1.1. The Karp Prize.

The Karp Prize is awarded by the Association for Symbolic Logic. The Karp Prize, established in 1973 in memory of Professor Carol Karp, is awarded for an outstanding paper or book in the field of symbolic logic. This award is made by the Association on the recommendation of the Association for Symbolic Logic Committee on Prizes and Awards. This prize is given for a "connected body of research, most of which has been completed in the time since the previous prize was awarded." The Prize is awarded every five years and consists of a cash award.

1.2. The 1993 Karp Prize to Ehud Hrushovski.

The 1993 Karp Prize was awarded to Ehud Hrushovski, Massachusetts Institute of Technology and Alex Wilkie, University of Oxford.

The Association for Symbolic Logic Committee on Prizes and Awards selected Ehud Hrushovski, MIT, and Alex Wilkie, Oxford, as the recipients of the 1993 Karp Prize. Hrushovski was honoured for his introduction of new methods in geometric stability theory; Wilkie was honoured for proving the model completeness of the field of real numbers with the exponential function. The two prizes were awarded at a special session at the Association for Symbolic Logic Annual Meeting in March, at which John Baldwin and Angus Macintyre gave talks which summarised the accomplishments for which Hrushovski and Wilkie were being honoured.

2.1. The Anna and Lajos Erdős Prize in Mathematics.

This prize was first awarded in 1977 and was, at that time, known as the Erdős Prize. The name was changed in 1996 to the Anna and Lajos Erdős Prize reflecting the original wish of Paul Erdős. Anna and Lajos Erdős were Paul Erdős's parents. The prize is awarded to an Israeli mathematician in an institution of higher education in Israel, in the fields of theoretical mathematics or computer science, who was under the age of 41 as of 31 May in the award year. The chairman of the jury making the award is appointed by the Committee of the Israel Mathematical Union. The prize is awarded at the Union's annual conference and the recipient is invited to lecture on their work to that conference.

2.2. The 1994 Erdős Prize to Ehud Hrushovski.

The Israel Mathematical Union awarded the 1994 Erdős Prize to Ehud Hrushovski (Hebrew University).

3.1. The Karp Prize.

The Karp Prize is awarded by the Association for Symbolic Logic. For more details see 1.1 above.

3.2. The 1993 Karp Prize to Ehud Hrushovski.

The recipient of the 1998 Karp Prize of the Association for Symbolic Logic was Ehud Hrushovski of the Hebrew University, Jerusalem, for his work on the Mordell-Lang Conjecture. This award was made by the Association on recommendation of the Association for Symbolic Logic Committee on Prizes and Awards.

4.1. The Rothschild Prize.

The Rothschild Prize was established in 1959 by Yad Hanadiv, a philanthropic foundation acting on behalf of the Rothschild family in Israel. Prizes are awarded to support, encourage and advance the Sciences and Humanities in Israel, and recognize original and outstanding published work in the following disciplines: Agriculture, Chemical Sciences, Engineering, Humanities, Jewish Studies, Life Sciences, Mathematics, Physical Sciences and Social Sciences. Prizes are awarded in two-year cycles; in each discipline, a Prize is awarded once in four years.

4.2. The 1998 Rothschild Prize to Ehud Hrushovski.

The 1998 Rothschild Prize was awarded to Professor Ehud Hrushovski, Mathematics.

5.1. The Michael Bruno Memorial Award.

The Michael Bruno Memorial Award is made by the Israel Institute for Advanced Studies. The Michael Bruno Memorial Award is granted annually to three outstanding mid-career Israeli scholars who have demonstrated an exceptional originality of mind, dedication and ground-breaking impact in their research, and who have succeeded in influencing and re-shaping their field of expertise. This award, a personal prize in an amount of 200,000 NIS, is both the highest recognition of past accomplishments and a vote of confidence in the laureates' future achievements and contributions to research and to Israeli academia.

5.2. The 2004 Michael Bruno Memorial Award to Ehud Hrushovski.

Ehud Hrushovski is a professor of Mathematics at The Hebrew University of Jerusalem. He received his Ph.D. from the University of California, Berkeley in 1986, where his Ph.D. thesis revolutionised stable model theory. He held positions at Princeton University and MIT before joining the faculty at The Hebrew University in 1994. Prof Hrushovski is a recipient of the Erdős Prize of the Israel Mathematical Union in 1994, the Rothschild Prize in 1998, and the Karp Prize of the Association for Symbolic Logic in in 1993 and 1998.

