Prehistory: the Kolozsv´arroots

New university in Transylvania Selected Passages from the History • The foundation of the Franz Joseph University in 1872. of the Szeged Mathematical School 1 • Math. departments: Kolozsv´arand the first years in Szeged • Elementary Mathematics • Higher Mathematics • Mathematical Physics Lajos Klukovits • A serious problem: there were no outstanding mathematician, prospective professors, in Transylvania. Bolyai Institute University of Szeged • The govermental solution: May 20, 2021. • the polyhistor Samuel Brassai and the aviatist ingeneer Lajos Martin were the first two, • two years later the young physicist M´orR´ethy.

Prehistory: the Kolozsv´arroots Kolozsv´ar:the the turn of the century. Scientific interests • Brassai: ranging from linguistics (eg. sanscrit) to astronomy, 3 famous new professors. • in Kolozsv´aryears mostly astronomy, Hungarian math. ´ language. 1. Gyula VALYI (1855-1913) • important deficiency: not recognized the ultimate importance • The most talanted student of R´ethy, of the Bolyai geometry. • in Berlin: lectures of Weierstrass, Kirchoff, Kronecker and Kummer. • Martin: theoretical questions of aviatics, • Main interest: partial differential equations, • R´ethy:the shape of the incompressible fluid rays. • first time regular course on the Appendix. • the first professor in Hungary lecturing on the Bolyai geometry, 2. Gyula FARKAS (1847-1930) • several constructions in hyperbolic geometry. • Mainly theoretical physicist, system of linear inequalities, • R´ethywas the first outstanding professor of theoretical physics • Farkas-lemma: fundamental in operation research. in Hungary. 3. Lajos SCHLESINGER (1864-1933) • One of his most important statemens: ...in our country before • Complex functions, system of differential equations, the two Bolyai, there were no outstanding mathematician; • several lecture notes. therefore almost all mathematical research have to be rooted in their results. Kolozsv´ar:The beginning of the XX. century. A new epoch, Hungary.

The status after Trianon Treaty.

New world famous professors. • A great part of the Hungarian professors and students were 1. Lip´otFEJER´ (1880-1959) forced to move outside Rumania. • Technical University Budapest: Gyula K˝onig,J´ozsefK¨ursch´ak • In 1919 they settled in Budapest. and Guszt´avRados, • In 1921 a newly estableshed university opened his first • Berlin: L. Fuchs, G. Frobenius and H.A. Schwarz, • 1900. Comptes Rendus (Paris): his famous theorem on semester in Szeged. summation of Fourier series via arithmetical means. • G¨ottingen:D. Hilbert ´esH. Minkowski The new beginning. • Paris: E. Picard and J. Hadamard. • 1905-19. Kolozsv´ar. • The Mathematical Seminar: the 3 original Kolozsv´ar 2. Frigyes RIESZ (1880-1956) mathematical departments and the Institute of Descriptive Geometry. 3. Alfr´edHAAR (1885-1933) • 3 professors: F. Riesz, A. Haar and R. Ortvay. • Assistants: Tibor Rad´oand L´aszl´oKALMAR.´

First years in Szeged. Selected results of F. Riesz. Frigyes RIESZ (1880-1956). The first fundamental discovery. • The spiritual leader. The first sentence fom his credo: Certain sets of functions can behave — using appropriate The duty of a professor is to transmite pure science like an operations — like a linear (vector) space. This led to the notion of antenna to high standard. Whether it is understood or not, it abstract function spaces. is not up to him. First famous theorem: the Riesz-Fischer theorem (1907). • Studies: • 1897 - 99: ETH Z¨urich, The space of the square integrable (in Lebesgue sence) functions is • 1899 - 1901: University of Budapest (Gy. K¨onig,J. K¨ursch´ak), isomorphic to the space of number sequences having finite sum of • 1903: G¨ottingen(D. Hilbert, H. Minkowski). the squares of its members. • 1902: PhD in Budapest, An equivalent form: A measurable function on [−π, π] is square • 1904 - 11: secondary school teacher in L¨ocse, Budapest. (Lebesgue) integrable iff the corresponding Fourier series converges 2 • 1911 - 19: professor in Kolozsv´ar,1916. corr.memb. HAS, in the space L . • 1919 - 21: professor in Budapest, • 1921 - 46: professor in Szeged, 1936. ord.memb. HAS This theorem led to the discovery of the abstract Hilbert space • 1947 - 56: professor in Budapest. fundamental in quantum mechanics. Selected results of F. Riesz. A summary of Riesz’s results, the famous book.

