Lychagin was elected

lychaginOn 11 December, 2008 Valentin V. Lychagin was unanimously elected as “socio corrispondente” (member by correspondent, член-корреспондент) of the Accademia Peloritana dei Pericolanti (Messina, Italy). Congratulations, Valentin!

Valentin T. Fomenko

On the occasion of his 70th birthday.
vtfomenko


15 июля 2008 года исполняется семьдесят лет доктору физико-математических наук, профессору по кафедре геометрии, Заслуженному деятелю науки Российской Федерации, академику РАЕН Валентину Трофимовичу Фоменко.

В.Т. Фоменко родился в хуторе Потапов Цимлянского района Ростовской области. В 1955 г. он поступает на физико-математический факультет Ростовского государственного университета, который оканчивает в 1960 г. С 1962 по 1974 гг. Фоменко работает в РГУ профессором, заведующим кафедрой математического анализа. В 1982—1988 гг. он занимает должность проректора по научной и учебной работе Волгоградского государственного университета, а с 1988 по 2001 гг. — проректора по научной работе Таганрогского государственного педагогического института. В периоды с 1975 по 1982 гг. и с 2001г. по настоящее время он — профессор, заведующий кафедрой алгебры и геометрии ТГПИ.

Фоменко — автор более двухсот научных работ, пяти учебных пособий и монографии. Он подготовил более 20 кандидатов наук, трое из которых защитили докторские диссертации. В соответствии с Указом Президента Российской Федерации от 16.09.1993 г. Президиум РАН присудил В.Т. Фоменко Государственную научную стипендию как выдающемуся ученому России. В 2007 г. на конкурсе “Золотой фонд отечественной науки и образования” возглавляемая им кафедра алгебры и геометрии удостоена диплома “Золотая кафедра”, а в 2008 г. ему самому присвоено почетное звание “Основатель научной школы”. Научная деятельность этой школы неоднократно поддерживалась грантами РФФИ, МО РФ, губернатора Ростовской области, ТГПИ, а также благотворительными обществами г. Таганрога. В настоящее время В.Т. Фоменко является научным руководителем фундаментального исследования по заданию Федерального агентства по образованию МО и НРФ (2006—2010 гг.).

Научная деятельность его школы представлена тремя направлениями в области дифференциальной геометрии “в целом”.

Первое направление — изгибание поверхностей в трехмерных евклидовых и римановых пространствах. Установлены условия существования непрерывных изгибаний поверхностей в предположении, что край поверхности скользит в процессе изгибания по заданной опоре (В.Т. Фоменко, Н.С. Казарян). Доказано, что множество всех решений уравнений Гаусса-Петерсона-Кодацци для односвязной поверхности положительной гауссовой кривизны в евклидовом пространстве есть связное аналитическое вложенное подмногообразие подходящего Банахова пространства, моделируемое в некотором банаховом пространстве аналитических функций (С.Б. Климентов). Указаны явные, удобные для приложений, формулы, связывающие решения этих уравнений и аналитические функции (В.Т. Фоменко). Установлены условия, при выполнении которых решение линеаризованной задачи теории изгибаний поверхностей может гарантировать решение нелинейной задачи изгибания поверхностей (Е.М. Колегаева, Н.С. Казарян). Указаны условия, обеспечивающие жесткость замкнутых склеенных поверхностей (Л.П. Фоменко).

Второе направление — деформации многомерных поверхностей в пространствах постоянной кривизны. Основным результатом, полученным в этом направлении, является выделение класса многомерных поверхностей, бесконечно малые изгибания которых описываются аналитическими функциями многих комплексных переменных (В.Т. Фоменко, А.Н. Зубков). Значительным результатом является доказательство изгибаемости многомерных склеенных поверхностей, представленных в виде декартова произведения поверхностей меньшей размерности (П.Е. Марков). Помимо изгибаний целесообразно рассматривать деформации поверхностей, при которых некоторые геометрические характеристики поверхности имеют наперед заданные значения вариаций. Эти условия накладывают ограничения на выбор поля деформации поверхности, описываемые, как правило, в виде дифференциальных уравнений. К настоящему времени достаточно полно изучены ареальные деформации поверхностей с сохранением гауссова образа поверхности (А.В. Забеглов, О.Н. Бабенко), а также деформации двумерных поверхностей в четырехмерном пространстве с сохранением грассманова образа поверхности (В.Т. Фоменко, И.А. Бикчантаев). Получены условия существования деформаций поверхностей с сохранением гауссова образа в римановом пространстве, при которых вариация элемента площади поверхности определяется кривизной поверхности, величиной нормального смещения и элементом площади поверхности с некоторым коэффициентом рекуррентности деформации (О.Н. Бабенко, В.В. Сидорякина, В.Т. Фоменко). Установлен закон распределения коэффициентов рекуррентности, порождающих деформации, совместимые с заданными внешними связями (В. Т. Фоменко, В.В. Сидорякина, А.И. Бодренко).

