Vladimir Arnold

Vladimir Arnold
Arnold in 2008
Nama asal	Владимир Арнольд
Lahir	(1937-06-12)12 Juni 1937 Odesa, Republik Sosialis Soviet Ukraina, Uni Soviet
Meninggal	3 Juni 2010(2010-06-03) (umur 72) Paris, Prancis
Kebangsaan	Uni Soviet, Rusia
Almamater	Universitas Negeri Moskow
Dikenal atas	ADE classification Arnold's cat map Arnold conjecture Arnold diffusion Arnold's rouble problem Arnold's spectral sequence Arnold tongue ABC flow Arnold–Givental conjecture Gömböc Gudkov's conjecture Hilbert's thirteenth problem KAM theorem Kolmogorov–Arnold theorem Liouville–Arnold theorem Topological Galois theory Mathematical Methods of Classical Mechanics
Penghargaan	Shaw Prize (2008) State Prize of the Russian Federation (2007) Wolf Prize (2001) Dannie Heineman Prize for Mathematical Physics (2001) Harvey Prize (1994) RAS Lobachevsky Prize (1992) Crafoord Prize (1982) Lenin Prize (1965)
Karier ilmiah
Bidang	Mathematics
Institusi	University Paris-Dauphine Institut Matematika Steklov Universitas Independen Moskow Universitas Negeri Moskow
Pembimbing doktoral	Andrey Kolmogorov
Mahasiswa doktoral	Alexander Givental Victor Goryunov Sabir Gusein-Zade Emil Horozov Yulij Ilyashenko Boris Khesin Askold Khovanskii Nikolay Nekhoroshev Boris Shapiro Alexander Varchenko Victor Vassiliev Vladimir Zakalyukin

Vladimir Igorevich Arnold (ejaan alternatif Arnol'd, bahasa Rusia: Влади́мир И́горевич Арно́льд, 12 Juni 1937 – 3 Juni 2010) adalah seorang matematikawan Soviet dan Rusia. Ia terkenal karena teorema Kolmogorov-Arnold-Moser mengenai teori kestabilan sistem terintegral, dan berkontribusi dalam beberapa bidang, termasuk teori geometri teori sistem dinamik, aljabar, teori katastrofik, topologi, geometri aljabar real, geometri simetris, topologi simetris, persamaan diferensial, mekanika klasik, mekanika klasik, pendekatan geometris diferensial untuk hidrodinamika, analisis geometris dan teori singularitas, termasuk mengajukan masalah klasifikasi ADE.

Hasil utama pertamanya adalah solusi dari masalah ketiga belas Hilbert pada tahun 1957 pada usia 19 tahun. Dia turut mendirikan tiga cabang matematika baru: Teori Galois topologi (dengan muridnya Askold Khovanskii), topologi simetris dan teori KAM.

Arnold juga dikenal sebagai seorang yang mempopulerkan matematika. Melalui kuliah-kuliahnya, seminar-seminar, dan sebagai penulis beberapa buku teks (seperti Metode Matematika Mekanika Klasik) dan buku-buku matematika populer, ia mempengaruhi banyak ahli matematika dan fisikawan. Banyak dari buku-bukunya diterjemahkan ke dalam bahasa Inggris. Pandangannya tentang pendidikan sangat bertentangan dengan pandangan Nicolas Bourbaki.

Biografi

Vladimir Igorevich Arnold lahir pada tanggal 12 Juni 1937 di Odesa, Uni Soviet (sekarang Odesa, Ukraina). Ayahnya adalah Igor Vladimirovich Arnold (1900-1948), seorang ahli matematika. Ibunya adalah Nina Alexandrovna Arnold (1909-1986, née Isakovich), seorang sejarawan seni Yahudi. Ketika masih bersekolah, Arnold pernah bertanya kepada ayahnya tentang alasan mengapa perkalian dua bilangan negatif menghasilkan bilangan positif, dan ayahnya memberikan jawaban yang melibatkan sifat_bidang bilangan riil dan pelestarian sifat distributif. Arnold sangat kecewa dengan jawaban ini, dan mengembangkan keengganan terhadap metode aksiomatik yang berlangsung sepanjang hidupnya. Ketika Arnold berusia tiga belas tahun, pamannya, Nikolai B. Zhitkov, yang merupakan seorang insinyur, memberitahunya tentang kalkulus dan bagaimana kalkulus dapat digunakan untuk memahami beberapa fenomena fisika, hal ini turut memicu ketertarikannya pada matematika, dan ia mulai mempelajari sendiri buku-buku matematika yang ditinggalkan ayahnya, yang meliputi beberapa karya Leonhard Euler dan Charles Hermite.

