The 1991 Alfred Jurzykowski Prize.

The Jurzykowski Prize was awarded to Henryk Iwaniec in 1991 by the Alfred Jurzykowski Foundation in New York.

2.1. The Wacław Sierpiński Medal.

The Sierpiński Medal is granted to outstanding mathematicians with Polish associations. The University of Warsaw, together with the Polish Mathematical Society, have awarded the medal since its inception in 1974. Celebration of the award includes the Wacław Sierpiński lecture, given by the laureate.

2.1. The 1996 Wacław Sierpinski Medal.

The 1996 Wacław Sierpinski Medal was awarded to Henryk Iwaniec (Rutgers University, New Brunswick, New Jersey, USA) on 22 April 1996 for his work on Complex prime numbers.

3.1. The Ostrowski Prize.

Since 1989 the Ostrowski Prize has been awarded every second year for an outstanding contribution in the field of mathematics. The Ostrowski Foundation, which awards the prize, was established by Professor A M Ostrowksi of Basel (1893-1986). He left his entire estate to the foundation and stipulated that the income should provide a prize for outstanding recent achievements in pure mathematics and the foundations of numerical mathematics. The aim of the Ostrowski Foundation is to promote the mathematical sciences. The prize consists of 50,000 Swiss Francs for each prize winner and the possibility to nominate a promising young candidate for a Postdoctoral Fellowship of 30,000 Francs.

3.2. The 2001 Ostrowski Prize.

The seventh Ostrowski Prize recognising outstanding mathematical achievement has been awarded to Henryk Iwaniec of Rutgers University; Peter Sarnak of Princeton University, the Institute for Advanced Study in Princeton, and Courant Institute at New York University; and Richard L Taylor of Harvard University.

The prize carries a monetary award of 150,000 Swiss francs (approximately US$87,000) and three fellowships of 30,000 Swiss francs. The jury, consisting of representatives from the universities of Basel, Jerusalem, and Waterloo and from the academies of Denmark and the Netherlands decided to divide the Ostrowski Prize into three equal parts. The ceremony took place in Jerusalem.

3.3. The 2001 Ostrowski Prize to Henryk Iwaniec.

Iwaniec's work is characterised by depth, profound understanding of the difficulties of a problem, and unsurpassed technique. He has made deep contributions to the field of analytic number theory, mainly in modular forms on $GL(2)$ and sieve methods. Particularly noteworthy are: his work (with W Duke and J Friedlander) on the question of "breaking convexity" for estimates of the growth of $L$-functions associated to modular forms, and his work (with J Friedlander) yielding an asymptotic formula for the number of primes up to $X$ that are the sum of a square and of a biquadrate (this is the first time one has been able to prove the existence of infinitely many primes in a prescribed very thin sequence), and his solution of Linnik's problem of the equidistribution of integral points on a two-dimensional sphere as the radius increases.

4.1. Frank Nelson Cole Prize in Number Theory.

The 2002 Frank Nelson Cole Prize in Number Theory was awarded at the 108th Annual Meeting of the American Mathematical Society in San Diego in January 2002.

The Cole Prize in Number Theory is awarded every three years for a notable research memoir in number theory that has appeared during the previous five years (until 2001, the prize was usually awarded every five years). The awarding of this prize alternates with the awarding of the Cole Prize in Algebra, also given every three years. These prizes were established in 1928 to honour Frank Nelson Cole on the occasion of his retirement as secretary of the American Mathematical Society after twenty-five years of service. He also served as editor-in-chief of the Bulletin for twenty-one years. The Cole Prize carries a cash award of $5,000.

4.2. The 2002 Frank Nelson Cole Prize in Number Theory.

The Cole Prize in Number Theory is awarded by the American Mathematical Society Council acting on the recommendation of a selection committee. For the 2002 prize, the members of the selection committee were: Benedict Gross, Carl Pomerance, and Paul Vojta (chair).

Previous recipients of the Cole Prize in Number Theory are: H S Vandiver (1931), Claude Chevalley (1941), H B Mann (1946), Paul Erdos (1951), John T Tate (1956), Kenkichi Iwasawa (1962), Bernard M Dwork (1962), James B Ax and Simon B Kochen (1967), Wolfgang M Schmidt (1972), Goro Shimura (1977), Robert P Langlands (1982), Barry Mazur (1982), Dorian M Goldfeld (1987), Benedict H Gross and Don B Zagier (1987), Karl Rubin (1992), Paul Vojta (1992), and Andrew J Wiles (1997).

The 2002 Cole Prize in Number Theory was awarded to Henryk Iwaniec and Richard Taylor. The text that follows presents, for each awardee, the selection committee's citation, a brief biographical sketch, and the awardee's response upon receiving the prize.

4.3. Citation for the 2002 Frank Nelson Cole Prize in Number Theory to Henryk Iwaniec.

The Frank Nelson Cole Prize in Number Theory is awarded to Henryk Iwaniec of Rutgers University for his fundamental contributions to analytic number theory. In particular, the prize is awarded for his paper (with J Friedlander) "The polynomial $X^{2} + Y^{4}$ captures its primes", in Ann. Math., which is the first paper ever to show that an integer polynomial with "sparse" range takes on infinitely many prime values. The method is robust, and already D R Heath-Brown has extended the method to certain cubic polynomials. In addition, the prize is awarded for the series of papers (with W Duke and J Friedlander) "Bounds for automorphic L-functions, I, II, III", in Invent. Math., and the paper (with B Conrey) "The cubic moment of central values of automorphic $L$-functions", in Ann. Math. In these papers, critical-line bounds for $L$-functions associated to certain modular forms were greatly improved by novel methods, including an amplification technique that provided the starting point for J W Cogdell, I Piatetskii-Shapiro, and P Sarnak to finally resolve Hilbert's eleventh problem (on representation by quadratic forms in a number field). And, the prize is awarded for the paper (with P Sarnak) "The nonvanishing of central values of automorphic L-functions and Landau-Siegel zeros", Israel J. Math., for the introduction of far-reaching averaging and mollification techniques for families of automorphic L-functions.