Третье направление исследований — внешняя геометрия погруженных многообразий. Для описания внешне-геометрического поведения поверхности вводятся новые геометрические характеристики поверхности, являющиеся инвариантами касательного или нормального расслоений поверхности: нормальная кривизна, нормальное кручение, относительное кручение, гауссово кручение поверхности в точке по заданному направлению (В.Т. Фоменко, И.И. Бодренко). Обращение в ноль одной из этих характеристик выделяет в пространстве класс поверхностей. Описание таких классов поверхностей представляет значительный интерес, так как они обобщают известные ранее классы поверхностей. Решению указанных задач посвящено большое количество работ у нас в стране и за рубежом. В 2004 году было дано полное описание поверхностей с нулевым нормальным кручением (В.Т. Фоменко).

Ректорат и кафедра алгебры и геометрии Таганрогского государственного педагогического института и Международный геометрический центр dω (Одесса) поздравляют Валентина Трофимовича с юбилеем и желают ему здоровья, счастья и радости новых открытий.

Н. В. Перчун

Boris Abramovich Rosenfeld (1917—2008)

photoRosenfeld

Geometric Center dω with the deep sorrow notifies that on April 5, 2008 the great geometer and historian of mathematics Boris Abramovich Rosenfeld passed away. He was 90 years old.

OBITUARY

Boris Rosenfeld, 90, of State College passed away on April 5, 2008 at his home. He was born on August 30, 1917 in St. Petersburg, Russia. He was the son of the late Abraham and Maria Rosenfeld. On April 7, 1946 he married Lucy Davidov in Moscow who survived him.

Boris Rosenfeld was a distinguished research mathematician and teacher and a world authority in the history of science, especially that of ancient Greece and medieval Middle East. He was as a full member of the International Academy of the History of Science.

He received his master’s degree, Ph.D., and the highest Doctor of Science degree from the Moscow State University. During the Second World War, he was on noncombat duties with the Soviet Army, which was fighting Nazi Germany in the alliance with Western democracies. From 1950 until his immigration to the United State in 1990 he held a succession of professorial and senior research appointments, including Azerbaijan State University in Baku, Pedagogical institutes at Zagorsk and Kolomna, and the Institute of History of Science and Technology of the USSR Academy of Science. At the age of 73 he was appointed as an adjunct professor at the Department of Mathematics at Penn State, which later became a joint appointment with departments of History and Philosophy. He retired from active teaching duties in 1995 at the age of 78. He continued active research work until shortly before his death, continuing to publish books and articles in international professional journals. One of his fundamental achievements was a comprehensive bibliography and commentary on all existing medieval Islamic manuscripts in mathematics, astronomy, and related areas published in Istanbul, Turkey in 2003. He completed the last and one of the most important endeavors of his life, a translation and scientific commentary of the classical treatise of Apollonius in 2007, being completely blind by then.

From 1940 until his death Boris Rosenfeld published over 400 scientific papers in professional journals and many monographs, several of which were highly influential and became standard sources in corresponding fields of mathematics and history of science. He has a large group of students and followers from his years of teaching and research in the Soviet Union. He supervised 82 Ph.D. dissertation, an unusually high number for his field. His last Ph.D. student Diana Rhodes received her Ph.D. from Penn State in 2005.

Apart from his wife, Boris Rosenfeld is survived by two daughters: Dr. Svetlana Katok of State College and Julia Rozenman of Great Falls, VA, their husbands, Dr. Anatole Katok and Dr. Michael Rozenman; five grandchildren: Dr. Elena Katok Bolton and her husband Dr. Gary Bolton, Boris Katok and his wife Sherrie Hashemi, Alexandra Rozenman and her husband Dr. Alexander Voronov, Dr. Mary Rozenman, and Danya Katok and her fiancee Nicholas Ahlbin; and two great-grandchildren, Uriel Bolton 6, and Zoa Katok 6.

A memorial service will be held on Wednesday, April 9 at 1pm in Koch Funeral Home, 2401 S. Atherton Str., State College. The burial at the Spring Creek Presbyterian Cemetery will follow.

