Masaki Kashiwara wins the 2025 Abel Prize for groundbreaking work in algebraic analysis, D-modules theory, and crystal bases in representation theory.Continue reading
Showing posts with label abel prize. Show all posts
Showing posts with label abel prize. Show all posts
The Abel Prize 2025: Masaki Kashiwara
Abel Prize 2020: Furstenberg and Margulis
for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics.
Hillel Furstenberg and Gregory Margulis invented random walk techniques to investigate mathematical objects such as groups and graphs, and in so doing introduced probabilistic methods to solve many open problems in group theory, number theory, combinatorics and graph theory. A random walk is a path consisting of a succession of random steps, and the study of random walks is a central branch of probability theory.
The works of Furstenberg and Margulis have demonstrated the effectiveness of crossing boundaries between separate mathematical disciplines and brought down the traditional wall between pure and applied mathematicssays Hans Munthe-Kaas, chair of the Abel committee. He continues:
Furstenberg and Margulis stunned the mathematical world by their ingenious use of probabilistic methods and random walks to solve deep problems in diverse areas of mathematics. This has opened up a wealth of new results, such as the existence of long arithmetic progressions of prime numbers, understanding the structure of lattices in Lie groups, and the construction of expander graphs with applications to communication technology and computer science, to mention a few.
Due to the corona pandemic there will not be a physical Laureate ceremony on May 19 this year. The honoring of the Abel Prize Laureates will be announced later.
press release
A prize for a theorem
Pierre de Fermat was born in 1601 in Beaumont-de-Lomagne in southwestern France. He was not even a mathematician; he was a civil servant who devoted himself to mathematics as a hobby. He was regarded as one of the most gifted self-taught mathematicians who ever lived.I think that this quotation from The girl who played with fire by Stieg Larson, the second novel from the series Millennium, was the perfect introduction to a post about the Fermat's Last Theorem and Andrew Wiles, who proofed it in 1993. And he awarded the Abel Prize just some days ago.
Simply a joke?
Pierre de Fermat by Bernarda Bryson
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos ejusdem nominis fas est dividere: cujus rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.(1)This is what Pierre de Fermat wrote in the margin of Arithmetica by Diofanto and it establish the impossibility to find solutions in the field of natural numbers for the diophantine equation(2) that generalizes the Pythagorean theorem: \[a^n + b^n = c^n\] Following the Pythagorean theorem, for $n = 2$ the Fermat's equation has solutions in the field of natural numbers, called Pythagorean triples. For $n$ greater than 2 the story, instead, becomes more complicated, so that the same Fermat, while writing what follows, never provided a full demonstration, but only for the special case of $n = 4$:
I have discovered a truly marvelous proof of this theorem, which can't be contained in the too narrow page margin.
Louis Nirenberg, the geometry and the Abel Prize
The Norwegian Academy of Sciences and Letters has decided to award the Abel Prize for 2015 to the American mathematicians John F. Nash Jr. and Louis Nirenberg “for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.” The President of the Academy, Kirsti Strøm Bull, announced the new laureates today 25 March. They will receive the Abel Prize from His Majesty King Harald at a ceremony in Oslo on 19 May.While Nash is known for his contribution to game theory with Nash equilibria, Nirenberg is
considered one of the outstanding analysts of the twentieth century. He has made fundamental contributions to linear and nonlinear partial differential equations and their application to complex analysis and geometry.His first result was the completion of the solution of a problem in differential geometry, starting from the 1916 work of Weyl (read On the work of Louis Nirenberg by Simon Donaldson, pdf).
The statement is very simple: an abstract Riemannian metric on the 2-sphere with positive curvature can be realised by an isometric embedding in $R^3$.In 1994 he received the Steele Prize by the American Mathematical Society. In that occasion, the Society writes (via MacTutor) a good summary of his activity:
Nirenberg is a master of the art and science of obtaining and applying a priori estimates in all fields of analysis. A minor such gem is the useful set of Garliardo-Nirenberg inequalities. A high point is his joint research with [Shmuel] Agmon and [Avron] Douglis on a priori estimates for general linear elliptic systems, one of the most widely quoted results in analysis. Another is his fundamental paper with Fritz John on functions of bounded mean oscillation which was crucial for the later work of [Charles] Fefferman on this function space. Nirenberg has been the centre of many major developments. His theorem with his student, Newlander, on almost complex structures has become a classic. In a paper building on earlier estimates of [Alberto] Calderón and [Antoni] Zygmund, he and [Joseph] Kohn introduced the notion of a pseudo-differential operator which helped to generate an enormous amount of later work. His research with [François] Trèves was an important contribution to the solvability of general linear PDEs. Some other highlights are his research on the regularity of free boundary problems with [David] Kinderlehrer and [Joel] Spuck, existence of smooth solutions of equations of Monge-Ampère type with [Luis] Caffarelli and Spuck, and singular sets for the Navier-Stokes equations with Caffarelli and [Robert] Kohn. His study of symmetric solutions of non-linear elliptic equations using moving plane methods with [Basilis] Gidas and [Wei Ming] Ni and later with [Henri] Berestycki, is an ingenious application of the maximum principle.I hope that you can appreciate the list of collaboration performed by Nirenberg: indeed he is one of the most collaborative mathematician in the word. In particular for this reason I'm really happy for the award to Nirenberg.
Read also: Interview with Louis Nirenberg (pdf)
Pierre Deligne and the Weil conjectures
for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fieldsOne of the most famous contribution by Deligne was the proof of one of the three Weil conjectures. These conjectures was stated by Andre Weil,
how to count the number of solutions to systems of polynomial equations over finite fields(1)In particular
Weil conjectured that such zeta-functions should be rational functions, should satisfy a form of functional equation, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the Riemann zeta function and Riemann hypothesis. The rationality was proved by Dwork (1960), the functional equation by Grothendieck (1965), and the analogue of the Riemann hypothesis was proved by Deligne (1974)
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