4.4. Henryk Iwaniec Biographical Sketch.

Henryk Iwaniec was born in Elblag, Poland, on 9 October 1947. He graduated from Warsaw University in 1971, and he received his Ph.D. there in 1972. From 1971 until 1983 he held various positions in the Institute of Mathematics of the Polish Academy of Sciences. In 1976 he defended his habilitation thesis. In the year 1976-77 he enjoyed a fellowship of the Accademia Nazionale dei Lincei at the Scuola Normale Superiore di Pisa. In 1979-80 he visited the University of Bordeaux. In 1983 he was promoted to professor. The same year he became member correspondent of the Polish Academy of Sciences.

Iwaniec left Poland in 1983 to take visiting positions in the USA at the Institute for Advanced Study in Princeton (1983-84 and 1985-86) and the University of Michigan at Ann Arbor (summer 1984), and he was the Ulam Distinguished Visiting Professor at Boulder (fall 1984). In January 1987 he assumed his present position as New Jersey State Professor of Mathematics at Rutgers University. He was elected to the American Academy of Arts and Sciences in 1995. He spent the year 1999-2000 as a distinguished visiting professor at IAS. Recently he became a citizen of the USA.

Iwaniec received first prizes in the Marcinkiewicz contests for student works in the academic years 1968-69 and 1969-70. In 1978 he received the State Prize from the Polish Government, in 1991 he received the Jurzykowski Award from the Alfred Jurzykowski Foundation in New York, and in 1996 he received the Sierpinski Medal. Iwaniec was an invited speaker at the International Congress of Mathematicians in Helsinki (1978) and in Berkeley (1986).

4.5. Response from Professor Iwaniec.

I thank from my heart the American Mathematical Society and the committee of the Cole Prize for selecting me for this award. My joy is even greater when I think that this is a significant award for professional accomplishments from beyond my native country, and in particular that this is coming from my new homeland in the USA. Less emotional, nevertheless important for me, is also the feeling of larger recognition of analytic number theory which the Cole Prize manifests in this case. Indeed, all the works cited for the prize are joint with many of my colleagues. Without their collaboration I cannot imagine how could I get that far. Yes, working together offers an immediate satisfaction from sharing ideas, but above all it is the only way we can cultivate in depth the modern analytic number theory.

Analytic number theory pursues hard classical problems of an arithmetical nature by means of best available technologies from any branch of mathematics, and that is its beauty and strength. Analytic number theory is not driven by one concept; consequently it has no unique identity. Fourier analysis was always present, but in the last two decades it has been expanded to nonabelian harmonic analysis by employing the spectral theory of automorphic forms. For example, applying this analysis implicitly we have established the asymptotic distribution of primes in residue classes in the range beyond the capability of the Grand Riemann Hypothesis. Moreover, along these lines, we were able to produce primes in polynomial sequences. To this end one needs to enhance the Dirichlet characters by more powerful cusp forms on congruence groups. In a different direction we performed amplified spectral averaging from which to deduce important estimates for individual values of $L$-functions and to apply the latter to questions of equidistribution of many arithmetical objects. Other fruitful resources for solving problems in analytic number theory were uncovered by exploiting the Riemann hypothesis for varieties. Connections of these problems with the profound theory of Deligne are by no means straightforward. Perhaps these brief words may give some idea of what the trends are in the subject today, or at least what we are doing there.

There are many colleagues to whom I owe my gratitude for inspiration and joint research over the last years; among them I would like to mention Enrico Bombieri, Brian Conrey, Jean-Marc Deshouillers, William Duke, John Friedlander, Etienne Fouvry, Philippe Michel, and Peter Sarnak.

5.1. The Leroy P Steele Prizes.

The Leroy P Steele Prizes were established in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein, and are endowed under the terms of a bequest from Leroy P Steele. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; most favourable consideration was given to papers distinguished for their exposition and covering broad areas of mathematics. In 1977 the Council of the American Mathematical Society modified the terms under which the prizes are awarded. In 1993, the Council formalised the three categories of the prize by naming each of them: (1) The Leroy P Steele Prize for Lifetime Achievement; (2) The Leroy P Steele Prize for Mathematical Exposition; and (3) The Leroy P Steele Prize for Seminal Contribution to Research.

5.2. The 2011 Leroy P Steele Prizes.

The 2011 American Mathematical Society Leroy P Steele Prizes were presented at the 117th Annual Meeting of the American Mathematical Society in New Orleans in January 2011. The Steele Prizes were awarded to Henryk Iwaniec for Mathematical Exposition, to Ingrid Daubechies for a Seminal Contribution to Research, and to John W Milnor for Lifetime Achievement.

5.3. Citation for the 2011 Mathematical Exposition Prize to Henryk Iwaniec.

Henryk Iwaniec is awarded the Leroy P Steele Prize for Mathematical Exposition for his long record of excellent exposition, both in books and in classroom notes. He is honoured particularly for the books Introduction to the Spectral Theory of Automorphic Forms (Revista Matemática Iberoamericana, Madrid, 1995) and Topics in Classical Automorphic Forms (Graduate Studies in Mathematics, 17, American Mathematical Society, Providence, RI, 1997). These books give beautiful treatments of the theory of automorphic forms from the author's perspective of analytic number theory. They have become classics in the field and are now a fundamental resource for students. The two books are complementary, with the first presenting the non-holomorphic theory of Maass forms for $GL(2)$ and the latter focusing on holomorphic modular forms.

