The Abel Prize 2013-2017

In what follows, I dwell on some major influences on my mathematical education. The account Luc Illusie gives of my work is much more systematic. I would like to begin by thanking him for it.

This is a report on the work of Pierre Deligne.

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I was born on the 21st of September, 1935 in Moscow to a family of scientists. My mother, Nadezka Kagan, was a virologist. She worked on vaccines against encephalitis and died in November of 1938 after she became infected by the vaccine on which she was working. My father was a professor of microbiology in one of Moscow’s medical institutions. He participated in World War II working as an epidimeologist from 1941 through 1945. He married again in 1940, and my stepmother, E.N. Levkovich, was also a famous virologist.

We review some of the most remarkable results obtained by Ya.G. Sinai and collaborators on the difficult problems arising in the theory of the Navier–Stokes equations and related models. The survey is not exhaustive, and it omits important results, such as those related to “Burgers turbulence”. Our main focus in on acquainting the reader with the application of the powerful methods of dynamical systems and statistical mechanics to this field, which is the main original feature of Sinai’s contribution.

Light is what a newborn baby sees first. It is no wonder that people were always interested in light propagation. Fundamental laws were discovered through the centuries. However, it was Ya. G. Sinai who first laid the foundation for the study of global properties of light propagation in media which contain reflectors (scatterers, mirrors, etc) and obtained fundamental results in this area.

In this article we will survey some of the contributions of Ya.G. Sinai to number theory and related fields. The multifacetedness of his work demonstrates Ya.G. Sinai’s vision of mathematics as a highly interconnected discipline, rather than a series of compartmented fields.

This section is devoted to Sinai’s advances in the entropy theory of dynamical systems and to some developments of his ideas. A history of dynamical entropy is also represented. When describing several events of this history, author’s personal recollection is partially used.

Sinai was never really far away from Mathematical Physics. Already his first papers on Kolmogorov–Sinai entropy and on the stability of Kolmogorov’s flow in 2D hydrodynamics (joint with L. Meshalkin, [19]) were very much in the areas which are closely related to Mathematical Physics. However, in his first research period, roughly in the late 1950s and the early 1960s, Sinai’s work was mostly concentrated around Ergodic Theory and Dynamical Systems. It is fair to say that his deep and lasting interest in Mathematical Physics started with his work on Statistical Mechanics in the late 1960s. This period culminated in the celebrated Pirogov–Sinai theory of phase transitions for ferromagnetic systems. After that Mathematical Physics was always one of the main themes of Sinai’s research. In general it was a period of very active interaction between mathematicians and physicists in the USSR.

Some principal contributions of Ya. Sinai to hyperbolic theory of dynamical systems, focusing mainly on constructions of Markov partitions and of Sinai–Ruelle–Bowen measures, are discussed. Some further developments in these directions stemming from Sinai’s work, are described.

In this chapter we present a brief survey of the rich and manifold developments of Sinai’s ideas, dating back to 1963, concerning his exact mathematical formulation of Boltzmann’s original ergodic hypothesis. These developments eventually lead to the 2013 proof of the so called “Boltzmann-Sinai Ergodic Hypothesis”.

Markov partitions designed by Sinai (Funct Anal Appl 2:245–253, 1968) and Bowen (Am J Math 92:725–747, 1970) proved to be an efficient tool for describing statistical properties of uniformly hyperbolic systems. For hyperbolic systems with singularities, in particular for hyperbolic billiards the construction of a Markov partition by Bunimovich and Sinai (Commun Math Phys 78:247–280, 1980) was a delicate and hard task. Therefore later more and more flexible and simple variants of Markov partitions appeared: Markov sieves (Bunimovich–Chernov–Sinai, Russ Math Surv 45(3):105–152, 1990), Markov towers (Young, Ann Math (2) 147(3):585–650, 1998), standard pairs (Dolgopyat). This remarkable evolution of Sinai’s original idea is surveyed in this paper.

On the distribution of the first positive sum for the sequence of independent random variables.

My beginning as a legally recognized individual occurred on June 13, 1928 in Bluefield, West Virginia, in the Bluefield Sanitarium, a hospital that no longer exists. Of course I can’t consciously remember anything from the first 2 or 3 years of my life after birth. (And, also, one suspects, psychologically, that the earliest memories have become “memories of memories” and are comparable to traditional folk tales passed on by tellers and listeners from generation to generation.) But facts are available when direct memory fails for many circumstances.

The life and work of John Forbes Nash, Jr.

I was born in Hamilton, Ontario, Canada, in 1925. My parents emigrated there from Ukraine, where my father was a Hebrew teacher. When my parents married they immediately crossed the border into Romania, illegally and were promptly arrested. Relatives managed to get them out of jail, and they slowly made their way across Europe, to Antwerp, Belgium, where they hoped to get immigration visas to the US. After a long time, during which my mother worked as a seamstress they decided to go to Canada, and my father continued teaching there.

In this set of notes I follow Nash’s four groundbreaking works on real algebraic manifolds, on isometric embeddings of Riemannian manifolds and on the continuity of solutions to parabolic equations. My aim has been to stay as close as possible to Nash’s original arguments, but at the same time present them with a more modern language and notation. Occasionally I have also provided detailed proofs of the points that Nash leaves to the reader.

Mathematics is the language of science, and partial differential equations are a crucial component: they provide the language we use to describe—and the tools we use to understand—phenomena in many areas including geometry, engineering, and physics.

I was born in a college room in Cambridge so my choice of profession was perhaps inevitable. My childhood was also spent in Cambridge except for the period from age two to six during which we lived in Nigeria. Apparently I resisted education at first, refusing to go to school for a whole term, but finally succumbed. Holidays then and later were spent on the estate and farm of a wonderful school that my maternal grandfather had created from nothing in Sussex.

This paper surveys the published mathematical works of Andrew Wiles up through the time he was awarded the Abel Prize.

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My mother died thirty years ago from a stroke. Her life had been sad. Her marriage was a failure. My father never wished to live with us, even when I was a baby. I was born in 1939 and my sister, Danièle, in 1938. My father sold the pharmacy he was running at 17 boulevard du Temple, Paris, and he enrolled in the Army in 1944. He was assigned to the Hospital Pharmacy Department at Rabat, Morocco. My mother stayed a full year with us in Paris. She finally decided to take us to Rabat against the wish of my father. In Rabat, the three of us were living in a single room of a low-cost hotel. My father was working, living, and sleeping at the hospital. We were used to his absence.

The mathematics of Yves Meyer cover a wide range of fields, as various as number theory, harmonic analysis, operator theory, partial differential equations, control theory, signal and image processing.This survey gives an introductory and self-contained overview of Yves Meyer’s contributions to these areas of research.