6.1. The Heinz Hopf Prize.

The Heinz Hopf Prize at ETH Zurich was established thanks to a donation from Dorothee and Alfred Aeppli. The CHF 30,000 award honours outstanding scientific achievements in the field of pure mathematics. The prize is awarded every two years on the occasion of the Heinz Hopf Lectures, which already have a long-standing tradition at ETH Zurich. The lectures are given by the laureates.

The Heinz Hopf Prize Committee evaluates the nominations for the prize and submits a recommendation to the Professors' Conference at the Department. The Committee's chairman calls for nominations, organises the Heinz Hopf Lectures and handles the financial aspects related to the prize.

6.2. Why Hopf Lectures? by Alfred Aeppli.

"To me, it is a great idea to gather from time to time in memory of Heinz Hopf, at the Swiss Federal Institute of Technology in Zurich (ETH), for some special lectures on current mathematical research. We remember, Heinz Hopf (1894-1971) was active at the ETH as one of the finest professors from 1931 to 1965 (see History). He will always be an inspiration to many of us in mathematics and in mathematical education at the university level. He was a mind at work who thrived in the world of disciplined curiosity full of intellectual challenges. He was a mathematical thinker who taught us to think. Once he had arrived in Switzerland in 1931 he was happy to live in the community provided by the Swiss higher education system. He remained a loyal member of the ETH until he died in 1971. Heinz Hopf demonstrated doing mathematics in talks, lecture courses, discussions and in his publications. I remember the introductory linear algebra course in 1947-48, the impression Heinz Hopf made on us students was outstanding and unforgettable. A student, by attending a course given by Hopf, got a good answer to the question "What is mathematics?" The notions had to be clear, the intellectual tools of the highest calibre, the arguments compelling and impeccable. There never was any doubt what could be accepted as valid.

Heinz Hopf helped create first rate thesis work in a number of cases over the years, e.g. Stiefel, Eckmann, Gysin and Samelson were among his Ph.D. students. He became the initiator of a mathematical school in Switzerland cultivating the development of ideas - new at the time - in mathematical disciplines, which can be called "geometric" in a broad sense. They include general and algebraic topology, differential geometry, manifolds, structures on manifolds, topological groups, algebraic geometry.

For Hopf, a specific well defined question was usually the starting point of a project. One example is particularly illuminating. At the time the higher homotopy groups had been introduced (by Hurewicz) it was noticed that they are abelian - like the homology groups -, and "therefore" the question was raised: can the higher homotopy groups be used to prove anything new of interest? In response to this question, Hopf produced the Hopf fibration of the 3-sphere by circles, the Hopf invariant and the homotopy classification of the continuous maps from the 3- to the 2-sphere. In those days it wasn't even clear that there are non-trivial (i.e. essential or non-contractible) continuous maps from the 3- to the 2-sphere. Hopf's answer was one of the first results in "modern" homotopy theory. Many ramifications followed, including notions like homology groups of a discrete group, Eilenberg-MacLane spaces, Moore spaces, Postnikov towers, obstructions to cross sections in fiber bundles.

Another fundamental contribution is Hopf's theorem on the homology groups of compact group manifolds. This stimulated the development of Hopf algebras with their applications in many places, e.g. in the theory of cohomology operations. Global properties of manifolds were of great interest to Hopf, including existence and classification of various structures on manifolds. He had an extraordinary ability to see interesting phenomena, e.g. the cartesian product of the 3-sphere with the circle was first recognised by Hopf as an example of a non-kaehlerian complex manifold. In classical differential geometry, Hopf's nicest result is his characterisation of the ordinary 2-sphere as the only compact surface of genus 0 in Euclidean space with constant mean curvature. Many more mathematical gems out of Hopf's workshop could be mentioned.

In conclusion: let us continue to do mathematics, and let us never forget those who did mathematics before us, among them Heinz Hopf."