Introduction to the Spectral Theory of Automorphic Forms begins with the basics of hyperbolic geometry and takes readers to the frontiers of research in analytic number theory. Many topics, such as the Kuznetsov formula and the spectral theory of Kloosterman sums, are covered for the first time in this book. It closes with a discussion of current research on the size of eigenfunctions on hyperbolic manifolds. By making these tools from automorphic forms widely accessible, this book has had a tremendous influence on the practice of analytic number theory.

Topics in Classical Automorphic Forms develops many standard topics in the theory of modular forms in a non-traditional way. Iwaniec's aim was "to venture into areas where different ideas and methods mix and interact." One standout part is the treatment of the theory of representation of quadratic forms and estimating sizes of Fourier coefficients of cusp forms. The breakthrough in the late 1980s in understanding representations by ternary quadratic forms originated with the seminal work of Iwaniec, which is described beautifully here.

5.4. Henryk Iwaniec Biographical Sketch.

Henryk Iwaniec graduated in 1971 from Warsaw University, got his Ph.D. the next year, and became professor at the Institute of Mathematics of the Polish Academy of Sciences before leaving for the United States in 1983. After taking several visiting positions in the United States (including a long-term appointment at the Institute for Advanced Study), in January 1987 he was offered a chair as New Jersey State Professor at Rutgers, the position he enjoys to this day.

Iwaniec's main interest is analytic number theory and automorphic forms. Prime numbers are his passion. His accomplishments were acknowledged by numerous invitations to give talks at conferences, including the International Congress of Mathematicians. Iwaniec is a member of the Polish Academy of Sciences, the American Academy of Arts and Sciences, the National Academy of Sciences, and the Polska Akademia Umiejetnosci.

Among several prizes Iwaniec has received are the Jurzykowski Foundation Award, the Sierpinski Medal, the Ostrowski Prize, and the Cole Prize in Number Theory.

Iwaniec teaches graduate students and collaborates with researchers from various countries. In 2005 he was honoured with the Doctorate Honoris Causa of Bordeaux University. In 2006 the town council of his native city made Iwaniec an Honorary Citizen of Elblag, a distinction he cherishes very proudly.

5.5. Henryk Iwaniec Response.

I thank the American Mathematical Society and the Committee of the Steele Prize for this award. I am very honoured and happy. This is a very meaningful recognition for me because the citation is telling not only about my fascination with the subjects but also about my attention to educating new generations of researchers. Modern analytic number theory takes ideas from the theory of automorphic forms and gives back new enhanced methods and results. While more arithmetical aspects of automorphic forms are covered relatively well in the literature, there is still not a sufficient exposition of analytic aspects. Hopefully more books will be written by other specialists that will address similar topics from many different directions. These two books selected by the Committee for the award came out of my teaching graduate courses and giving presentations in workshops, so inevitably they contain some of my favourite ways of handling the problems. I am glad that my choices and writing style are well received. If indeed these works do have "influence on the practice of analytic number theory," I will be most happy.

6.1. The Shaw Prizes.

In 2002, under the auspice of Run Run Shaw, a visionary philanthropist, the Shaw Prize Foundation was established. The inaugural Shaw Prize was presented two years later in 2004. The Shaw Prize consists of three annual awards, namely the Prize in Astronomy, the Prize in Life Science and Medicine, and the Prize in Mathematical Sciences. Each of these awards carries the amount of 1 million dollars.

The Shaw Prize honours individuals, regardless of race, nationality, gender, and religious belief, who are currently active in their respective fields and who have recently achieved distinguished and significant advances, who have made outstanding contributions in academic and scientific research or applications, or who in other domains have achieved excellence. The Shaw Prize is dedicated to furthering societal progress, enhancing quality of life, and enriching humanity's spiritual civilisation.

The Shaw Prize Certificate presented to each laureate is mounted elegantly on a dark brown leather folder. Within the folder, the left side features a decorative rendering, displaying the Shaw Prize medal in relief, and the motto of the Prize, "for the benefit of humankind", in gold engraving. The decoration comes in three distinct colours, each representing one of the award categories. For mathematics, the colour is red. On the certificate, "The Shaw Prize in Mathematical Sciences", engraved in gold, is displayed prominently. Below is the name of the laureate, meticulously handwritten by a local calligrapher, followed by the citation and the date of the award. Each certificate is signed by the Chair of the Board of Adjudicators and the Chair of the Shaw Prize Council.

6.2. The 2015 Shaw Prize in Mathematical Sciences.

The Shaw Prize in Mathematical Sciences 2015 is awarded to Gerd Faltings, Managing Director at Max Planck Institute for Mathematics in Bonn, Germany, and Henryk Iwaniec, New Jersey Professor of Mathematics at Rutgers University, USA, for their introduction and development of fundamental tools in number theory, allowing them as well as others to resolve some longstanding classical problems.

6.3. About Henryk Iwaniec.

Henryk Iwaniec was born in 1947 in Poland and is currently New Jersey Professor of Mathematics at Rutgers University, USA. He graduated from the University of Warsaw in Mathematics in 1971, and obtained his PhD there in 1972. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. Before being appointed Professor of Mathematics at Rutgers University in 1989, he held visiting positions at the Institute for Advanced Study, Princeton, the University of Michigan, and the University of Colorado at Boulder.

Faltings and Iwaniec were awarded the 2015 Shaw Prize for their contributions to number theory. Number theory concerns whole numbers, prime numbers, and polynomial equations involving them. The central problems are often easy to state but extraordinarily difficult to resolve. Success, when it is achieved, relies on tools from many fields of mathematics. This is no coincidence since some of these fields were introduced in attempts to resolve classical problems in number theory. Faltings and Iwaniec have developed many of the most powerful modern tools in algebra, analysis, algebraic and arithmetic geometry, automorphic forms, and the theory of zeta functions. They and others have used these tools to resolve longstanding problems in number theory.