Riesz-Fr´echetor Riesz representation theorem. Any element of the dual space of a Hilbert space H can be uniquely written in a form of inner product in H. LEC¸ONS d’ANALYSE FONCTIONELLE (1952) This theorem characterizes the linear functionals of a Hilbert space. More notable results. • A joint work with his former student and college B´elaSZOKEFALVI-NAGY,˝ 1. The general notion of the topological space. translated into several languages: English, Deutch, Russian 2. A new construction of the Lebesgue integral without measure Japonese Chinese and finally Hungarian. theory using almost everywhere convergent series of step • It served and serve for generations of reseachers and students functions. as a textbook and lecture note in real functions and functional 3. A new characterization of subharmonic functions: a road to analyses. potential theory. 4. Lattice ordered linear spaces.

A remarkable step in the math. life in Szeged. The second famous professor: Alfr´edHAAR.

Studies. • A dinner in G¨ottingen(1929): Kuno Klebelsberg (minister of • 1903: Technical University Budapest: chemistry, education in Hungary) and Richard Courant. • 1904: mathematics in G¨ottingen:C. Charatheodory, D. • Courant told the minister on famous Szeged mathematicians: Hilbert, F. Klein, E. Zermelo. Riesz Haar, Ker´ekj´art´o,Rad´o,Kalm´ar and Lipka. • 1909: PhD supervised by Hilbert, his thesis: Zur Theorie der • Traveling home Klebelsberg ordered a special financial aid to orthogonalen Functionsystemen (Math. Annalen 1910). improve the circumstances of the Szeged Math. Seminar. • In 1929 the Mathematical Seminar and the Institute of Professional activities. Geometry joined as BOLYAI INSTITUTE. • The Department of Mathematical Physics became an • Privatdozent in G¨ottingen, 1907-12. independent institute under the name of Institute of • Extraordinary professor in Kolozsv´ar1912-17. Theoretical Physics. The leader was Zolt´anBAY. • Ordinary professor in Kolozsv´ar 1917-19. • Ordinary professor in Szeged 1921-33. Selected research topics. Foundation of a periodical: the story of the ”ACTA”.

G¨ottingenand Kolozsv´aryears. • Riesz’s and Haar’s initiative: to lounch a mathematical • Summations of series. periodical of high standard, based on civil support. • • General orthonormal system of functions. The original title: Acta Litterarum ac Scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae Sectio • Variational calculus. Scientiarum Mathematicarum, later Acta Scientiarum • Partial differential equations and their applications to Mathematicarum (shortly Acta Sci. Math. Szeged). Editors: elasticity theory. F. Riesz and A. Haar. • Shortly became a leading math. periodical. Szeged years: Analysis on groups. • Some authors from volume 1 (1922): J. Egerv´ary, A Haar, C. • Introducing an invariant measure on locally compact groups: Jordan, B. Ker´ekj´art´o,J. K¨ursch´ak,J. von Neumann, A. ⇒ Haar measure. Ostrowki, T. Rad´o,G. Szeg˝o,F. Riesz, M. Riesz, B. • A famous paper: Der Massbegriff in der Theorie der Sz˝okefalvi-Nagy. kontinuirlichen Gruppen (Math. Ann. 1933.) • The latest volume, vol. 85 published in 2019. • His concepts led to set up an abstract theory of commutative • This periodical has played a tremendous role in the harmonic analysis (J. von Neumann, Pontryagin) enrichment of the math. library.

Founder of the Szeged Mathematical School in the city Complete the ”First Triumvirate”: B´elaKEREKJ´ ART´ O´ Pantheon. (1898-1946).

Scientific carieer. • Math. studies in Budapest University 1916-20. • Doctorate 1920, • Extraordinary professor 1922 Szeged, full professor 1925-38 Szeged, • Visiting professor G¨ottingen,Barcelona and Princeton 1923-25. • Member of HAS 1934, 1945. • Professor of the Budapest University 1938. B´elaKEREKJ´ ART´ O´ (1898-1946). B´elaKEREKJ´ ART´ O´ (1898-1946). Two courses in G¨ottingen

• Topology, Mathematische Betrachtungen zur Kosmologie. • An enlarged version in book form of the first course: Vorlesungen ¨uber Topologie 1923. • The first work in this kind, inspiring much later research in Two famous results. geometry. 1. A homeomorphic mapping betwwen two Jordan-curve can be • Reception of the book: contradictory. extended to the whole plain. • S. Lefschetz in his 1924 book: 2. The proof of the reverse of the famous Jordan-curve-theorem. This production, from the pen of a young Hungarian mathematician, who is beginning to be known for his contributions to analysis situs, is wellcome for several reason. • Two main topics from the book: 1. Topology of the plain and its curves centering around the Jordan curve theorems. 2. Structure of regions and their boundaries, the general closed curves.

The end of the ”First Triumvirate”.

1. Haar died in 1933. 2. Ker´ekj´art´omoved to Budapest in 1938. 3. New famous scientists came into limelight: • L´aszl´oKALMAR´ (1905-76.) • L´aszl´oREDEI´ (1900-80.) • B´elaSZOKEFALVI-NAGY˝ (1913-91.) Thank you for your attention. 4. After World War II they formed the SECOND TRIUMVIRATE, 5. but their period will be the content of a future talk.