In his long fight with eye diseases and approaching blindness Boris Rosenfeld has benefited from superior skills of several outstanding eye surgeons and from two cornea transplants. In recognition of that, memorial contributions may be made to the Gift of Life Family House

См. также статью ученика Бориса Абрамовича М. П. Замаховского, написанную по случаю 90-летия Розенфельда и опубликованную в журнале Математическое Просвещение, Серия 3, 2007, выпуск 11, стр. 15—20.

I. S. Krasil'shchik

On the occasion of his 60th birthday.

I. S. Krasil’shchik
I. S. Krasil’shchik

Joseph Semenovich Krasil'shchik was born on February 10, 1948 in Moscow (USSR).

He attended one of the best Moscow high schools with specialization in Physics and Mathematics. The school (#52) had rich ties with Moscow State University and he entered its Department of Mechanics and Mathematics in 1966 after graduating the school with honors. Among the department faculty at that time there was a number of excellent mathematicians: Kolmogorov, Manin, Arnold, Novikov, Kurosh, Gelfand to name but a few. In the university Krasil'shchik specialized in geometry and topology under the direct of A.M. Vinogradov. He graduated from the university in 1971.

Being an active member of the famous Vinogradov’s seminar, he followed his mentor when the latter switched his scientific studies to applications of algebraic topology and differential geometry to differential equations. And it became the principal area of research for Krasil'shchik ever since. He made a considerable contribution in the field.

From the very first papers till now a subject of constant interest of Krasil'shchik is the algebraic aspects of differential calculus. His works in this field include Hamiltonian formalism in (super)commutative algebras; algebraic study of differential equations; generalization of the δ-Poincaré lemma; algebraic theory of Frölicher-Nijenhuis, Richardson-Nijenhuis, and Schouten brackets.

Joseph Krasil'shchik and his teacher and collaborator Alexander Vinogradov are the principal founders of the nonlocal theory in geometry of differential equations. The central notion of this theory—a covering—is a key to decipher the structure of Bäcklund transformations, zero-curvature representations, Miura transformation, recursion operators, nonlocal symmetries, conservation laws, and Hamiltonian structures, Estabrook-Wahlquist algebras, etc.

In 1972–1989 Krasil'shchik worked as a Researcher at the All-Union Institute for Scientific and Technology Information (USSR Academy of Sciences). There he obtained his Ph.D. degree in Information Science. Yet the institute loose discipline allowed finding time for studies in abstract mathematics. Krasil'shchik publications of that period dealt with hamiltonian cohomologies of canonical algebras and nonlocal symmetries. These articles laid a foundation for his further research in the field.

From 1989 to 2003 Krasil'shchik worked at the Moscow Institute for Municipal Economy, first as senior lecturer and later as professor. After he obtained the Doctor of Science degree in Physics and Mathematics at the Moscow State University, 1997 he also took a position of the full professor at the Independent University of Moscow and still holds it.

It was a period of a great scientific activity. The publications by Krasil'shchik treats such topics as cohomology background in geometry of PDE, deformations and integrable systems, Bäcklund transformations, the connection between integrability and supersymmetry, etc. He was invited to short-time visiting positions at University of Tromso, Norway; University of Twente, the Netherlands; Erwin Schrodinger Inst. for Math. Physics, Austria; University of Seville, Spain; the Lille University, France; Bar-Ilan University, Israel; Universities of Florence, Salerno, and Lecce, Italy, etc. He took part (as a plenary lecturer) in numerous conferences around the world; coauthored five monographs in a collaboration with A.V. Bocharov, V.N. Chetverikov, S.V. Duzhin, N.G. Khor'kova, P.H.M Kersten, V.V. Lychagin, A.V. Samokhin, Yu.N.Torkhov, A.M. Verbovetsky, A.M. Vinogradov.

In 1990th Krasil'shchik began a long, fruitful collaboration with Paul Kersten from Twente University in Enschede, Holland. Together, they developed the theory of deformations of differential equations into a tool for computing major invariants of differential equations, such as recursion operators and Hamiltonian operators. Krasil'shchik’s C-cohomology, Kersten’s REDUCE package, and nonlocal theory all together boiled down to a beautiful geometric theory of recursion operators of PDEs, which was summarized in the monograph “Symmetries and recursion operators for classical and supersymmetric differential equations” (2000).