6.4. The contribution of Henryk Iwaniec.

Iwaniec's work concerns the analytic side of Diophantine analysis, where the goal is usually to prove that equations do have integral or prime solutions, and ideally to estimate how many there are up to a given size.

One of the oldest techniques for finding primes is sieve theory, originating in Erastosthenes' description of how to list the prime numbers. Iwaniec's foundational works and breakthroughs in sieve theory and its applications form a large part of this active area of mathematics. His proof (with John Friedlander) that there are infinitely many primes of the form $X^{2} + Y^{4}$ is one of the most striking results about prime numbers known; the techniques introduced to prove it are the basis of many further works. The theory of Riemann's zeta function - and more generally of $L$-functions associated with automorphic forms - plays a central role in the study of prime numbers and diophantine equations. Iwaniec invented many of the powerful techniques for studying $L$-functions of automorphic forms, which are used widely today. Specifically, his techniques to estimate the Fourier coefficients of modular forms of half-integral weight and for estimating $L$-functions on their critical lines (the latter jointly with William Duke and John Friedlander) have led to the solution of a number of longstanding problems in number theory, including one of Hilbert's problems: that quadratic equations in integers (in three or more variables) can always be solved unless there is an "obvious" reason that they cannot.

In a series of papers remarkable both in terms of its concept and novel techniques, Iwaniec together with different authors (Ètienne Fouvry and then Enrico Bombieri and John Friedlander) established results about the distribution of primes in arithmetic progressions which go beyond the notorious Riemann Hypothesis. This opened the door to some potentially very striking applications. Yitang Zhang's much celebrated recent result on bounded gaps between primes relies heavily on the works of Iwaniec et al. Iwaniec's work mentioned above, together with his many other technically brilliant works, have a central position in modern analytic number theory.

Mathematical Sciences Selection Committee
The Shaw Prize

24 September 2015   Hong Kong

6.5. Henryk Iwaniec autobiography.

I was born on October 9, 1947 in Elblag, Poland, eight hours before my identical twin brother Tadeusz. We grew up close in a loving family with no academic tradition.

Our interest in mathematics began in a technical high school (which focused on steam turbines) where we were successful in mathematical Olympiads. Instead of continuing our education in engineering we entered the Mathematics Department of Warsaw University in 1966. However, we no longer shared our ideas in mathematics as we wished to be more independent intellectually.

At the end of my first undergraduate year I was fortunate to attract the attention of Professor Andrzej Schinzel, who invited me to attend and to give talks in his number theory seminar at the Mathematics Institute of the Polish Academy of Sciences. Early on I was fascinated by analytic number theory because of the large variety of tools, which are used to establish results of an arithmetical flavour. I was particularly impressed by the work of Yu V Linnik. I started alone on sieve methods while Professor Schinzel provided general advice, and also helped with editorial matters.

A year before I graduated in 1971 I had two papers accepted for publication, one of which became my master thesis and the other my PhD thesis, which I defended in the spring of 1972 without entering graduate school (the requirement of passing an exam on Marxism philosophy delayed my thesis defence!).

From 1971 to 1983 (the year I left Poland with my family for the United States) I was employed at the Mathematics Institute of the Polish Academy of Sciences in Warsaw, at which time I was promoted to Professor and elected corresponding member of the Polish Academy of Sciences.

During this period I spent one year in Pisa (1976/77) at the Scuola Normale Superiore where I had the opportunity to expand my interest in sieve methods under the influence of Enrico Bombieri and in collaboration with John Friedlander. Later in the USA my collaboration with John evolved into many other topics in number theory (exponential and character sums, distribution of prime numbers, etc.). Our numerous joint results constitute the essential bulk of my total work.

Next, my time spent at the University of Bordeaux in 1979/80 and the subsequent visits during 1980-82 were critical for directing my interest to the field of automorphic forms. There began my collaboration with Jean-Marc Deshouillers. We developed new estimates for the Fourier coefficients of cusp forms in the spectral aspect and for sums of Kloosterman sums. All of these are basic tools in modern analytic number theory; they open doors for using non-abelian harmonic analysis to study natural numbers.

At the same time in Bordeaux I worked with Étienne Fouvry on primes in arithmetic progressions. We succeeded in establishing equidistribution results over residue classes, which surpass those implied by the Riemann hypothesis. Later we improved the range of these results jointly with Bombieri and Friedlander at the Institute for Advanced Study (AIS) at Princeton.

My first years in the USA (September 1983-December 1986) were spent at the IAS with two semester breaks for visits to the University of Michigan and as a Distinguished Ulam Visiting Professor to the University of Colorado at Boulder. I also made a few trips to Stanford University to work with Peter Sarnak. At the Institute it was like being in paradise to have free time to think of mathematics, to discuss and to share ideas with members of the School of Mathematics, distinguished visitors, as well as with the faculty of Princeton University. Such an atmosphere easily boosts the aspirations of everybody. I was driven to the uncharted waters of the Riemann hypothesis for varieties (Deligne's result) directing some of it into powerful tools for analytic number theory. Jointly with Friedlander and Fouvry, and with assistance from Birch, Bombieri and Katz, crucial improvements were made in asymptotic formulas for some arithmetic functions, which are fundamental to the theory of prime numbers. The Riemann hypothesis for varieties is also used later in my work on $L$-functions (jointly with Brian Conrey) and on Hecke eigenvalues (jointly with William Duke). It is gratifying that my mathematical "children", Étienne Fouvry, Emmanuel Kowalski and Philippe Michel travel much further conceptually in their "Theory of Trace Functions", and produce very strong results which unify many applications.

