The Abel Prize 2018-2022

I was born and passed the first two decades of my life in the vicinity of Vancouver, British Columbia, an area now overwhelmed by immigration from the Orient, above all, China and India. More precisely, I was born in New Westminster in 1936, spent the first few years of my childhood on the shore about seventy miles to the north, in Lang Bay close to Powell River, returned to New Westminster to begin school, then moved to White Rock, where I passed my adolescence, and then went to Vancouver, of which New Westminster is now a suburb, to the University for five years, leaving in 1958 for graduate school, never to return except for short visits. I recall first the geography of the area, then the circumstances there in the nineteenth century and the beginning of the twentieth as well as the circumstances of my family and me. It is an area that has seen changes that, although peaceful have been definitive: a semi-rural, even partly rural environment, with a population that, apart from a visible, but small, indigenous component, still had close ties to the Old Country, generally meaning Great Britain and Ireland, and to Eastern Canada, has become more urban, much more cosmopolitan, and undoubtedly much more sophisticated, and apparently, much wealthier.

Master of Science, University of British Columbia, 1958

PhD, Yale University, 1960

I was born during World War II as the first of four children of an artist mother and an engineer father. My parents had come of age during the depression, which shaped their lives and my early years. My father worked for the Aluminum Company of America, and my parents moved from New Jersey to Ohio during the war at least in part because the aircraft industry was relocated to the midwest.

In this article we discuss some of Karen Uhlenbeck’s most prominent mathematical results. Uhlenbeck’s publications range across many mathematical areas, including differential geometry and geometric analysis, elliptic and hyperbolic partial differential equations and integrable systems. In this article we only attempt to describe some part of this range. The main omissions are that we say nothing about her work on wave and Schroedinger maps, and very little about integrable systems.

M.A., Brandeis University, 1966

Ph.D., Brandeis University, 1968

I was born in Berlin in 1935 shortly after the rise of Nazism in Germany. My parents were both born in Germany, their parents having emigrated from Poland and Russia respectively. My father, largely self-taught, served as an advocate’s assistant prior to his marriage, and afterwards he was manager of a furniture store. One of my few memories of Germany was the morning after Kristallnacht in November 1938, when the synagogue adjacent to our basement apartment was vandalized, and our own apartment seriously damaged. As Jews without longstanding ties to Germany, we were given expulsion orders. After unsuccessfully applying to various countries for refuge, we were admitted for temporary residence in England, with plans to continue on to the U.S. My father was particularly eager to immigrate to the U.S. where he hoped to join my mother’s brother who owned a poultry farm in New Jersey. Concerned that immigration authorities in the U.S. would not allow him entry on account of an ailment he had contracted, he underwent surgery in a London hospital which unfortunately did not succeed. The surgery brought about his death, and he was buried in a London cemetery, leaving my mother to manage with two children, myself and an older sister. We were fortunate that my uncle in New Jersey was able to arrange for our immigration to the U.S. where we arrived in 1940, and our first year in the new country was spent on my uncle’s farm.

I was born on February 24, 1946, in Moscow, an only child. This was shortly after the end of World War II, and the living conditions, in general, were not so good. But my family was relatively well off, and my parents protected me from all troubles which were around. I lived in Moscow until August of 1991 when I moved to the USA. Since then I have lived in New Haven, Connecticut.

In the following article we have attempted to give some account of the extraordinary work of Hillel Furstenberg and its impact on modern mathematics. His influence goes far beyond his published papers. He shared his ideas freely and many of them appear in papers written by others. No attempt has been made to be exhaustive, but we have tried to mention his major works.

A.E. was supported by an Investigator grant from the Simons Foundation. D.F. was supported by NSF grant DMS-1906107 and DMS-2208430. D.K. was supported by NSF grant DMS-1900560 and the Institute for Advanced Study School of Mathematics.

[1] Positive harmonic functions on nilpotent groups. Soviet Math. Dokl., 7:241– 244. Also available in Dokl. Akad. Nauk SSSR, 166:1054–1057 (in Russian).