In 2004–2007 Krasil'shchik had a position of a full professor at the Moscow State Technical University of Civil Aviation. Since 2007 he is a full professor at the Russian State University for the Humanities. In 2007 he also had a short-time visiting positions at University of Angers, France. In this period Krasil'shchik’s research concentrates mainly on Hamiltonian and symplectic structures, recursions, and hierarchies. The specter of application includes KdV-mKdV system, dispersionless Boussinesq type equation, Monge-Ampère equation, supersymmetric KdV equation, Camassa-Holm equation, etc. Overall, the list of his papers include more than 70 papers.

In recent years, Krasil'shchik undertook a project to extend these results to (non)local Hamiltonian and symplectic operators. He carries out this project in collaboration, by now, with V. Golovko, S. Igonin, P. Kersten, A. Verbovetsky, and R. Vitolo.

Krasil'shchik has a prominent place in the mathematical community. For many years he heads a famous “Krasil'shchik’s” research seminar at the Independent University of Moscow. It produces ideas, aspirants, international cooperation; and it spreads the elevated spirit of a pure mathematical research at the times of collapsing mathematical education. Krasil'shchik is a member of Moscow Mathematical Society and American Mathematical Society. He is an editor to eight volumes of collected works.

Outside mathematics he is a passionate angler. His other hobbies are extreme water tourism in wilderness of Siberia, Polar Urals, etc., and also classical music and jazz. His family life is a source of lasting happiness.

We, friends and colleagues of Joseph Semenovich Krasil'shchik wish him good health, happiness and new trophies: in fishing and mathematics.

V. Goldberg, S. Igonin, B. Kruglikov, A. Kushner, V. Lychagin, V. Rubtsov, A. Samokhin, A. Verbovetsky, R. Vitolo.

Mathematical publications of Joseph Krasil’shchik

Books

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Primenenie nelinejnykh differentsial'nykh uravnenij v grazhdanskoj aviatsii, MIIGA, Moscow, 1977 (Russian).

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Geometriya nelineinykh differentsialnykh uravnenii, MIEM, Moscow, 1982 (Russian).

A. M. Vinogradov, I. S. Krasil'shchik, and V. V. Lychagin, Vvedenie v geometriyu nelineinykh differentsialnykh uravnenii, Nauka, Moscow, 1986 (Russian).

I. S. Krasil'shchik, V. V. Lychagin, and A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon and Breach, 1986.

A. V. Bocharov, A. M. Verbovetsky, A. M. Vinogradov (editor), S. V. Duzhin, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, N. G. Khor'kova, and V. N. Chetverikov, Simmetrii i zakony sokhraneniya uravnenij matematicheskoj fiziki, Factorial, Moscow, 1997 (Russian), Second edition 2005.

I. S. Krasil'shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education & Sciences, Opava, 1998, arXiv:math/9808130.

A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor'kova, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, A. M. Verbovetsky, and A. M. Vinogradov (editor), Symmetries and conservation laws for differential equations of mathematical physics, AMS, 1999.

I. S. Krasil'shchik and A. M. Verbovetsky, Gomologicheskie metody v geometrii differentsial'nykh uravnenij, IUM, Moscow, 1999, Lecture notes, (Russian).

I. S. Krasil'shchik and P. H. M. Kersten, Symmetries and recursion operators for classical and supersymmetric differential equations, Kluwer, 2000.

I. S. Krasil'shchik, P. H. M. Kersten, M. Marvan, and Verbovetsky A. M., Gomologicheskie metody geometrii differentsial'nykh uravnenij, MCCME, Moscow, 2008 (Russian), to appear.

Book Translations

Zh. Pommare, Sistemy uravnenii s chastnymi proizvodnymi i psevdogruppy Li, Mir, Moscow, 1983 (Russian), Translated from the English by A. V. Bocharov, M. M. Vinogradov and I. S. Krasil'shchik, Translation edited and with a preface and an appendix by A. M. Vinogradov.

A. M. Vinogradov, Cohomological analysis of partial differential equations and secondary calculus, Translations of Mathematical Monographs, vol. 204, AMS, 2001, Translated from the Russian manuscript by Joseph Krasil'shchik.

Jet Nestruev, Smooth manifolds and observables, Graduate Texts in Mathematics, vol. 220, Springer, 2003, Joint work of A. M. Astashov, A. B. Bocharov, S. V. Duzhin, A. B. Sossinsky, A. M. Vinogradov, and M. M. Vinogradov. Translated from the 2000 Russian edition by A. B. Sossinsky, I. S. Krasil'shchik, and S. V. Duzhin.