I love working in collaboration with others. Peter Sarnak and I established (among many other things) statistical results concerning the central values of families of automorphic L-functions, which shed new light on the Riemann hypothesis. Moreover, we revealed new connections with the notorious exceptional character issues. Brian Conrey, Kannan Soundararajan and I established that a large percentage of zeros of L-functions rest on the critical line. John Friedlander and I developed extra axioms for sieve theory which allows one to break the parity barrier, and hence to produce prime numbers in sparse sequences. Algebraic and analytic number theory coexists beautifully in my recent work with John Friedlander, Barry Mazur and Karl Rubin on the spin of prime ideals.

In January 1987 I moved from the IAS to Rutgers University accepting the position of New Jersey State Professor of Mathematics. Teaching graduate students at Rutgers is a great pleasure; it relaxes me from the more intense pursuit of research, and it convinces me that the future of my beloved subject, analytic number theory, is bright.

24 September 2015 Hong Kong.

6.6. Henryk Iwaniec's Shaw lecture.

Let me say something first about myself, about how I became a mathematician and, perhaps, why I became a mathematician. Then I will come back to mathematics to entertain you a little bit with some perplexing problems in number theory. So I was born in Elblag, an industrial city in not the central part of Poland and I think I can say I was fortunate to grow up with my twin brother who was 8 hours younger. We grew up together for quite a while until we entered school, I was 6. There was really no known tradition, academic tradition, in the family. Nevertheless my parents were really stimulating, gave encouragement and gave great attention to knowledge, and that was sufficient for us to get interested in science rather early. Not really mathematics at that time and so there was no particular incentive to do mathematics in our elementary school.

We entered a technical high school in the town which specialised in technical things so we learned a great deal about machines and technology, particular about steam turbines for power plants. That was quite an experience which I think had some effects, trade-offs, on my attitudes toward mathematics. For example a mathematician likes to see the beauty in the mathematical statements but for me that is not enough. You really can see more deeply. You know intrinsic beauty after you really establish a result so you can see how you got it, the tools, the use of machinery, is part of the beauty. It's not just a realisation of your goal. Anyway the atmosphere in the high school was fantastic. I mean there was no competition, nobody was really claiming to be number one. Some people were good in one field and some in another field of interest. How did we get interested in mathematics? Well because of mathematical contests. First mathematical circle in science and then in mathematical Olympiads where we were successful so it boosts your ambition. Our teacher couldn't really do much for us but he distinguished us in the class by excusing us from taking tests. So that's also encouraging, something like that, but having a brother, a twin brother, it was a wonderful thing for us to stimulate each other. We could ask questions ourselves and look for answers together. By the way never competing, we actually never competed, and so we went to university.

The mathematics department was at the University of Warsaw but the interesting part of it is that by the regulations of the government we would not be eligible to enter Warsaw because we lived faraway from Warsaw. But winning the Olympiad gave us this privilege of choosing where to study, also to enter without taking the entrance exams. It was a blessing because I don't think I would have passed the mandatory exam in the Russian language. Anyway when we entered university we decided to work with mathematics in different fields because we wanted to be intellectual independent which is, in my opinion, extremely natural when you grow up together as a twin brother, at some point you would like to split. But of course we exchanged our experiences all the time and never never ever competed and I think my brother is as happy as I am that I'm getting the Shaw Prize.

Right, so at the University I was fortunate that I received attention of some sort from the Academy very early, at the end of the first year undergraduate year, and so I specialised very early. It just makes my story short to say that I already had two papers accepted for publication before I graduated and so I never really wrote a PhD thesis, you know, because one of these was accepted and actually, you know, I could get my PhD degree before I got a Master's Degree but there was a little slight delay because there's a requirement that I had to pass a Marxist philosophy exam which delayed my defence for six months. So I mastered that philosophy in six months. Anyway another interesting episode I would like to mention from that period is the teachers, the professors, realised that for the group of good students it would be better if they set a special programme for the students because the regular programmes were really too easy. So they created a special group, something with a dozen out of 300 admitted at that time, so we followed from the very beginning a special programme. It was an excellent idea. Also we were really regular guys, we didn't want to be something special but a few of us had an idea of taking exams in advance and pass one year and be ahead of our colleagues. But in Poland the system is so rigid, you know, that this very courageous idea required approval of the default from the Dean. The Dean didn't object but he gave us a wonderful lesson. He said that we were foolish because we were going voluntarily to cut off some of the best years in our life, that is the college years. So we listened to him and we finished on time and I tell this story sometimes to my students because I think there is no need to be, you know, fast and forget the normal life. We had lots of interesting other things to do like hobbies. It actually gave me a lot of time to do my beloved subject number theory which was not a part of the syllabus and also I started dating a girl, my wife, etcetera. So I think there is no rush for such things and there are lots of things from my life could be said maybe on another occasion so let me say a little bit about mathematics.

I don't really have time to tell you what I'm really doing in number theory so I have chosen for this presentation something a little bit to entertain, you know, but also a very perplexing subject that is about ineffectiveness in mathematics. I mean, mathematics is considered as a very very precise science and this is true but it doesn't mean that there is no controversy. I'm not talking about something with proofs that are incomplete, I'm suddenly talking about the precise arguments which are still controversial. I mean every statement and hypothesis requires rigorous proofs as you know yet there are some results that hard to accept even with the rigorous proofs. I have in mind the Banach-Tarski paradox. For example you can even subdivide, you know, a ball, say a gold ball, into two equal balls, so if it's made from gold then you can improve the budget of the government by doing this. The problem with this paradox is its ineffective construction using, you know, certain arguments that mathematicians are arguing whether to accept as valid or not. And in number theory this - jump instantly to this - we have some beautiful examples of ineffective results and sometimes we would like to accept this as a valid, you know, argument or sometimes not.

