R. Remmert – författare

1. The classical theorem of Mittag-Leffler was generalized to the case of several complex variables by Cousin in 1895. In its one variable version this says that, if one prescribes the principal parts of a merom orphic function on a domain in the complex plane e, then there exists a meromorphic function defined on that domain having exactly those principal parts. Cousin and subsequent authors could only prove the analogous theorem in several variables for certain types of domains (e. g. product domains where each factor is a domain in the complex plane). In fact it turned out that this problem can not be solved on an arbitrary domain in em, m ~ 2. The best known example for this is a "notched" bicylinder in 2 2 e . This is obtained by removing the set { (z , z ) E e 11 z I ~ !, I z 1 ~ !}, from 1 2 1 2 2 the unit bicylinder, ~ :={(z , z ) E e llz1 < 1, lz1 < 1}. This domain D has 1 2 1 2 the property that every function holomorphic on D continues to a function holo­ morphic on the entire bicylinder. Such a phenomenon never occurs in the theory of one complex variable. In fact, given a domain G c e, there exist functions holomorphic on G which are singular at every boundary point of G.

:So eine Illrbeit witb eigentIid) nie rertig, man muli iie fur fertig erfHiren, wenn man nad) 8eit nnb Umftiinben bas moglid)fte get an qat. (@oetqe,

Felix Hausdorff gehört zu den herausragenden Mathematikern der ersten Hälfte des 20. Jahrhunderts. Eine Gesamtausgabe seiner Werke galt lange als Desideratum. Die auf 8 Bände veranschlagte Edition wird Hausdorffs gesamtes publiziertes Opus enthalten, ferner eine Reihe bemerkenswerter Stücke aus dem umfangreichen wissenschaftlichen Nachlaß. Alle Texte werden von Fachleuten auf den einzelnen Gebieten sorgfältig kommentiert; an dieser Arbeit sind mehr als 20 Mathematiker, Mathematikhistoriker, Astronomen, Philosophen und Literaturwissenschaftler aus vier Staaten beteiligt. Der vorliegende Band IV enthält Hausdorffs Arbeiten zur Analysis, Algebra und Zahlentheorie, darunter die klassischen auch heute noch vielzitierten Texte zu Hausdorff-Maß und Hausdorff-Dimension und zum Hausdorffschen Kugelparadoxon. Aus dem Nachlaß werden 19 Faszikel publiziert, ferner einige interessante Briefe.

Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus 9 immediately attach a g­ dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. th In the second half of the 19 century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds.

Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus 9 immediately attach a g­ dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. th In the second half of the 19 century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds.

:So eine Illrbeit witb eigentIid) nie rertig, man muli iie fur fertig erfHiren, wenn man nad) 8eit nnb Umftiinben bas moglid)fte get an qat. (@oetqe,

Felix Hausdorff gehört zu den herausragenden Mathematikern der ersten Hälfte des 20. Jahrhunderts. Eine Gesamtausgabe seiner Werke galt lange als Desideratum. Die auf 8 Bände veranschlagte Edition wird Hausdorffs gesamtes publiziertes Opus enthalten, ferner eine Reihe bemerkenswerter Stücke aus dem umfangreichen wissenschaftlichen Nachlaß. Alle Texte werden von Fachleuten auf den einzelnen Gebieten sorgfältig kommentiert; an dieser Arbeit sind mehr als 20 Mathematiker, Mathematikhistoriker, Astronomen, Philosophen und Literaturwissenschaftler aus vier Staaten beteiligt. Der vorliegende Band IV enthält Hausdorffs Arbeiten zur Analysis, Algebra und Zahlentheorie, darunter die klassischen auch heute noch vielzitierten Texte zu Hausdorff-Maß und Hausdorff-Dimension und zum Hausdorffschen Kugelparadoxon. Aus dem Nachlaß werden 19 Faszikel publiziert, ferner einige interessante Briefe.

...Je mehr ich tiber die Principien der Functionentheorie nachdenke - und ich thue dies unablassig -, urn so fester wird meine Uberzeugung, dass diese auf dem Fundamente algebraischer Wahrheiten aufgebaut werden muss (WEIERSTRASS, Glaubensbekenntnis 1875, Math. Werke II, p. 235). 1. Sheaf Theory is a general tool for handling questions which involve local solutions and global patching. "La notion de faisceau s'introduit parce qu'il s'agit de passer de donnees 'locales' a l'etude de proprietes 'globales'" [CAR], p. 622. The methods of sheaf theory are algebraic. The notion of a sheaf was first introduced in 1946 by J. LERAY in a short note Eanneau d'homologie d'une representation, C. R. Acad. Sci. 222, 1366-68. Of course sheaves had occurred implicitly much earlier in mathematics. The "Monogene analytische Functionen", which K. WEIERSTRASS glued together from "Func- tionselemente durch analytische Fortsetzung", are simply the connected components of the sheaf of germs of holomorphic functions on a RIEMANN surface*'; and the "ideaux de domaines indetermines", basic in the work of K. OKA since 1948 (cf. [OKA], p. 84, 107), are just sheaves of ideals of germs of holomorphic functions.Highly original contributions to mathematics are usually not appreciated at first. Fortunately H. CARTAN immediately realized the great importance of LERAY'S new abstract concept of a sheaf. In the polycopied notes of his Semina ire at the E. N. S.

Of making many books there is no end; and much study is a weariness of the flesh. Eccl. 12.12. 1. In the beginning Riemann created the surfaces. The periods of integrals of abelian differentials on a compact surface of genus 9 immediately attach a g­ dimensional complex torus to X. If 9 ~ 2, the moduli space of X depends on 3g - 3 complex parameters. Thus problems in one complex variable lead, from the very beginning, to studies in several complex variables. Complex tori and moduli spaces are complex manifolds, i.e. Hausdorff spaces with local complex coordinates Z 1, ... , Zn; holomorphic functions are, locally, those functions which are holomorphic in these coordinates. th In the second half of the 19 century, classical algebraic geometry was born in Italy. The objects are sets of common zeros of polynomials. Such sets are of finite dimension, but may have singularities forming a closed subset of lower dimension; outside of the singular locus these zero sets are complex manifolds.

From the German preface of R. Remmert: "When kings build their kingdom, there is work for the draymen. Kiyoshi Oka was a king. His kingdom was the function theory of several complex variables. He solved problems which were believed to be unsolvable; he developed methods whose audacity brought the admiration of the mathematical world. Oka gave new life to complex analysis." This book comprises Oka's ten Memoires with comments by Henri Cartan.