Books and Collections Editing

P. H. M. Kersten and I. S. Krasil'shchik (eds.), Geometric and algebraic structures in differential equations, Kluwer, 1995, Papers from the Workshop on Algebra and Geometry of Differential Equations held in Enschede, 1993, Reprint of Acta Appl. Math. 41 (1995), no. 1-3.

I. S. Krasil'shchik and A. M. Vinogradov (eds.), Algebraic aspects of differential calculus, Kluwer, 1997, Acta Appl. Math. 49 (1997), no. 3.

Marc Henneaux, Joseph Krasil'shchik, and Alexandre Vinogradov (eds.), Secondary calculus and cohomological physics, Contemporary Mathematics, vol. 219, AMS, 1998.

B. P. Komrakov, I. S. Krasil'shchik, G. L. Litvinov, and A. B. Sossinsky (eds.), Lie groups and Lie algebras. Their representations, generalisations and applications, Mathematics and its Applications, vol. 433, Kluwer, 1998.

I. S. Krasil'shchik and A. M. Vinogradov (eds.), Geometrical aspects of nonlinear differential equations, Kluwer, 1999, Acta Appl. Math. 56 (1999), no. 2-3.

Joseph Krasil'shchik (ed.), Symmetries of differential equations and related topics, Kluwer, 2002, Acta Appl. Math. 72 (2002), no. 1-2.

Joseph Krasil'shchik (ed.), Geometry of PDE in action: zero-curvature representations, recursion operators, and control systems, Kluwer, 2004, Acta Appl. Math. 83 (2004), no. 1-2.

I. S. Krasil'shchik and B. S. Kruglikov (eds.), Algebra and geometry of PDEs, Springer, 2008, Acta Appl. Math. 101 (2008), no. 1-3.

Papers and other publications

1974

I. S. Krasil'shchik, Gamil'tonov formalizm v kanonicheskikh kol'tsakh, Voronezh Winter Math. School, Voronezh State Univ., 1974, pp. 23–24 (Russian).

1975

A. M. Vinogradov and I. S. Krasil'shchik, What is Hamiltonian formalism?, Russ Math. Surv. 30 (1975), no. 1, 177–202, Russian original: Uspehi Mat. Nauk 30 (1975), no. 1, 173–198; also in Integrable systems: selected papers, London Math. Soc. Lect. Note Ser., 60, 1981, 241–266.

1980

I. S. Krasil'shchik, Hamiltonian cohomology of canonical algebras, Sov. Math. Dokl. 21 (1980), 625–629, Russian original: Dokl. Akad. Nauk SSSR 251 (1980), 1306–1309.

A. M. Vinogradov and I. S. Krasil'shchik, A method of calculating higher symmetries of nonlinear evolutionary equations, and nonlocal symmetries, Sov. Math. Dokl. 22 (1980), 235–239, Russian original: Dokl. Akad. Nauk SSSR 253 (1980), 1289–1293.

1984

A. M. Vinogradov and I. S. Krasil'shchik, On the theory of nonlocal symmetries of nonlinear partial differential equations, Sov. Math. Dokl. 29 (1984), 337–341, Russian original: Dokl. Akad. Nauk SSSR 275 (1984), 1044–1049.

I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal symmetries and the theory of coverings: an addendum to Vinogradov’s “Local symmetries and conservation laws” [Acta Appl. Math. 2 (1984), 21–78], Acta Appl. Math. 2 (1984), 79–96.

1986

I. S. Krasil'shchik, Pochemu preobrazovaniya Beklunda obrazuyut gruppu?, Proceedings of the seminar on the algebra and geometry of differential equations (Moscow), MSU, 1986 (Russian).

1987

I. S. Krasil'shchik, Del'ta-lemma Puankare, Mimeographed notes, 1987.

1988

I. S. Krasil'shchik, Schouten bracket and canonical algebras, Global analysis—studies and applications, III, Lecture Notes in Math., vol. 1334, Springer, 1988, pp. 79–110, Russian original: Global'nyj analiz i matematicheskaya fizika, Nov. Global'nom anal., Voronezh, 1987, 73–94.

1989

I. S. Krasil'shchik and A. M. Vinogradov, Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta Appl. Math. 15 (1989), 161–209.

1991

I. S. Krasil'shchik, Supercanonical algebras and Schouten brackets, Math. Notes 49 (1991), no. 1, 50–54, Russian original: Mat. Zametki 49 (1991), no. 1, 70–76.

1992

I. S. Krasil'shchik, Algebry deformatsij differentsial'nykh uravnenij i operatory rekursii, Proceedings XXX MIKKHiS Scientific Conference (Moscow), MIKKHiS, 1992, pp. 18–21 (Russian).