6.3. The 2019 Heinz Hopf Prize to Ehud Hrushovski.

The 2019 Heinz Hopf Prize goes to the Israeli mathematician Professor Ehud Hrushovski for his outstanding contributions to model theory and their application to algebra and geometry. Born in 1959, he earned his PhD from the University of California at Berkeley in 1986, then held positions at Princeton University, MIT and the Hebrew University of Jerusalem, among others, before being appointed Merton Professor of Mathematical Logic at Oxford University. He has been a plenary speaker at the 1998 International Congress of Mathematicians in Berlin and received a number of awards for his seminal works, including in 1993 (together with Alex Wilkie) and 1998 the Karp Prize of the Association of Symbolic Logic, which is awarded only every five years. ETH Zurich now awards the 2019 Heinz Hopf Prize to Hrushovski for his outstanding contributions to model theory and their application to algebra and geometry.

Model theory is a relatively young subject. It has been developed systematically since the early 1950s, but has roots in older subjects, including logic, universal algebra and set theory. Put simply, model theory applies logic to study structures in pure mathematics. Mathematical structures define operations and relations on a set of objects. Within such a framework, statements can be made about a structure that can be either true or false - depending on the interpretation in a given context. This is not a flaw, but instead a consequence of the different mathematical languages that can be used to express ideas. Model theory provides tools to abstract the problem. This allows, on the one hand, obtaining new information about the mathematical structure studied and, on the other hand, developing general theorems from interesting property of a theory, which help in turn to understand mathematics itself on a very general level.

Surprising and beautiful contributions

In the course of his career, Hrushovski has made substantial contributions to both aspects, developing model theory as a subject and applying its tools to classical problems in various fields of mathematics, often in ways that are described as surprising and beautiful by those in the field. Among his most important contributions are the 1988 construction of a new strongly minimal set disproving a 1984 conjecture of Boris Zilber, and his work with Zilber on Zariski geometries and his proof of the Mordell-Lang conjecture for functions fields in 1996, his work with Zoé Chatzidakis on the model theory of difference fields and a new proof of the Manin-Mumford conjecture in 2001, his work with Ya'acov Peterzil and Anand Pillay on NIP ("not the independence property") structures, and his 2012 paper on approximate subgroups, which had a key role in subsequent work of Emmanuel Breuillard, Ben Green and Terence Tao.

Honouring the memory of Heinz Hopf.

ETH Zurich awards the Heinz Hopf Prize every two years for outstanding scientific work in the field of pure mathematics. The prize honours the memory of Professor Heinz Hopf (1894-1971), who came to ETH Zurich in 1931 to succeed the mathematician, physicist and philosopher Hermann Weyl. Hopf remained at ETH until 1965, where he made important contributions to various fields of mathematics, reflected in numerous entities named after him, including Hopf invariants, Hopf algebras and Hopf links. But he left his mark as a teacher, too: "A student, by attending a course given by Hopf, got a good answer to the question 'What is mathematics?' The notions had to be clear, the intellectual tools of the highest calibre, the arguments compelling and impeccable." Thus spoke his former student Alfred Aeppli (1928-2008) recalling his experience of Hopf's introductory course in linear algebra, which he attended in 1947-48.

Later in his life, Alfred Aeppli and his wife Dorothee made a generous donation in memory of his teacher, which became the basis to establishing the Heinz Hopf Prize. ETH Zurich awards the prize since 2009 on the occasion of the long-standing biennial Heinz Hopf Lectures, which are now given by the Hopf Prize laureate. In addition, the award carries a prize money of CHF 30,000.

With the 2019 Hopf Prize, exceptional contributions in the wider field of logic are honoured, whereas the latest previous awards were for work in the areas of geometric analysis (2017 with Richard Schoen) and algebraic geometry (2015 with Claire Voisin). Preceding the award ceremony on Monday, 28 October 2019, Ehud Hrushovski will give the first Heinz Hopf lecture 2019.

Joël Mesot, President of the ETH Zurich, handed over the prize to Ehud Hrushovski at the award ceremony on 28 October 2019. Urs Lang, Chair of the Heinz Hopf committee and Lou van den Dries, professor at the University of Illinois, honoured the work of Ehud Hrushovski.

6.4. Oxford University congratulates Ehud Hrushovski winning the 2019 Heinz Hopf Prize.

On Monday, 28 October, Ehud Hrushovski received the 2019 Heinz Hopf Prize. With the award, ETH Zurich honours the contributions of the Oxford mathematician to model theory - both to developing the subject, and to applying it to diverse areas of mathematics.