Ketika menjadi mahasiswa Andrey Kolmogorov di Universitas Negeri Moskow dan masih remaja, Arnold menunjukkan pada tahun 1957 bahwa fungsi kontinu apa pun dari beberapa variabel dapat dikonstruksi dengan fungsi dua variabel yang terbatas, dengan demikian memecahkan Masalah ketiga belas Hilbert. Ini adalah Teorema representasi Kolmogorov-Arnold.

Setelah lulus dari Universitas Negeri Moskow pada tahun 1959, ia bekerja di sana hingga 1986 (profesor sejak 1965), dan kemudian di Institut Matematika Steklov.

Ia menjadi akademisi Akademi Ilmu Pengetahuan Uni Soviet (Akademi Ilmu Pengetahuan Rusia sejak tahun 1991) pada tahun 1990. Arnold dapat dikatakan telah memprakarsai teori topologi simetris sebagai sebuah disiplin ilmu yang berbeda. Dugaan Arnold tentang jumlah titik tetap dari Simplekskormofisme Hamiltonian dan perpotongan Lagrangian juga menjadi motivasi dalam pengembangan homologi Floer.

Pada tahun 1999, ia mengalami kecelakaan sepeda yang serius di Paris, yang mengakibatkan cedera otak traumatis. Dia sadar kembali setelah beberapa minggu tetapi mengalami amnesia dan untuk beberapa waktu bahkan tidak bisa mengenali istrinya sendiri di rumah sakit, Dia kemudian membuat pemulihan yang baik.

Arnold bekerja di Steklov Mathematical Institute di Moskow dan di Université Paris-Dauphine hingga kematiannya. Hingga 2006 ia dilaporkan memiliki indeks kutipan tertinggi di antara para ilmuwan Rusia, dan indeks-h sebesar 40. Murid-murid PhD-nya termasuk Alexander Givental, Victor Goryunov, Sabir Gusein-Zade, Emil Horozov, Yulij Ilyashenko, Boris Khesin, Askold Khovanskii, Nikolay Nekhoroshev, Boris Shapiro (matematikawan)|Boris Shapiro]], Alexander Varchenko, Victor Vassiliev, dan Vladimir Zakalyukin.

Di kalangan mahasiswa dan koleganya, Arnold juga dikenal memiliki selera humor yang tinggi. Sebagai contoh, suatu kali dalam seminarnya di Moskow, pada awal tahun ajaran, ketika ia biasanya merumuskan masalah-masalah baru, ia berkata:

Ada prinsip umum bahwa orang bodoh dapat mengajukan pertanyaan yang tidak dapat dijawab oleh seratus orang bijak. Sesuai dengan prinsip ini, saya akan merumuskan beberapa masalah.

Kematian

Arnold died of acute pancreatitis on 3 June 2010 in Paris, nine days before his 73rd birthday. He was buried on 15 June in Moscow, at the Novodevichy Monastery.

In a telegram to Arnold's family, Russian President Dmitry Medvedev stated:

The death of Vladimir Arnold, one of the greatest mathematicians of our time, is an irretrievable loss for world science. It is difficult to overestimate the contribution made by academician Arnold to modern mathematics and the prestige of Russian science.

Teaching had a special place in Vladimir Arnold's life and he had great influence as an enlightened mentor who taught several generations of talented scientists.

The memory of Vladimir Arnold will forever remain in the hearts of his colleagues, friends and students, as well as everyone who knew and admired this brilliant man.

Popular mathematical writings

Arnold is well known for his lucid writing style, combining mathematical rigour with physical intuition, and an easy conversational style of teaching and education. His writings present a fresh, often geometric approach to traditional mathematical topics like ordinary differential equations, and his many textbooks have proved influential in the development of new areas of mathematics. The standard criticism about Arnold's pedagogy is that his books "are beautiful treatments of their subjects that are appreciated by experts, but too many details are omitted for students to learn the mathematics required to prove the statements that he so effortlessly justifies." His defense was that his books are meant to teach the subject to "those who truly wish to understand it" (Chicone, 2007).

Arnold was an outspoken critic of the trend towards high levels of abstraction in mathematics during the middle of the last century. He had very strong opinions on how this approach—which was most popularly implemented by the Bourbaki school in France—initially had a negative impact on French mathematical education, and then later on that of other countries as well. Arnold was very interested in the history of mathematics. In an interview, he said he had learned much of what he knew about mathematics through the study of Felix Klein's book Development of Mathematics in the 19th Century —a book he often recommended to his students. He studied the classics, most notably the works of Huygens, Newton and Poincaré, and many times he reported to have found in their works ideas that had not been explored yet.