- Mathematics is a very precise science.
Every statement and hypothesis require rigorous proofs.

- Yet, some results are hard to accept as absolutely true (Banach-Tarski Paradox).

- In Analytic Number Theory we have important estimates which are not effective (Thue, Roth, Heilbronn, Deuring, Landau, Siegel).

The best place to start would probably be the issue of exceptional character which we consider as a ghost.
...


6.7. Discussion with Professor Henryk Iwaniec at Rutgers after winning the Shaw Prize.

Henryk Iwaniec has been in New Jersey since 1983, first at the Institute for Advanced Study at Princeton and now for more that thirty years on the staff of Rutgers University. He came from Poland with his wife Kasha and two young daughters.
Kasha Iwaniec:-

We planned to stay maximum three years, but it somehow became permanent.

Henryk Iwaniec:-

We're not really for political reason here even though it was a factor since life here was much much nicer. I've had a great career. Much better in the United States and offers were coming there.

Stephen Miller, Vice-chair Mathematics, Rutgers University:-

He's meant the world to this Department, He's trained so many great students here, he's taught great courses, he's written amazing books. When he touches a subject then there is a clarity that comes through and its there for everyone else to see.

Dorian Goldfeld, Mathematics, Columbia University:-

There was one Einstein and he's the analogous thing in the mathematical world.

Henryk Iwaniec:-

One third plus one quarter is what? Seven twelfths, right.

He was very interested in primes. Primes are always just divisible by one and themselves like 2, 3, 5, 7, 11, 13, etc.
Henryk Iwaniec:

I did like to solve problems of prime numbers and then I have some successes. I think my best one is jointly with John Friedlander from Toronto. We proved there are infinitely many prime numbers which are representable as the sum of a square and a 4th power of integers.

Here's a simple demonstration of how it works. For the square use the number 1

1 times 1 is 1.

For the fourth power use 2

2 times 2 times 2 times 2 is 16.

1 plus 16 is 17 and 17 is prime. This works infinitely but not often.

I mean the number of integers which are representable as the sum of the square plus a fourth power are very very rare and now finding primes in this rare sequence is even harder.

It's something unbelievable to prove that they're infinity primes in such a sparse sequence.

Prime numbers are used in the so-called crypto systems to give us internet security, to use credit cards for instance.

For encryption you need primes of very special shape in order to claim that they are very hard to find. Because you know you want to do this to be very hard. You couldn't have any kind of authentication security on the internet without prime number theory.

Henryk's work plays a role in that. When you talk about breaking a crypto system you're talking about a thing which may take years to do. You have to understand things theoretically and he's developed the tools to do that.
Henryk Iwaniec: -

When you see people using your concepts and without even, you know, mentioning it because nobody even knows where it actually originated from but you may see a trace of you a little bit that's a great pleasure.

He creates these remarkable results and tools that other people can apply to problems.

Daniel Goldston, a mathematician across the country in San Jose, remembers lecturing one day with Henryk in attendance.

Daniel Goldston:-

I think I gave a talk once in 1990 and he says, "Well, I have something I should tell you about," and so he proceeded to spend like an hour writing on the board completely solving this really hard problem that sort of arose out of my talk. I said, "It's okay if I take notes on This." I wrote them down and I kept these notes for ten years because I wanted to use what he had told me to solve something. He had more than a hundred results of proofs of things that were considered extremely difficult whereas, you know, a lot of really outstanding mathematicians have just a couple or maybe three or four. At the same time he's busy, you know, writing books and sort of developing the tools so that everyone else can also do these results.

The work is done here in his home office.
Henryk Iwaniec:-

I like working at night late. Night is usually when great inspirations come. From my contemplations at night I contemplate a lot before I start writing notes.

Henryk's twin brother Tadeusz is also a mathematician. They inspired and challenged each other in high school where they were so far ahead of other students that when the teacher came into the class for an exam:
Henryk Iwaniec:-

He said "you both leave the class" and I said "okay we have a test." We didn't even write tests. At Warsaw University doing my undergraduate study I had accepted for publication a paper which became my master thesis and is my Ph.D.

At University he met his wife-to-be Kasha, also a mathematics Ph.D.
Kasha Iwaniec:-

We can understand each other, you know. Maybe I don't understand details of his work but I understand at least what kind of work he does.

Henryk appreciates her art.
Kasha Iwaniec:-

I invite him to exhibitions.

Henryk Iwaniec:

Yes, yes, she invites me so I have contact with other kinds of people and so there is a nice exchange of feelings.

At 67 Henryk wants more work.
Henryk Iwaniec:-

I don't intend to retire. You can be very productive, prolific also, later.

7.1. The Stefan Banach Medal.

The Stefan Banach Medal is awarded to individuals by the Presidium of the Polish Academy of Sciences (PAN) in recognition of outstanding contributions to the development of mathematical sciences. The Stefan Banach Medal was established in 1992, marking the centenary of Stefan Banach's birth.

7.2. The 2015 Stefan Banach Medal.

Henryk Iwaniec was awarded the 2015 Stefan Banach Medal of the Polish Academy of Sciences.

8.1. The Doob Prize.

The Doob Prize was established by the American Mathematical Society in 2003 and endowed in 2005 by Paul and Virginia Halmos in honour of Joseph L Doob (1910-2004). Paul Halmos (1916-2006) was Doob's first PhD student. Doob received his PhD from Harvard in 1932 and three years later joined the faculty at the University of Illinois, where he remained until his retirement in 1978. He worked in probability theory and measure theory, served as American Mathematical Society president in 1963-1964, and received the American Mathematical Society Steele Prize in 1984 "for his fundamental work in establishing probability as a branch of mathematics and for his continuing profound influence on its development." The Doob Prize recognises a single, relatively recent, outstanding research book that makes a seminal contribution to the research literature, reflects the highest standards of research exposition, and promises to have a deep and long-term impact in its area. The book must have been published within the six calendar years preceding the year in which it is nominated. Books may be nominated by members of the Society, by members of the selection committee, by members of American Mathematical Society editorial committees, or by publishers. The prize of US$5,000 is given every three years.

