# Autobiography of Ehud Hrushovski

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 indispensible 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