I. S. Krasil'shchik, Differential operators of constant growth and Jacobi structures of infinite order, Publ. IRMA Lille 30 (1992), no. XI, 1–21.

I. S. Krasil'shchik, Some new cohomological invariants for nonlinear differential equations, Differential Geom. Appl. 2 (1992), 307–350.

1993

I. S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations. I, Russ. Math. 37 (1993), no. 1, 25–35, Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1993, no. 1, 27–37.

I. S. Krasil'shchik, Some new cohomological invariants of nonlinear differential equations. II, Russ. Math. 37 (1993), no. 2, 52–65, Russian original: Izv. Vyssh. Uchebn. Zaved. Mat. 1993, no. 2, 54–68.

I. S. Krasil'shchik, Lie algebra structures for the symmetries of differential equations possessing recursion operators, ESI Preprint 47 (1993).

I. S. Krasil'shchik, An algebraic model for characteristics of differential equations, ESI Preprint 48 (1993).

1994

I. S. Krasil'shchik and P. H. M. Kersten, Deformations and recursion operators for evolution equations, Geometry in partial differential equations, World Sci., 1994, pp. 114–154, Also: Memorandum of the Twente University, 1992, no. 1104.

1995

I. S. Krasil'shchik, Hamiltonian formalism and supersymmetry for nonlinear differential equations, ESI Preprint 257 (1995).

I. S. Krasil'shchik, Notes on coverings and Bäcklund transformations, ESI Preprint 260 (1995).

I. S. Krasil'shchik and P. H. M. Kersten, Graded differential equations and their deformations: a computational theory for recursion operators, Acta Appl. Math. 41 (1995), 167–191.

P. H. M. Kersten and I. S. Krasil'shchik, Graded Frölicher-Nijenhuis brackets and the theory of recursion operators for super differential equations, The interplay between differential geometry and differential equations, Amer. Math. Soc. Transl. Ser. 2, vol. 167, AMS, 1995, pp. 91–129, Also: Memorandum of the Twente University, 1993, no. 1147.

1996

I. S. Krasil'shchik, A supersymmetry approach to Poisson structures over differential equations, Differential geometry and applications, Masaryk Univ., Brno, 1996, pp. 381–391.

I. S. Krasil'shchik, Poisson structures on nonlinear evolution equations, Memorandum of the Twente University 1320 (1996).

1997

I. S. Krasil'shchik, Calculus over commutative algebras: a concise user guide, Acta Appl. Math. 49 (1997), 235–248.

I. S. Krasil'shchik, Poincaré δ-lemma for smooth algebras, Acta Appl. Math. 49 (1997), 249–255.

I. S. Krasil'shchik, Characteristics of linear differential operators over commutative algebras, Acta Appl. Math. 49 (1997), 257–269.

I. S. Krasil'shchik, Algebraicheskie metody v teorii integriruemykh sistem, MSU, Moscow, 1997, Dr. Sci. Thesis (Russian).

1998

I. S. Krasil'shchik, Symmetries and recursion operators for soliton equations, Nonlinearity and geometry, PWN, Warsaw, 1998, pp. 141–156.

Joseph Krasil'shchik, Cohomology background in geometry of PDE, Secondary calculus and cohomological physics, Contemp. Math., vol. 219, AMS, 1998, pp. 121–139.

I. S. Krasil'shchik, Algebras with flat connections and symmetries of differential equations, Lie groups and Lie algebras. Their representations, generalisations and applications, Math. Appl., vol. 433, Kluwer, 1998, pp. 407–424.

1999

Joseph Krasil'shchik in collaboration with Barbara Prinari, Lectures on linear differential operators over commutative algebras. (The 1st Italian diffiety school, July, 1998), Diffiety Inst. Preprint Series 1 (1999), DIPS 1/99.

I. S. Krasil'shchik, Cohomological approach to Poisson structures on nonlinear evolution equations, Lobachevskii J. Math. 3 (1999), 127–145 (electronic).

Joseph Krasil'shchik and Michal Marvan, Coverings and integrability of the Gauss-Mainardi-Codazzi equations, Acta Appl. Math. 56 (1999), 217–230, arXiv:solv-int/9812010.

2000

Joseph Krasil'shchik, Integrability and supersymmetry, RIMS Kokyuroku 1150 (2000), 147–152.

Joseph Krasil'shchik, On one-parametric families of Bäcklund transformations, Diffiety Inst. Preprint Series 1 (2000), DIPS 1/2000.