7.1. The Pólya Prize.

The Pólya Prize is awarded by the London Mathematical Society. It is awarded every 2 years out of 3, when De Morgan Medal is not awarded. The Prize was first awarded in 1987. It is named after George Pólya, who was a member of the society for over 60 years. The prize is awarded:-

... in recognition of outstanding creativity in, imaginative exposition of, or distinguished contribution to, mathematics within the United Kingdom.

It cannot be given to anyone who has previously received the De Morgan Medal.

7.2. Short citation for the 2021 Pólya Prize to Ehud Hrushovski.

Professor Ehud Hrushovski FRS, of the University of Oxford, is awarded a Pólya Prize for his profound insights that transformed very abstract model-theoretic ideas into powerful methods in well-established classical areas of geometry and algebra.

7.3. Long citation for the 2021 Pólya Prize to Ehud Hrushovski.

Professor Ehud Hrushovski FRS, of the University of Oxford, is awarded a Pólya Prize for his profound insights that have transformed very abstract model-theoretic ideas around definability into powerful methods that have had spectacular success in diverse areas of geometry and algebra, such as diophantine geometry, several varieties of motivic mathematics, p-adic integration, rigid geometry, permutation groups and algebraic groups, and approximate groups. This creates an amazing complex of deep connections.

Hrushovski obtained his PhD from Berkeley in 1986. Model theory had, in the previous twenty years, developed rapidly, with two outstanding achievements in the mid 1960s being Morley's work on omega-stable theories (with a prominent role for indiscernibles and emerging geometric notions, for example around dimension.), and the Ax-Kochen-Ershov work on Henselian fields (followed by Ax's work on the model theory of finite fields).

Already in the 1980's Shelah and Zilber were inspirational figures in model theory. Shelah had his eyes on general classification of models of theories, while Zilber was intent on a geometrical approach to theories, based on definability considerations such as interpretability. An idea that links both approaches is that of stable theories. Stability generalises Morley's omega-stability, linked to algebraically closed fields, thus connected to geometric notions. In particular it links to the notion of irreducible curve, with the general notion being called strong minimality. In stable theories in general there are various notions of a geometric and combinatorial nature, often forbidding to a novice. Hrushovski, by exceptional efforts both mathematical and methodological, has revealed the fertility of the geometric theory for various sophisticated classical mathematical subjects.

His thesis concerned groups interpretable in stable theories, and can be seen as a vast generalisation of the fundamental coordinatisation theorem of projective geometries. Zilber had made a bold and beautiful conjecture about strongly minimal sets. Hrushovski gave an intriguing family of counterexamples. Hrushovski and Zilber reflected deeply, repaired, and then proved the conjecture, by imposing extra conditions. This allowed Hrushovski to make a startling contribution to diophantine geometry, with a proof of the characteristic p version of a conjecture of Mordell-Lang.

He wrote a rich and beautiful book with Cherlin, connecting permutation group theory and associated geometry to basic problems in the model theory of finite structures. Hrushovski has had a major influence on profound recent work on approximate groups by Breuillard, Green and Tao, via stability-theoretic insights.

Much of his recent work is connected to valued fields and pseudofinite structures. With Haskell and MacPherson he gave a comprehensive study of definable equivalence relations in $p$-adic fields (such fields are not stable). This deep analysis is of basic importance in extending greatly Denef's work which uses model theory to analyse the structure of $p$-adic Poincaré series. Issues of uniformity in $p$ are crucial for the work of Denef and Loeser on a powerful theory of motivic integration. Hrushovski and Kazhdan, in turn, developed a different motivic integration using ideas of geometric model theory, and obtained new results in algebraic geometry, as well as a quite new perspective on model theory of henselian fields.

Hrushovski and Loeser gave a model-theoretic version of Berkovich spaces, using the refined model theory to make the topology closer to classical topologies. O-minimality (which is nowadays very important for diophantine geometry) is deeply involved, as it is in the Hrushovski-Kazhdan work. More recent novel model theoretic work, with Ducros, connects complex and nonarchimedean integrals.

8.1. 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 one 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.

8.2. The 2022 Shaw Prize in Mathematical Sciences.

The 2022 Shaw Prize in Mathematics was awarded jointly to Noga Alon and Ehud Hrushovski for their remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences.

