I was born on December 18 1953 in Johannesburg, South Africa. My parents were pharmacists, who by example set high moral standards and a work ethic that has served my two brothers and myself throughout our lives. I spent most of my time at school playing competitive chess becoming one of the top players in Southern Africa. I was eager to move to Europe and play chess on the international circuit, but luckily for me, my father insisted that I go to University first and study Mathematics and Physics, the subjects that came naturally to me. As soon as I was introduced to modern and abstract mathematics at the University of Witwatersrand, I was smitten by its beauty and challenges, and I have never looked back. Even today, learning of a new idea that solves a problem, or developing a tool that leads to the solution of a central problem is what draws me and I think many others, to mathematics.
I went to Stanford to do a PhD with Paul Cohen who in 1963 revolutionized set theory solving the first of Hilbert’s problems from 1900. By the time I arrived at Stanford in 1976,Paul’s interests had turned to another of Hilbert’s problems ― the Riemann Hypothesis, and not surprisingly his ideas led him to study Atle Selberg’s works. Selberg had thought in depth about this problem and had developed powerful tools to study related problems in the theory of numbers. I was very fortunate that Paul took me on as a not completely unequal partner in filling in the details of some of Selberg’s work. My mathematical taste and style is due largely to these two singular mathematicians. While both of them worked by themselves, I have always enjoyed and benefited greatly from working jointly with others. Any achievements that I can claim owe immeasurably to my co-authors as well as achievements of others. For me fundamental problems rather than fields of specialization drive research, and progress is often decisive when unexpected disciplines are combined. My collaborations with Ralph Phillips, Alexander Lubotzky, Ilya Piatetski-Shapiro, Henryk Iwaniec and Nicholas Katz were transformative for me in this and other ways.
My mathematical horizons were opened broadly after completing my PhD in 1980 when I moved to the Courant Institute. In particular, my view that there is little difference between mathematics and applied mathematics, was solidified. I moved back to Stanford University in 1984 and then to Princeton University in 1991 where I continue to serve as a Professor in the Mathematics department. From 2001–2005 I was half time at the Courant Institute, and from 2007–2024 half time as Professor at the Institute for Advanced Study in Princeton moving there recently to emeritus status.
At an International number theory conference in Hangzhou in 2005, an answer to one of the many interesting questions that were posed was provided by solutions to the classical (1879) Markoff Diophantine equation, with a twist that the solutions should satisfy certain divisibility properties. The standard approach to such problems is to apply a “sieve”, which is an elaborate inclusion/exclusion counting procedure. However, in this exotic setting there were no tools at the time. My longtime collaborator Alexander Gamburd and I started then and continue today to develop this theory, now known as the “Affine Sieve”. We were quickly joined by the brilliant Jean Bourgain (a former Shaw Prize winner and who passed away way too young) and we were able to overcome the novel challenges that present themselves in the simplest settings. One such was to show that certain related combinatorial structures are “expanders”. This property has wide applications in engineering as it allows for the construction of sparse but highly connected communication networks. Lubotzky, Phillips and I had used sophisticated number theoretic methods to construct optimal such expanders known as “Ramanujan Graphs”. Interestingly in this exotic setting the roles are reversed and a key input in the affine sieve is established using tools from combinatorics and computer science. Alireza Golsefidy and I completed a general theory of the affine sieve in the linear setting. We made crucial use of related developments at that time by Helfgott, Varju, Pyber-Szabo and Breuillard-Green-Tao. The soil on which the affine sieve is built is an orbit of an affine action and what makes it exotic is when this group is number theoretically deficient or “Thin”. A flourishing Dynamical and Diophantine theory of thin matrix groups has been developed more recently. For nonlinear affine actions, the theory is still at its infancy, though progress has been achieved for the original Markoff equation by Bourgain, Gamburd, Amit Ghosh and myself, and by William Chen.
I first met my wife Helen Nissenbaum in a logic class at the University of Witwatersrand in 1972. She is a Philosopher and did her PhD at Stanford as well. We were married in San Francisco in 1977. She is the Andrew H and Ann R Tisch Professor at Cornell Tech where she is in the Information Science Department. That I could flourish in my long mathematical pursuits has only been possible thanks to Helen’s continued support, understanding and partnership. The same goes for our wonderful three daughters Dana, Zoe and Ann who mostly allow me to operate cluelessly.
12 November 2024 Hong Kong
