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Wykład Łojasiewicza

Serdecznie zapraszamy na tegoroczny wykład im. Stanisława Łojasiewicza.

Maksym Radziwiłł (UT Austin)

Recent developments in analytic number theory


I will survey recent developments in analytic number theory with an emphasis on their relationships with areas outside number theory.

Uniwersytet Jagielloński, Wydział Matematyki i Informatyki

ul. Łojasiewicza 6, 30-348 Kraków

Poniedziałek, 15 maja 2023, sala 0004, godz. 16:00.

Wykład będzie transmitowany na żywo na naszym kanale YouTube.

Warsztaty towarzyszące odbędą się w dniach 16-19 maja 2023.

Stanisław Łojasiewicz was born on October 9, 1926 in Warsaw. As a teenager he was particularly interested in music and mathematics, and eventually he decided to become a mathematician. He completed his mathematical studies in 1945-47 at the Jagiellonian University in Cracow, and from that time his career was connected with the Jagiellonian University, as well as with the Mathematical Institute of the Polish Academy of Sciences. The magnitude of his mathematical achievements sets him in the rank of the most outstanding mathematicians of the twentieth century. His printed output counts 70 positions and concerns differential equations, theoretical mechanics, differential analysis, distribution theory and analytic geometry.

At the beginning he was under influence of Tadeusz Ważewski; hence, he was dealing with differential equations, mainly ordinary, studying in particular asymptotical effects. He defended his PhD thesis entitled Sur l'allure asymptotique des intègrales du système d'équations differentielles au voisinage de point singulier in 1950. Then he turned his interests towards distribution theory, the systematic description of which was created in the first half of fifties by Laurent Schwartz. First works of Łojasiewicz in this direction concerned the fixation of variables in distributions. During his first stay abroad, in Paris in 1957 he achieved a great success solving the problem of division of distributions by analytic functions posed by Schwartz. His solution, published in 1958 in CRAS had a great significance for the development of the theory of partial differential equations, differential analysis and analytic geometry. Independently, at the same year 1958 Lars Hörmander published his result about division of distributions by polynomials. Łojasiewicz's solution was based on some foundamental property of real analytic function which is now widely known as Łojasiewicz's inequality. The carefull analysis of the method of the proof of this inequality led Stanisław Łojasiewicz to creating a new geometry, i.e. semianalytic geometry, which studies the subsets of euclidean spaces described locally by a finite number of analytic equalities and inequalities, and initiated its generalization called now subanalytic geometry, developed by Gabrielov, Hironaka and others, dealing with proper images of semianalytic sets by analytic maps. Since many mathematical problems lead to studying such objects, semi- and subanalytic geometry became a useful tool in many branches of analysis as approximation theory, control theory, and also in logic in connection with model theory.

In 1958-60 Stanisław Łojasiewicz visited universities in Kingston (Ontario), in Chicago, in Berkeley and the Institute of Advanced Studies in Princeton. In 1962 he obtained professorship at the Jagiellonian University. At the same year he was invited by Aldo Andreotti to Pisa where he elaborated his fameous theorem on triangulation of semianalytic sets. In 1964, invited by UNESO , he gave a course on division problem and triangulation at the University if Buenos Aires. Years 1964-65 Łojasiewicz spent at the University of Paris, where he presented his first systematic elaboration of semianalytic geometry published as a well-known monograph Ensembles semi-analytiques. During his next stay in France, in 1967-68 at the Institut des Hautes Etudes Scientifiques, he was dealing with problems of differential analysis, first of all in connection with the Malgrange-Mather Preparation Theorem. The result of his studies was a beautiful and surprisingly short proof of this important theorem, which was presented at the Singularity Symposium at Liverpool in 1970, together with other results about Whitney fields and determinacy of jets. In 1970 Stanisław Łojasiewicz was invited to deliver a lecture about semianalytic geometry at the International Congress of Mathematicians in Nice.

In 1971 Stanisław Łojasiewicz was elected to Polish Academy of Sciences as a correspondent member and became its member in 1980. In 1983 he was elected to the Pontifica Academia Scientiarum. In seventies he was continuing his lively collaboration with mathematicians from Italy (University of Pisa), and from France. In the academic year 1976/77 he was working at the Instituto de Matematica Pura et Aplicada in Rio de Janeiro. From late seventies he was studying mainly subanalytic geometry, as well as complex analytic geometry to which he devoted his monograph An introduction to complex analytic geometry. This monograph as well as two others An introduction to the theory of real functions and mentioned above Ensembles semi-analytiques are examples of mastery in mathematical precision and conciseness.

Stanisław Łojasiewicz died on November 14, 2002.

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