I. S. Krasil'shchik, Deformations and integrable systems, Proc. Conf. Differential Equations and Related Topics dedicated to 100th Anniversary of birthday of I. G. Petrovskii, MSU, Moscow, 2001.

2002

Sergei Igonin and Joseph Krasil'shchik, On one-parametric families of Bäcklund transformations, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, Tokyo, 2002, pp. 99–114, arXiv:nlin/0010040.

Paul Kersten and Joseph Krasil'shchik, Complete integrability of the coupled KdV-mKdV system, Lie groups, geometric structures and differential equations—one hundred years after Sophus Lie, Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, 2002, pp. 151–171, arXiv:nlin/0010041.

Joseph Krasil'shchik, Geometry of differential equations: a concise introduction, Acta Appl. Math. 72 (2002), 1–17.

Iosif Krasil'shchik, A simple method to prove locality of symmetry hierarchies, Diffiety Inst. Preprint Series 9 (2002), DIPS 09/2002.

P. H. M. Kersten and I. S. Krasil'shchik, From recursion operators to Hamiltonian structures. The factorization method, Memorandum of the Twente University 1624 (2002), Lectures delivered at the MRI Spring School “Frobenius Manifolds in Mathematical Physics”.

P. H. M. Kersten, I. S. Krasil'shchik, and A. M. Verbovetsky, An extensive study of the N=1 supersymmetric KdV equation, Memorandum of the Twente University 1656 (2002).

2003

S. Igonin, P. H. M. Kersten, and I. Krasil'shchik, On symmetries and cohomological invariants of equations possessing flat representations, Differential Geom. Appl. 19 (2003), 319–342, arXiv:math/0301344.

P. H. M. Kersten, I. S. Krasil'shchik, and A. M. Verbovetsky, A geometric approach to Hamiltonian structures for evolution equations, Proc. Int. Conf. Kolmogorov and Contemporary Mathematics (Moscow), 2003, pp. 815–816.

I. S. Krasil'shchik, A. M. Verbovetsky, and P. H. M. Kersten, Nonlocal Hamiltonian, symplectic and recursion structures for N=1 supersymmetric KdV equation, Proc. Int. Conf. Kolmogorov and Contemporary Mathematics, 2003, pp. 817–818.

I Krasil'shchik, The long exact sequence of a covering: three applications, Diffiety Inst. Preprint Series 6 (2003), DIPS 6/2003.

2004

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, Hamiltonian operators and l*-coverings, J. Geom. Phys. 50 (2004), 273–302, arXiv:math/0304245.

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, (Non)local Hamiltonian and symplectic structures, recursions and hierarchies: a new approach and applications to the N=1 supersymmetric KdV equation, J. Phys. A 37 (2004), 5003–5019, arXiv:nlin/0305026.

P. Kersten, I. Krasil'shchik, and A. Verbovetsky, On the integrability conditions for some structures related to evolution differential equations, Acta Appl. Math. 83 (2004), 167–173, arXiv:math/0310451.

Paul Kersten, Iosif Krasil'shchik, and Alexander Verbovetsky, Nonlocal constructions in the geometry of PDE, Symmetry in nonlinear mathematical physics. Part 1, Inst. Math. NAS Ukr., Kiev, 2004, pp. 412–423.

P. H. M. Kersten, I. Krasil'shchik, and A. Verbovetsky, The Monge-Ampère equation: Hamiltonian and symplectic structures, recursions, and hierarchies, Memorandum of the Twente University 1727 (2004).

2005

I. S. Krasil'shchik, Prostoj metod dokazatel'stva lokal'nosti simmetrij evolyutsionnykh uravnenij, Scientific Bulletin of MSTUCA 91 (2005), 12–19 (Russian).

2006

Paul Kersten, Iosif Krasil'shchik, and Alexander Verbovetsky, A geometric study of the dispersionless Boussinesq type equation, Acta Appl. Math. 90 (2006), 143–178, arXiv:nlin/0511012.

P. H. M. Kersten and I. S. Krasil'shchik, The Cartan covering and complete integrability of the KdV-mKdV system, Constructive Algebra and Systems Theory (B. Hanzon and Hazewinkel M., eds.), Royal Netherlands Academy of Arts and Sciences, 2006, pp. 251–265.

2007

A. M. Verbovetsky, V. A. Golovko, and I. S. Krasil'shchik, Skobka Li dlya nelokal'nykh tenej, Scientific Bulletin of MSTUCA 114 (2007), 9–23 (Russian).