John Friedlander and Henryk Iwaniec were awarded the Joseph L Doob Prize at the 123rd Annual Meeting of the American Mathematical Society in Atlanta, Georgia, in January 2017 for their book Opera de Cribro, published in 2010 as volume 57 of the American Mathematical Society Colloquium Publications.

8.2. The 2017 Doob Prize.

John Friedlander and Henryk Iwaniec were awarded the Joseph L Doob Prize at the 123rd Annual Meeting of the American Mathematical Society in Atlanta, Georgia, in January 2017 for their book Opera de Cribro, published in 2010 as volume 57 of the American Mathematical Society Colloquium Publications.

8.3. Citation for the 2017 Doob Prize.

This citation is written by Professor Enrico Bombieri.

This monograph by two top masters of the subject is dedicated to the study of sieves in number theory and to their applications. Its Latin title could be translated literally as "A Laborious Work Around the Sieve," but the Latin has a conciseness easily missed in any translation.

The Eratosthenes sieve, going back to the 3rd century BCE, was a simple but efficient method to produce a table of prime numbers, and for a long time it was the only way to study the mysterious sequence of primes, at least experimentally. It was only in 1919 that the Norwegian mathematician Viggo Brun obtained the first quantitative results of the correct order of magnitude for the density of sifted sequences by combining the sieve with ideas from combinatorics. From another direction, the introduction of complex variable methods by Hardy, Ramanujan, and Littlewood and of techniques of harmonic analysis by Vinogradov helped to obtain the correct conjectures about the distribution of prime numbers of special type and of their fine distribution, such as the study of the sequence of gaps between prime numbers.

For a long time Brun's method and its refinements by Buchstab and many others were the only tools at the mathematician's disposal for obtaining unconditional results on the arithmetical structure of sequences of integers until, in 1950, Selberg put forward a new, simple, elegant method to study such questions. Selberg's method and Brun's combinatorial method were independent of each other and gave rise to new deep results on the arithmetic structure of special sequences. In the 1950s and early 1960s the new ideas of Linnik and Rényi gave origin to the so-called Large Sieve, particularly apt to the study of the distribution of sequences of integers in arithmetic progressions.

In the next thirty years many very deep results on classical questions previously considered to be inaccessible were obtained. Suffice it here to mention the asymptotic formula for the number of primes representable as the sum of a square and of a fourth power, obtained by Friedlander and Iwaniec in 1998, and a similar result by Heath-Brown in 2001 for the number of primes which are the sum of a cube and of twice a cube. So it was time for a new book dealing not only with the sieves per se but, in fact, with the very deep new techniques needed for the applications. The first nine chapters of this monograph deal with the sieves, followed by three chapters dedicated to the optimisation of parameters. The next ten chapters are dedicated to specific problems, including several milestone results. The last three chapters, which are a most original contribution to this monograph, deal with the future by raising new questions, giving partial answers, and indicating new ways of approaching the problems.

Two long appendices deal with technical results of general application. The bibliography, with 161 entries, is a major complement to this work. Everything is well written, the motivations of the arguments are well explained, and the numerous examples help the student to understand the subject in depth. These features distinguish this unique monograph from anything that had been written before on the subject and lift it to the level of a true masterpiece.

8.4. Biographical Sketch of Doob Prize winner: Henryk Iwaniec.

Henryk Iwaniec was born on 9 October 1947, in Elblag, Poland. He graduated from Warsaw University in 1971, and he received his PhD in 1972. In 1976 he defended his habilitation thesis at the Institute of Mathematics of the Polish Academy of Sciences, where he held various positions from 1971 until 1983. In 1983 he was promoted to extraordinary professor (which is one step below the ordinary professor) and was elected to member correspondent of the Polish Academy of Sciences. Henryk Iwaniec spent the year 1976-1977 at the Scuola Normale Superiore di Pisa and the year 1979-1980 at the University of Bordeaux I. He left Poland in 1983 to take visiting positions in the United States: at the Institute for Advanced Study in Princeton (1983-1984), at the University of Michigan in Ann Arbor (summer 1984), as Ulam Distinguished Visiting Professor at Boulder University (fall 1984), and again at the Institute for Advanced Study in Princeton (January 1985-December 1986). Iwaniec was appointed as New Jersey State Professor of Mathematics at Rutgers University, where he has held this position from January 1987 until the present.

Iwaniec was elected to the American Academy of Arts and Sciences in 1995, to the National Academy in 2006, and to the Polska Akademia Umiejetnosci in 2006 (foreign member). He received the Docteur Honoris Causa of Bordeaux University in 2006. Iwaniec twice received first prizes in the Marcinkiewicz contest for student works in the academic years 1968-1969 and 1969-1970. Among several other prizes he received are the Alfred Jurzykowski Award (New York, 1991); the Wacław Sierpinski Medal (Warsaw, 1996); the Ostrowski Prize (Basel, 2001, shared with Richard Taylor and Peter Sarnak); the Frank Nelson Cole Prize in Number Theory (American Mathematical Society, 2002, shared with Richard Taylor); the Leroy P Steele Prize for Mathematical Exposition (American Mathematical Society, 2011); the Stefan Banach Medal (Polish Academy of Sciences, 2015); and the Shaw Prize in Mathematical Sciences (Hong Kong, 2015, shared with Gerd Faltings). Henryk Iwaniec was an invited speaker at the International Congresses of Mathematicians in Helsinki (1978), Berkeley (1986), and Madrid (2006).