8.3. About Ehud Hrushovski.

Ehud Hrushovski was born in 1959 in Israel and is currently Merton Professor of Mathematical Logic, University of Oxford, UK and a Fellow of Merton College, Oxford University, UK. He obtained his Bachelor's degree and PhD in Mathematics from the University of California, Berkeley, USA in 1982 and 1986 respectively. He was an Instructor (1987-1988) and Visiting Assistant Professor (1988-1989) at Princeton University, USA. He joined the Massachusetts Institute of Technology (MIT), USA where he was successively Assistant Professor (1988-1991), Associate Professor (1992-1994) and Full Professor (1994). While working at MIT, he also served as an Assistant Professor (1991-1992) and became a Full Professor (1994-2017) at the Hebrew University of Jerusalem, Israel. He moved to the University of Oxford in 2016, where he has been appointed Merton Professor of Mathematical Logic (2016-). He is a member of the Israel Academy of Sciences and Humanities and the American Academy of Arts and Sciences.

8.4. The contribution of Noga Alon and Ehud Hrushovski.

The Shaw Prize in Mathematical Sciences 2022 is awarded in equal shares to Noga Alon, Professor of Mathematics at the University of Princeton, USA and Ehud Hrushovski, Merton Professor of Mathematical Logic, University of Oxford, UK for their remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences.

Noga Alon works in the broad area of discrete mathematics. He introduced new methods and achieved fundamental results which entirely shaped the field. Among a long list of visible results with applications, one can extract the following contributions. With Matias and Szegedy he pioneered the area of data stream analysis. With Milman he connected the combinatorial and algebraic properties of expander graphs. With Kleitman he solved the Hadwiger-Debrunner conjecture (1957). In his "combinatorial Nullstellensatz" he formulated in a special case an explicit version of Hilbert's Nullstellensatz from algebraic geometry which is widely applicable for discrete problems. This led to a proof (1995) of the Dinitz conjecture on Latin squares by Galvin and further generalisations. With Tarsi he bounded the chromatic number of a graph. With Nathanson and Ruzsa he developed an algebraic technique yielding a solution to the Cauchy-Davenport problem in additive number theory. His book with Spencer on probabilistic methods became the essential basic manual on probability, combinatorics and beyond.

Ehud Hrushovski works in the broad area of model theory with applications to algebraic-arithmetic geometry and number theory. Among a long list of visible results with applications, one can extract the following contributions. He introduced the group configuration theorem as a vast generalisation of Zilber's and Malcev's theorems, which became a powerful tool in geometric stability theory and eventually enabled him to solve the Kueker's conjecture for stable theories. With Pillay he proved a structure theorem on groups which led him to then prove the Mordell-Lang conjecture in algebraic geometry in positive characteristic. This came as a big surprise. He disproved a conjecture by Zilber on strongly minimal sets, introducing a method which became an essential technique for estimating complexity. He wrote with Chatzidakis a theory of difference fields which, he showed later, has striking applications to dynamics in geometry over finite fields, and was for example a key tool to solve the Gieseker conjecture on the structure of D-modules over finite fields. He found a proof of the Manin-Mumford conjecture (Raynaud's theorem) using his tools ultimately stemming from logic. He gave algorithms to compute Galois groups of linear differential equations. Finally, he developed a theory of integration in valued fields and non-archimedean tame geometry, starting from his work with Kazhdan (2006) and finishing with his work with Loeser (2016).

Mathematical Sciences Selection Committee
The Shaw Prize

24 May 2022 Hong Kong

8.5. Essay on the 2022 Shaw Prize.

Historians consider mathematics as one of the oldest branches of science, if not of all human intellectual activities. They date tally numbers back to 23,000 BCE. Several systems of numerals were developed by the Romans, the Indians, the Chinese, the Mesopotamians … till we adopted the Hindu-Arabic numeral system in the 14th century. The numbers {1, 2, …} are part of our everyday life. A collection of objects is called countable if we can count it as {1, 2, …}. Discrete mathematics is the part of mathematics which studies properties that can be counted, in opposition to continuous mathematics which studies continuous or differentiable functions on spaces, which was initiated by Leibniz and Newton in the 17th century. These days we often associate discrete mathematics with codes and computer science, e.g., in relation with breaking the German codes in WWII, or more recently in the development of smart phones.

