## Abel acceptance speech by Gregory Margulis

In 2020, our colleague Gregory Margulis and longtime department friend Hillel Furstenberg shared the Abel Prize. The prize ceremony took place this May 25-26, online, together with the 2021 winners Laszlo Lovasz (also a one-time Yale colleague) and Avi Wigderson. We reproduce below Prof. Margulis’ acceptance speech. The Abel website can be reached at this link [1], and the slides for Prof. Margulis’ prize lecture are here. [2]** **

**Acceptance speech**

Ladies and Gentlemen.

I am very grateful to the Abel Committee and the Norwegian Academy of Science and Letters who awarded me the Abel Prize. It is a great honor. It is a double honor for me to share the prize with one of my mathematical heroes Hillel Furstenberg who greatly influenced my research. I am also very honored to be among the previous winners of the Abel Prize. I should especially mention my thesis advisor Yakov Sinai and also Jacques Tits who supported me for several decades.

Niels Henrik Abel was one of the greatest mathematicians of the nineteenth century. As Felix Klein writes in his “Lectures on the development of mathematics in XIX century”, “In Abel we see a great, very original genius … who completely dedicated himself to problems of purest, most abstract mathematics. Maybe only his early death did not allow him to apply his talent in other parts of mathematics”. (It is a rather imprecise English translation of the Russian translation of the German original.)

In late 1950s, a great Norwegian mathematician Atle Selberg began a program of study of the structure of some special class of discrete subgroups, called lattices. A few years later Selberg and Piatetski-Shapiro stated a conjecture about arithmeticity of lattices in higher rank semisimple Lie groups. I proved this conjecture using various approaches. It took about seven years from approximately 1967 to 1974. In the subject of discrete subgroups of Lie groups, I also would like to emphasize my proof of the normal subgroup theorem. In theoretical computer science I am best known for giving explicit constructions of expanders. I also made contributions to other topics and fields in mathematics, mostly in collaboration with other mathematicians. Among these contributions I want to mention the work in the following areas. (I) Distribution of values of quadratic forms at integral points. It started when I proved the Oppenheim conjecture in the mid 1980s and continued until very recently. (ii) Diophantine approximation on manifolds which includes the proof of Baker-Sprindzuk conjectures. (iii) Dynamics, including homogeneous dynamics, theory of Anosov flows, and rigidity of group actions on manifolds. (iv) Probability, including random graphs and random walks on homogeneous spaces. (v) Discrete groups of affine transformations. This is some kind of list of contributions which I consider quite important.

I had a quite long career in mathematics. My first paper was published in 1966, and I am still active in mathematical research. I always tried to work on difficult problems, sometimes unsuccessfully and sometimes successfully. Success was achieved in some cases by more or less straightforward approach and in other cases by discovering new connections between various fields of mathematics.

I was influenced by many mathematicians. I should especially mention (besides Sinai and Tits) Piatetski-Shaprio, Kazhdan, and Vinberg during my early career and Furstenberg, Mostow and M.S.Raghunathan during later years.

I am very grateful to my many collaborators and to my former graduate students. The work with graduate students was a very rewarding experience.

One of my former graduate students, Hee Oh is here. She is a vice president of the American Mathematical Society, and she represents AMS at this event.

Finally I would like to thank my wife of 48 years for help and support.

Let me finish with a funny note. When there was a call from the Norwegian Academy of Scince and Letters, my wife took the phone and she said “What prize?”. And she decided it was a scam and hang up the phone. The second time, instead of Abel Prize, she heard “Nobel Prize” and she said “There is no Nobel Prize in mathematics”. And third time, it worked.