8.5. Response from John Friedlander and Henryk Iwaniec.

We are grateful to the American Mathematical Society and to the Joseph L Doob Prize Selection Committee for having chosen our book Opera de Cribro for this award.

We are, in particular, gratified by the recognition that this prize brings to the (beloved by us) subject of our book. The study of sieve methods in number theory began its modern history with the works of Viggo Brun just about one hundred years ago. Brun's works were of an elementary (though not at all easy) combinatorial nature, yet led to theorems about prime numbers that still today have found no other source of proof. The first few following decades saw further development of the sieve mechanisms themselves given by many people, most notably Atle Selberg. Beginning in the 1970s, the subject entered into a new period during which it has become possible to incorporate into the sieve structure deep results coming from several of the main sources which power modern analytic number theory more generally. These include, most frequently, harmonic analysis, both classical and automorphic; algebraic tools of various types; and arithmetic geometry. But anything is fair game. Basically, the modern sieve takes from mathematics anything it can use, and the more surprising the source, the more intensely the beauty shines through.

We also greatly appreciate the timing of the Joseph L Doob Prize. Although we spent five years working intensively on our Opera, it of course actually incorporates works of the authors dating back over a considerably longer period of time. This prize represents to us a milestone of our collaboration almost precisely forty years after it began in Pisa reading the preprint of The Asymptotic Sieve, written by Enrico Bombieri.

9.1. The Casimir Funk Natural Sciences Award.

Casimir Funk (1884-1967), for whom the Polish Institute of Arts and Sciences' Natural Sciences Award is named, made important contributions to the fields of hormone research, enzymology and chemical synthesis. However, he is best known for his pioneering research that lead to the discovery of vitamins and for defining the role of vitamins in nutrition. Funk postulated that nutritional deficiency diseases "can be prevented and cured by the addition of certain preventive substances which we call vitamins. The term "vitamin" (coined by Funk from "vita", Latin for life, and "amine") has become accepted as the description of a group of functionally related but structurally distinct substances. His hypothesis has had a major impact on the direction of research in a field that, before his discoveries, was filled with controversy. His work has guided the developing science of nutrition and had an impact on biochemistry and medicine.

The Casimir Funk Natural Sciences Award was first made by the Polish Institute of Arts and Sciences in America in 1995. The award honours an outstanding scientist of Polish origin (Polish born or of Polish ancestry) living and working in the United States or Canada.

9.2. The 2020 Casimir Funk Natural Sciences Award.

Henryk Iwaniec, New Jersey Professor of Mathematics, has been named the 2020 winner of the Casimir Funk Natural Sciences Award from the Polish Institute of Arts and Sciences in America, for his outstanding contributions to analytic number theory.

Henryk Iwaniec is only the second mathematician to have been awarded the prize, the first being Benoit Mandelbrodt.

9.3. The 2020 Casimir Funk Award presentation and lecture.

On 12 May 2021, Henryk Iwaniec delivered the lecture "Mathematical concepts, some research questions and a bit of history." It was presented as part of the KF Collegium of Eminent Scientists Lecture Series and was combined with the presentation of the Casimir Funk Natural Sciences Award to Professor Iwaniec by the President of the Polish Institute of Arts and Sciences Professor Robert Blobaum.

In his lecture, Professor Iwaniec illustrates the ideas that drive old and modern mathematics by answering a few notorious questions: is mathematics a science or an art, or do we discover or create the rules of mathematics. He describes a mathematical argument that leads to an operation that is inconsistent with common sense (a magic duplication of a ball for free). Examples of ineffective results are given. Questions about the distribution of prime numbers are discussed. Professor Iwaniec also speaks about the history of mathematics, in particular about the achievements of Polish mathematicians.

Henryk Iwaniec, Ph.D. is a Polish-American mathematician, and, since 1987, New Jersey State Professor at Department of Mathematics, Rutgers University. Iwaniec studied at the University of Warsaw, where he got his Ph.D. in 1972 under Andrzej Schinzel. He then held positions at the Institute of Mathematics of the Polish Academy of Sciences until 1983 when he left Poland. He held visiting positions at the Institute for Advanced Study, University of Michigan, and the University of Colorado Boulder before being appointed Professor of Mathematics at Rutgers University.

Iwaniec studies both sieve methods and deep complex-analytic techniques, with an emphasis on the theory of automorphic forms and harmonic analysis. In 1997, Iwaniec and John Friedlander proved that there are infinitely many prime numbers of the form $a^{2} + b^{4}$. Results of this strength had previously been seen as completely out of reach: sieve theory - used by Iwaniec and Friedlander in combination with other techniques - cannot usually distinguish between primes and products of two primes, say.

In 2001 Iwaniec was awarded the seventh Ostrowski Prize. The prize citation read, in part, "Iwaniec's work is characterised by depth, profound understanding of the difficulties of a problem, and unsurpassed technique. He has made deep contributions to the field of analytic number theory, mainly in modular forms on $GL(2)$ and sieve methods."

Professor Iwaniec became a fellow of the American Academy of Arts and Sciences in 1995. He was awarded the fourteenth Frank Nelson Cole Prize in Number Theory in 2002. In 2006, he became a member of the National Academy of Science and received the Leroy P Steele Prize for Mathematical Exposition in 2011. In 2012 he became a fellow of the American Mathematical Society. In 2015 he was awarded the Shaw Prize in Mathematics.

9.4. Henryk Iwaniec's Casimir Funk Award lecture.

Henryk Iwaniec delivered the Casimir Funk Award lecture "Mathematical concepts, some research questions and a bit of history" on 12 May 2021. The lecture attempts to present Henryk Iwaniec's view of mathematics to a general audience.

You can read our transcript of part of this lecture at THIS LINK.