Noga Alon has made profound contributions to discrete mathematics with notable applications to theoretical computer science. Remarkably, some of his results also interact with algebraic geometry, which started in the 4th century BCE with the Greeks drawing in the sand the different shapes of conics, the intersections of cones with planes, and with algebraic topology, which started with Euler realising in the 18th century that a walk through the city of Königsberg that crosses each of its seven bridges once and only once is impossible. We mention two of Alon's many results, but his output also touches other domains such as graph theory, probability theory, and complexity theory.

The Nullstellensatz in algebraic geometry, due to Hilbert at the end of the 19th century, describes all polynomial functions that vanish on the zero set of finitely many polynomial functions. It is perhaps the most foundational result in algebraic geometry. Alon's 1999 combinatorial Nullstellensatz studies the special case in which the zero set consists of a large box. He is then able to control precisely several invariants attached to Hilbert's solutions. This ingenious formulation has led to powerful results in extremal combinatorics, graph theory, additive number theory and combinatorial geometry.

Helly's theorem at the beginning of the 20th century shows that given an infinite family of compact convex sets in the $d$-dimensional Euclidean space, their intersection is non-empty if all intersections of $d + 1$ among them is non-empty. It is one of the core theorems in convex set theory. The 1957 Hadwiger-Debrunner $(p, q)$-problem, solved in 1992 by Alon and Kleitman, concerns a difficult generalisation of this theorem where instead one assumes that amongst any $p$ of the sets there are $q$ with a non-empty intersection. The solution was a tour de force and required the development of tools that found additional applications in discrete and computational geometry.

Mathematical logic is the branch of mathematics which is arguably the closest to philosophy. Through the work of Gödel and Gentzen among others, it developed at the beginning of the 20th century the foundations of various areas of mathematics and shaped their axiomatisation. Within it, model theory, starting in the 1950s with the work of Tarski, studies the formal language which underlies a mathematical structure.

Ehud Hrushovski has made profound contributions to model theory with applications to a broad list of topics in algebraic geometry and group theory, as well as in combinatorics and number theory. Among Hrushovski's whole output we mention two results that touch the first two areas mentioned.

The study of Euler's proof on the Königsberg bridges led to the concept of a topology on a space as the data of closed sets with certain properties. It goes back to Cauchy in the 18th century and Gauss in the 19th century. The intuition comes from the Euclidean space in which we live, where it is possible to separate points by small neighbourhoods, a property eventually singled out by Hausdorff in the first half of the 20th century. Zariski, at approximately the same time, defined a much coarser topology on algebraic spaces simply by declaring that the zero-sets of polynomial functions are the closed sets. For this topology, nowadays called the Zariski topology, the separation property does not hold. Hrushovski and Zilber in 1993 characterised the Zariski topology through a collection of combinatorial and elementary properties based on the notion of dimension. Remarkably, this new vision of algebraic geometry enabled Hrushovski to prove the Mordell-Lang conjecture in positive characteristic: on algebraic spaces defined by polynomial functions with coefficients in generalised Galois congruence fields, and on which it makes sense to add points, one can characterise the subspaces on which it still makes sense to add points.

The notion of a group, due to Galois in the first half of the 19th century, is central in all branches of mathematics. It describes the symmetries of a mathematical object. It is a key concept in Galois theory equating the theory of field extensions with the one of their group of symmetries. Group theory is one of the most studied areas in mathematics, for example finite groups, topological groups, algebraic groups, among them linear groups etc. The subsets of a group which respect the symmetries are called subgroups. Approximate subgroups are those subsets which miss by very little (in a precise way) the symmetry property. They were defined and studied in additive combinatorics, a new branch of combinatorics, starting in 2012. Hrushosvski drew parallels between those approximate subgroups and certain structures in model theory which enabled him to solve a conjecture of Green on the structure of certain approximate subgroups. These results played a crucial role in the proof of a fundamental theorem of Breuillard, Green and Tao on the structure of approximate groups.

Noga Alon and Ehud Hrushovski have made remarkable contributions to discrete mathematics and model theory with interactions with algebraic geometry, topology and computer science. The methods they developed have become the basis of many further developments in the areas of mathematics that they have profoundly shaped.

