... poet, author, teacher, translator, and literary scholar.

I was very struck by certain mathematical things. I remember very well learning the proof of commutativity of multiplication, being shown a rectangle divided into small squares. I remember even being sceptical about it, being unsure whether one was counting the corners of the squares, or the actual squares, whether there was a cheat there. And I remember the real line being introduced, and the revelation of some connection between numbers and something drawn. On the whole, though, it was pretty dreary and boring. It was all about learning things by heart. But from time to time, I was struck by certain ideas.

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.

... overwhelmed by the newness and beauty of the mathematical edifice I was beginning to glimpse.

I feel very lucky to have been a student of Leo Harrington's. His depth and breadth of vision are on a scale that I did not know existed. But he is able to know what is major without making anything seem minor, and to create a sense of excitement at every step of the way. Learning from him has been a great pleasure.

I was supported financially by a Berkeley Graduate Fellowship in 1982/83 and a National Science Foundation Graduate Fellowship in 1983/84-1985/86.

... was a huge revolution [which] turned [model theory] from a nice collection of ideas to a really deep area with its own systematic methodology at a completely different order of magnitude. This is stability theory.

... his introduction of new methods in geometric stability theory.

The easiest concrete example of model theory to explain is Tarski's study of Greek geometry, which is a rich subject. It appears to be about certain figures in the plane, cut out by some algebraic equations. It's a great achievement to do one theorem at a time, to prove the Pythagorean theorem or the great many theorems proven by the Greeks, each one a separate achievement. At a certain point one gets the feeling that there's also a domain, something closed there, where you could describe the whole family of theorems and maybe systematise them. So we have Tarski's theorem about the decidability of elementary geometry. Elementary geometry means that any question we can ask about algebraic shapes, including conic sections, hyperbolas, parabolas, or also more complicated ones, in fixed dimension and fixed degree, can be answered systematically. Descartes had a feeling that it would work, that he could somehow do analytic geometry and answer questions like that, with algebraic manipulations. Tarski proved that you can systematically answer all these things. That sheds a different light on the whole field; it's one whole level of abstraction up from a single elementary theorem in geometry. And surprisingly it reflects back once you have a global understanding of the area, that you can also tackle concrete questions using that - there were specific questions to which people were looking for answers - about complicated algebraic shapes, singularities and so on. Having the systematic understanding which Tarski had is the beginning of model theory. It gives, e.g., ways to prove effectiveness results but also to prove numerical bounds of the questions. Sometimes a specific question is so difficult that you can only get a handle on them via a more abstract and wider view.

Professor Ehud Hrushovski is a Professor of Mathematics at the Hebrew University of Jerusalem. Formerly, Hrushovski was the Professor of Mathematics at the Massachusetts Institute of Technology. He received his Ph.D. from the University of California, Berkeley in 1986, where his Ph.D. thesis revolutionised stable model theory. He has applied methods and theorems from model theory to other parts of mathematics, particularly arithmetic algebraic geometry, including the Mordell-Lang conjecture for function fields and a case of the Jacobi conjecture for difference fields. Additionally, he has advanced model theory, making algebra appear inherently rather than being imposed.

Ehud Hrushovski is a mathematical logician, concerned with mapping the interactions and interpretations among different mathematical worlds. He studied in UC Berkeley, worked in Princeton, Rutgers, MIT and Paris and for twenty five years at the Hebrew University in Jerusalem. He is now based in Oxford University.

Guided by the model theory of Robinson, Shelah and Zilber, Hrushovski 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 took part in the creation of geometric stability, simplicity theory in finite dimensions, and establishing the role of definable groups within first order model theory.

Hrushovski co-authored papers with forty-five collaborators. He has received a number of awards including the Karp, Erdős and Rothschild prizes and the 2019 Heinz Hopf prize.

"Contemporary symbolic logic can produce useful tools - though by no means omnipotent ones - for the development of actual mathematics, more particularly for the development of algebra, and it would appear, of algebraic geometry." This statement (with a reference to still older roots) was made by Abraham Robinson in his 1950 address to the International Congress of Mathematicians. Instances of such uses of logic include the correction and proof by Ax-Kochen of a p-adic conjecture of Artin's, and the Denef - Van den Dries proof of a p-adic analytic conjecture of Serre. The internal development of model theory since the 70's has led to entirely new techniques, that, combined with the older ones, have begun to find applications to Diophantine geometry. It is the purpose of this talk to explain these methods and connections.

The present applications use only the finite-dimensional part of model theory (in a sense to be explained). Shelah and his followers created a theory of much greater generality (superstability, supersimplicity) incorporating many of the features of the finite dimensional case. One hopes that future applications will use this power. This exposition will limit itself to the finite-dimensional heartland (finite Morley rank, S1-rank).

Instead of defining the abstract context for the theory, I will present some of its results in a number of special, and hopefully more familiar, guises: compact complex manifolds, ordinary differential equations, difference equations, highly homogeneous finite structures. Each of these has features of its own, and the transcription of the general results is not routine; they are nonetheless readily recognisable as instances of a single theory. The current applications to Diophantine geometry arise by way of the difference and differential "examples". §2 and §6 will describe the model theory behind these results, and the prospects and difficulties lying ahead.

Throughout his career, starting with his Berkeley Ph.D. thesis, geometric stability theory (definable groups, minimal types, internality and binding groups, orthogonality, canonical bases, imaginaries, often in unstable contexts) has been a guiding theme; an example is his 2000 paper with Hart and Laskowski The uncountable spectra of countable theories, a culmination of classification theory over countable languages. "Hrushovski constructions" first appeared in his talks in 1988 to give counterexamples to conjectures of Lachlan and Zilber but the ideas since then have yielded countless other important examples, as well as Zilber's pseudo-exponential field. His 1996 paper Zariski geometries with Zilber exhibits a natural context where Zilber's trichotomy conjecture holds (so is a counterpart to the Hrushovski constructions), and was a key ingredient to his subsequent work on diophantine geometry. Early drafts of his monograph with Cherlin, Finite structures with few types, as well as work on PAC structures, gave versions of the independence theorem which underpins simple theories.

On the more applied side of model theory, Hrushovski's work with Chatzidakis and Peterzil on ACFA opened up difference algebra as an area of model-theoretic applications, reinforced by his manuscript on the nonstandard Frobenius. Hrushovski startled not just the model theory community when he found applications of large chunks of geometric stability theory in Diophantine geometry: he obtained for example a model-theoretic proof of the geometric Mordell-Lang conjecture, in all characteristics (a new result for function fields in characteristic p), and a new proof of the Manin-Mumford conjecture, with explicit bounds. A series of joint papers with Pillay in the late 2000s (one paper also with Peterzil, another also with Simon) yielded new ways of thinking about NIP theories model-theoretically, proved the Pillay conjecture on definable groups in o-minimal expansions of ordered fields, exhibited the significance of Keisler measures, and found model-theoretic applications of Vapnik-Chervonenkis theory; it has yielded many further developments in definable dynamics by Hrushovski and others. His 2009 paper on approximate subgroups led to the Breuillard-Green-Tao classification of finite approximate subgroups and to many other riches. Over the last 25 years, the model theory of valued fields has been a major theme of his work, with ideas from stability theory feeding into our understanding of algebraically closed valued fields, with applications in motivic integration (Hrushovski and Kazhdan), nonarchimedean tame topology and Berkovich space (Hrushovski and Loeser), and zeta functions for groups (Hrushovski, Martin, and Rideau-Kikuchi).