J. Krasil'shchik, Nonlocal geometry of PDEs and integrability, Symmetry and perturbation theory (G. Gaeta, R. Vitolo, and Walcher S., eds.), World Sci., 2007, pp. 100–108.

I. S. Krasil'shchik, Estestvennye nakrytiya i integriruemye sistemy, Symmetries: theoretical and methodological aspects, Astrakhan Univ., Astrakhan, 2007, pp. 46–53 (Russian).

2008

V. A. Golovko, I. S. Krasil'shchik, and A. M. Verbovetsky, Variational Poisson-Nijenhuis structures for partial differential equations, Theor. Math. Phys. 154 (2008), 227–239, Russian original: Teor. Mat. Fiz. 154 (2008), 268–282.

V. Golovko, P. Kersten, I. Krasil'shchik, and A. Verbovetsky, On integrability of the Camassa-Holm equation and its invariants, Acta Appl. Math. 101 (2008).

Maks A. Akivis

On the occasion of his 85th birthday and 65 years of scientific activity.

M.A. Akivis
M.A. Akivis

One of the great contemporary geometers, Professor, Dr. Maks Aizikovich
Akivis celebrated his 85th birthday and 65 years of scientific activity on January 5, 2008. On this occasion we want to honor the continuing scholarly productivity of Akivis whose scientific activity prior to 1993 and through 1998 was recognized in articles Maks Aizikovich Akivis published in Uspekhi Mat. Nauk 48 (1993), no. 3 (291), 213–216; in Webs and Quasigroups, 1993, pp. 4–8; and in Webs and Quasigroups, 1998/1999, pp. 7–11. In the last publication, a complete list of Akivis’ publications prior to 1999 and the list of Ph. D. theses written under his supervision were published.

Maks Aizikovich Akivis was born on January 5, 1923 in Novosibirsk, USSR.
While he was in high school, he demonstrated outstanding mathematical talent and ability. In 1940 he entered the Faculty of Mechanics and Mathematics of Moscow State University. During the Great Patriotic War (WWII) his studies were interrupted from 1942 to 1945 while he served in the Soviet Army and participated in the liberation of Prague and the capture of Berlin. For his patriotic service he was awarded many orders and medals.
After WWII he resumed his studies at Moscow State University. However,
he was not able to graduate normally: he was expelled when he was a fifth
year student for “ideological reasons”. These “ideological reasons” also kept
Akivis, one of the best students of the Faculty of Mechanics and Mathematics,
from becoming a graduate student of Moscow State University. Not until 1958 was he able to defend his Ph. D. thesis which summarized the results of his undergraduate work. In 1964 Akivis defended a second (Doctor of Science) dissertation, and one year later, in 1965 he became a full professor.
Professor I. M. Gelfand, who taught at Moscow State University in the 1940s,
recalls: “M. A. Akivis and E. B. Dynkin were my best students at Moscow
State University in the 1940s. Akivis choose Differential Geometry as his field
of research. Unfortunately, the end of his career as a student was darkened
by the conditions in the Soviet Union at that time, and he could not pursue
the normal graduate studies. However, he was able to overcome all difficulties and became one of the best scientists of his generation in the area of classical differential geometry.”
From 1956 to 1960 Akivis taught at the Tula Mechanical Institute, and from
1960 to 1994 he was a Professor in the Department of Mathematics at Moscow Institute of Steel and Alloys.

Since 1948 Akivis published more than 150 scientific books and papers. His results in multidimensional projective and conformal differential geometry, in web theory and in the theory of differential geometric structures are fundamental. Many of these results are classical and are cited in numerous papers.

The first Akivis papers were devoted to the T-pairs of complexes (three-parameter families) of straight lines in a three-dimensional projective space.

His advisor S. P. Finikov suggested that he apply the notion of harmonic intersection of ruled surfaces which had been introduced by  E. Cartan to the theory of congruences (two-parameter families) of straight lines of a three-dimensional projective space. Akivis solved this problem brilliantly: he found a new geometric property of the T-pairs of congruences introduced by Finikov and extended his results to the pairs of complexes of straight lines. As he showed, the T-pairs of complexes of straight lines he discovered are transferred by the Plücker mapping into a configuration of a five-dimensional projective space consisting of a tangentially degenerate two-dimensional submanifold and a three-dimensional submanifold carrying a net of conjugate lines. In the 1950s and 1960s Akivis devoted a series of papers to the projective theory of submanifolds of the type indicated above. As a result, he created a new area of projective differential geometry which continues to be developed successfully to this day.

M. A. Akivis (PDF)