29 September 2022 Hong Kong

8.6. Ehud Hrushovski autobiography.

I grew up in West Jerusalem, not a large town in the 1960s, and sharply bordered in space and in time. My mother was a psychologist, my father a theoretician of literature. The apartment was full of books, almost all in languages I could not read; works of literature or the human sciences. In retrospect, I can remember some very meaningful contacts with mathematics in those early years; but I was not conscious of this pattern at the time. Certainly, it did not occur to me that mathematics may exist as a profession.

I finished high school at seventeen; this gave me a year before the compulsory three-year Israeli army service. Influenced by my father, I was excited about the prospects for a theory of the understanding of natural language, for elucidating the mystery of metaphors. I applied to study Mathematics and Philosophy at Oxford, thinking of the mathematical part as merely a sensible preparation. But at Oxford I was overwhelmed by the newness and beauty of the mathematical edifice I was beginning to glimpse. When I went back to school in Berkeley in 1980, I knew it was mathematics that I wanted to study.

As I write this, my youthful dreams about language and metaphor seem far away. But perhaps I did not stray infinitely far! If one had to say today what single category is most closely associated with model theory, the reply would surely be theories in first-order languages, with interpretations - exhibiting hidden similarities of structure - as their connecting morphisms.

In 1982, I began working towards a PhD at Berkeley; Leo Harrington became my supervisor. I completed it in 1985/86 as an exchange student in Paris. Saharon Shelah had recently completed his search for the "main gap", drawing robust dividing lines of order and disorder among first order theories; I made contributions to the stable part, showing in particular that many theoretical phenomena were governed by definable groups. I spent the next three years in Rutgers and Princeton, deeply influenced by Gregory Cherlin. Totally categorical theories formed the innermost region of Shelah's classification; there, Zilber had shown that certain specific mathematical theories, originating in linear algebra over a finite field, have a relation to the whole that is not simply illustrative but formative: in some sense they generate that class, and even the simplest general properties cannot be understood without this recognition.

Cherlin had given another proof of Zilber's theorem, relating it to the newly achieved classification of the finite simple groups into finitely many families. But Zilber's theorem corresponded to only two of these. Cherlin and I later extended the entire theory of total categoricity to a class representing all families of simple groups over a given finite field. Stability had seemed the indispensable bedrock for Shelah's theory; only on that basis could one define the deeper notions of regular types, orthogonality, domination, and further concepts of geometric stability. And yet here it turned out that the upper stories of this house could be transferred intact to an unstable setting, with entirely different, group-theoretic derivations of their basic properties. This was to me a revelation, later multiply confirmed, not only about the specific subject but about the nature of mathematics.

Zilber further conjectured that within a wider class of theories, algebraic geometry plays a similarly fundamental role. Since my days in Berkeley and Paris, this conjecture fascinated me; there was nothing I wanted to prove more. But at some point I began to suspect it may not be true, and constructed a counterexample. The dimension-governed method that I used found many applications; a recent one by David Evans resolved a basic question in structural Ramsey theory.

In 1990 I took up my first tenure-track position in MIT, and later alternated between MIT and the Hebrew University in Jerusalem. I met Zilber, who was eventually able to leave the Soviet Union. Together, we proved his conjecture under additional topological assumptions. I used this to show it held true in the world of ordinary differential equations, in any characteristic. That in turn led to the solution of the Mordell-Lang conjecture for function fields; a purely algebro-geometric proof, by Roessler, came twenty years later. It was extraordinary to see to what extent model theory predicts the landscape of algebraic ODE's, starting with nothing more than the Leibniz rule.

I moved fully to Jerusalem around 1994. I married Merav there, and our son David was born in 2005. I engaged in studies of these ideas within a number of theories related to geometry: valued fields, in three extended collaborations with Haskell and Macpherson, Kazhdan and Loeser; difference equations, in many papers with Zoé Chatzidakis, as well as one showing that the Frobenius automorphism of arithmetic geometry holds a critical place in the wide theory; definable measures in NIP theories, with Peterzil and Pillay, and later in general. Each of these turned out to have meaningful applications within the field under study. A very long range project with Itay Ben Yaacov attempts to incorporate global aspects of geometry.

In 2016 we moved to Oxford. I often teach Maths and Philosophy students, closing a circle with my first year of university. Recently, I have again been working on questions of fundamental model theory, rather than a specific theory, but they too have applications to approximate subgroups and approximate lattices.