Master of Science, Yeshiva University, 1955

PhD, Princeton University, 1958

Doctor of Science, Minsk, 1983

I was born in Budapest in 1948. This was the year when the Stalinist regime seized total power. From my early childhood, I remember my parents’ fear of the secret police, and then in particular the uprising in 1956, the street fight that went on under our windows.

It is customary for Abel prize winners to write a brief biography in these volumes, and I have done the same below. Writing it, the shortcomings of such an account quickly became apparent. Condensing a person, result, event, experience, or a lesson learned (and there are quite a few of these below) into a sentence or two, leaves so much to be desired and feels like doing an injustice to their meaning for me. But, it had to be done, so here goes.

This is an exposition of the contributions of László Lovász to mathematics and computer science written on the occasion of the bestowal of the Abel Prize 2021 to him. Our survey, of course, cannot be exhaustive. We sketch remarkable results that solved well-known open and important problems and that – in addition – had lasting impact on the development of subsequent research and even started whole new theories. Although discrete mathematics is what one can call the Lovász home turf, his interests were, from the beginning of his academic career, much broader. He employed algebra, geometry, topology, analysis, stochastics, statistical physics, optimization, and complexity theory, to name a few, to contribute significantly to the explosive growth of combinatorics; but he also exported combinatorial techniques to many other fields, and thus built enduring bridges between several branches of mathematics and computer science. Topics such as computational convexity or topological combinatorics, for example, would not exist without his fundamental results. We also briefly mention his substantial influence on various developments in applied mathematics such as the optimization of real-world applications and cryptography.

This is an overview of some of the works of Avi Wigderson, 2021 Abel prize laureate. Wigderson’s contributions span many fields of computer science and mathematics. In this survey we focus on four subfields: cryptography, pseudorandomness, computational complexity lower bounds, and the theory of optimization over symmetric manifolds. Even within those fields, we are not able to mention all of Wigderson’s results, let alone cover them in full detail. However, we attempt to give a broad view of each field, as well as describe how Wigderson’s papers have answered central questions, made key definitions, forged unexpected connections, or otherwise made lasting changes to our ways of thinking in that field.

[2] The Complexity of the Hamiltonian Circuit Problem for Maximal Planar Graphs. Technical Report 298, Department of EECS, Princeton University, February 1982.

Dr.Rher.Nat., Eötvös Lor´and University, 1971

Dr.Math.Sci., Hungarian Academy of Sciences, 1977

MSE, Princeton University, 1981

MA, Princeton University, 1982

PhD, Princeton University, 1983

When we first learned long division at age eight I had fun working out examples where the problem was put at the top of the page and the long division algorithm computation ran diagonally down the page to finish exactly in the tiny vicinity of the lower right hand corner. I wanted to make the biggest problem that fit on one page. This was a curious metaphor of one’s serious practical limitations today in simulating 3D fluid motion on computers.

Before going into Dennis Sullivan’s work on dynamics, we would like to share some of our reminiscences on the remarkable way in which he influenced a huge number of mathematicians, including the two of us. Both while at IHES and at CUNY, Dennis had an office which came with an anteroom. Our impression is that he would spend most of his time in this anteroom, talking about mathematics with whoever he had invited or whoever was around. Quite often while listening to somebody, he would end up giving a new spin or a new interpretation to what they had been saying. Similarly, he would explain what he was working on, trying out new ideas, and also often explaining results of others. Spending time with him was always an incredible experience.

Dennis Sullivan’s early work, despite being strongly motivated by geometric problems, had a strong algebraic nature, certainly much more so than his later work on foliations, Kleinian groups, and dynamics where geometry and analysis play central roles. The aim of this essay is to try to explain to the non-expert some of this early work and indicate some of its impact to the current moment.

[1] Triangulating Homotopy Equivalences. Ph.D. Thesis, Princeton University.

PhD, Princeton University, 1966