Karya matematika

Lihat pula: Kestabilan Tata Surya

Arnold bekerja pada teori sistem dinamik, teori bencana, topologi, geometri aljabar, geometri simetris, persamaan diferensial, mekanika klasik, hidrodinamika, dan teori singularitas. Michèle Audin menggambarkannya sebagai "seorang geometer dalam arti yang paling luas" dan mengatakan bahwa "dia sangat cepat dalam membuat hubungan antara berbagai bidang".

Hilbert's thirteenth problem

See also: Kolmogorov–Arnold representation theorem

The problem is the following question: can every continuous function of three variables be expressed as a composition of finitely many continuous functions of two variables? The affirmative answer to this general question was given in 1957 by Vladimir Arnold, then only nineteen years old and a student of Andrey Kolmogorov. Kolmogorov had shown in the previous year that any function of several variables can be constructed with a finite number of three-variable functions. Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering the Hilbert's question when posed for the class of continuous functions.

Moser and Arnold expanded the ideas of Kolmogorov (who was inspired by questions of Poincaré) and gave rise to what is now known as Kolmogorov–Arnold–Moser theorem (or "KAM theory"), which concerns the persistence of some quasi-periodic motions (nearly integrable Hamiltonian systems) when they are perturbed. KAM theory shows that, despite the perturbations, such systems can be stable over an infinite period of time, and specifies what the conditions for this are.

In 1964, Arnold introduced the Arnold web, the first example of a stochastic web.

Singularity theory

In 1965, Arnold attended René Thom's seminar on catastrophe theory. He later said of it: "I am deeply indebted to Thom, whose singularity seminar at the Institut des Hautes Etudes Scientifiques, which I frequented throughout the year 1965, profoundly changed my mathematical universe." After this event, singularity theory became one of the major interests of Arnold and his students. Among his most famous results in this area is his classification of simple singularities, contained in his paper "Normal forms of functions near degenerate critical points, the Weyl groups of A_k,D_k,E_k and Lagrangian singularities".

Fluid dynamics

See also: Arnold–Beltrami–Childress flow and Beltrami vector field § Beltrami fields and complexity in fluid mechanics

In 1966, Arnold published "Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits", in which he presented a common geometric interpretation for both the Euler's equations for rotating rigid bodies and the Euler's equations of fluid dynamics, this effectively linked topics previously thought to be unrelated, and enabled mathematical solutions to many questions related to fluid flows and their turbulence.

Real algebraic geometry

In the year 1971, Arnold published "On the arrangement of ovals of real plane algebraic curves, involutions of four-dimensional smooth manifolds, and the arithmetic of integral quadratic forms", which gave new life to real algebraic geometry. In it, he made major advances in the direction of a solution to Gudkov's conjecture, by finding a connection between it and four-dimensional topology. The conjecture was to be later fully solved by V. A. Rokhlin building on Arnold's work.

Symplectic geometry

The Arnold conjecture, linking the number of fixed points of Hamiltonian symplectomorphisms and the topology of the subjacent manifolds, was the motivating source of many of the pioneer studies in symplectic topology.

Topology

According to Victor Vassiliev, Arnold "worked comparatively little on topology for topology's sake." And he was rather motivated by problems on other areas of mathematics where topology could be of use. His contributions include the invention of a topological form of the Abel–Ruffini theorem and the initial development of some of the consequent ideas, a work which resulted in the creation of the field of topological Galois theory in the 1960s.

Theory of plane curves

According to Marcel Berger, Arnold revolutionized plane curves theory. Among his contributions are the Arnold invariants of plane curves.

Other

Arnold conjectured the existence of the gömböc.

Arnold generalized the results of Isaac Newton, Pierre-Simon Laplace, and James Ivory on the shell theorem, showing it to be applicable to algebraic hypersurfaces.

Bacaan lebih lanjut

• Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "Tribute to Vladimir Arnold", Notices of the American Mathematical Society, March 2012, Volume 59, Number 3, pp. 378–399.
• Khesin, Boris; Tabachnikov, Serge (Coordinating Editors). "Memories of Vladimir Arnold", Notices of the American Mathematical Society, April 2012, Volume 59, Number 4, pp. 482–502.
• Boris A. Khesin; Serge L. Tabachnikov (2014). Arnold: Swimming Against the Tide. American Mathematical Society. ISBN 978-1-4704-1699-7. 
• Leonid Polterovich; Inna Scherbak (7 September 2011). "V.I. Arnold (1937–2010)". Jahresbericht der Deutschen Mathematiker-Vereinigung. 113 (4): 185–219. doi:10.1365/s13291-011-0027-6. 
• "Features: "Knotted Vortex Lines and Vortex Tubes in Stationary Fluid Flows"; "On Delusive Nodal Sets of Free Oscillations"" (PDF). EMS Newsletter (96): 26–48. June 2015. ISSN 1027-488X.