29 September 2022 Hong Kong

8.7. Acceptance speech by Professor Ehud Hrushovski on 19 October 2022.

I'm immensely happy and proud to receive with Noga Alon the Shaw Prize Mathematical Sciences. I'm very grateful to the Shaw Family and the Shaw Prize Foundation for their vision in creating this extraordinary prize and overall by the list of mathematicians who were the previous recipients.

In the mid-1980s when I joined the field Saharon Shelah had just completed his classification theory program designed to answer an apparently abstract question to determine the number of different models a given theory can have in uncountable cardinalities. This guided his discovery of a sequence of profound dichotomies allowing for increasingly deep structured theories. Other ideas were injected by Boris Zilber governing the fine structure of theories where this number is always one. A different part of the discipline pioneered by Tarski and Robinson put many significant areas of mathematics on an axiomatic footing in contrast with Gödel's undecidability results for arithmetic. It turned out that these could be combined to shed light on questions as concrete as solutions to polynomial equations.

A number of extended collaborations were very meaningful to my life in mathematics. Even beyond direct joint work, I was very lucky to join groups guided first and foremost by common respect for the depth and beauty of our subject matter. This was true of the cohesive and highly active global model theory community, of the logic group in Oxford that I now belong to, and of the Mathematics Department in the Hebrew University in Jerusalem that was my home for 25 years.

To a graduate student, the geometry of mathematics appears tree-like, hyperbolic. It takes years to learn the basics of your field; other branches diverge at the start and seem forever inaccessible. But mathematics is made from ideas, and is only weakly reflected by formal statements. The exponential growth of the tree of propositions is tempered by an equal number of interconnections and interpretations. It is at first a great surprise, and always a great joy, to find the same idea or the same question shared by many fields with entirely different foundations.

I'd like to view this prize as a recognition of the part taken by model theory in this common movement. I feel very fortunate to have been chosen as it's representative.

8.8. Oxford University congratulates Ehud Hrushovski winning the 2022 Shaw Prize.

Merton Professor of Mathematical Logic Ehud (Udi) Hrushovski has been announced as the joint winner of the 2022 Shaw Prize in Mathematical Sciences for his "remarkable contributions to discrete mathematics and model theory with interaction notably with algebraic geometry, topology and computer sciences". He shares the prize with Noga Alon, Professor of Mathematics at Princeton University.

Professor Hrushovski commented:

It is an amazing honour to join, along with Noga Alon, the extraordinary group of mathematicians who have received the Shaw prize; a list beginning with Shiing-Shen Chern and Merton's Andrew Wiles. Model theory studies mathematical structures from the point of view of language; born of philosophy, it has attained considerable mathematical depth enabling interaction with many fields. This quality was created and maintained by many people, including my predecessors in Merton, Dana Scott, Angus Macintyre, and Boris Zilber.

I was very happy that several of my close collaborators were mentioned in the citation, including Boris, whose ideas have shaped much of my mathematical life. I view the prize as a recognition by the mathematical community of the achievements of my field, and am deeply grateful and proud to be chosen as the representative.

Before coming to Oxford, Udi Hrushovski studied at UC Berkeley, and he has worked at Princeton, Rutgers, MIT, Paris, and for 25 years at the Hebrew University of Jerusalem.

Professor Hrushovski's work is concerned with mapping the interactions and interpretations among different mathematical worlds. Guided by the model theory of Robinson, Shelah and Zilber, he investigated mathematical areas including highly symmetric finite structures, differential equations, difference equations and their relations to arithmetic geometry and the Frobenius maps, aspects of additive combinatorics, motivic integration, valued fields and non-Archimedean geometry. In some cases, notably approximate subgroups and geometric Mordell-Lang, the metatheory had impact within the field itself, and led to a lasting involvement of model theorists in the area. He also took part in the creation of geometric stability and simplicity theory in finite dimensions, and in establishing the role of definable groups within first order model theory. He has co-authored papers with 45 collaborators and has received a number of awards including the 1998 Karp Prize, the 1994 Erdős Prize, the 1998 Rothschild prize, and the 2019 Heinz Hopf prize.

There are three annual Shaw Prizes, in Astronomy, Life Science & Medicine, and Mathematical Sciences, each of which is accompanied by an award of US$1.2 million. The prizes, established in 2002 in Hong Kong and first presented by the Shaw Prize Foundation in 2004, honour living individuals 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.