Daniel Marinus Kan

Daniel Marinus Kan (2005)

Daniel Marinus Kan (born August 4, 1927 - † August 4, 2013 in Newton , Massachusetts , USA ) was a mathematician who worked in the field of homotopy theory. Over the course of the last five decades of his life, he has written or co-authored dozens of essays and monographs in this field .

Early life

Kan grew up in Amsterdam in a liberal Jewish family . In 1941 he had to move from Barlaeus Gymnasium to the Jewish Lyceum. In the summer of 1943 the family was deported, first to Westerbork and then to Bergen-Belsen . Shortly after the liberation, both parents died of typhus .

After graduating from Barlaeus Gymnasium, he studied mathematics at the University of Amsterdam on the advice of L. E. J. Brouwer . There Johannes de Groot aroused an interest in topology in him . In 1951 he emigrated to Israel and initially worked as an assistant scientist at the Weizmann Institute as part of a project to search for oil deposits. He married Nora Poliakof, who, like him, came from Amsterdam and had survived Bergen-Belsen. The couple had four children.

Career

Kan received his doctorate in 1955 under Samuel Eilenberg at the Hebrew University of Jerusalem . He has taught at MIT since the early 1960s .

Its importance for the beginnings of modern homotopy theory is perhaps comparable to that of Saunders Mac Lane for homological algebra , insofar as he consistently used methods of category theory . His most famous work is the abstract formulation of adjointness of functors from 1958.

Kan also made contributions to the theory of simplicial sets and simplicial methods in topology in general . He gave a combinatorial definition of homotopy groups for Kan complexes (semisimplicial complexes that have the Kan expansion property), which provides the usual - topologically defined - homotopy groups of the space for the singular chain complex of a topological space .

In 1976, together with Bill Thurston , he proved Kan and Thurston's theorem that for every path-connected topological space there is a discrete group such that the Eilenberg-MacLane space is a good approximation , i.e. H. there is a continuous mapping which induces an isomorphism in singular homology . ${\displaystyle X}$ ${\displaystyle \pi }$ ${\displaystyle K(\pi ,1)}$ ${\displaystyle X}$ ${\displaystyle K(\pi ,1)\rightarrow X}$

His PhD students include William G. Dwyer and Aldridge Bousfield (a spectral sequence is named after Bousfield and Kan).

Fonts

• On css complexes. In: Amer. J. Math. , 79, 1957, pp. 449-476.
• A combinatorial definition of homotopy groups. In: Ann. of Math. (2) 67, 1958, pp. 282-312.
• Adjoint functors. In: Trans. Amer. Math. Soc. , 87, 1958, pp. 294-329.
• with Bousfield: The homotopy spectral sequence of a space with coefficients in a ring. In: Topology , 11, 1972, pp. 79-106.
• with Bousfield: Homotopy limits, completions and localizations. In: Lecture Notes in Mathematics , Vol. 304. Springer-Verlag, Berlin / New York, 1972. v + 348 pp.
• with Thurston : Every connected space has the homology of a . ${\displaystyle K(\pi ,1)}$In: Topology , 15, 1976, no. 3, pp. 253-258.
• with Dwyer: Function complexes in homotopical algebra. In: Topology 19, 1980, no. 4, pp. 427-440.

See also

• Kan extension
• Kan fiber
• Kan complex

Web links

• Daniel Marinus Kan in the Mathematics Genealogy Project (English)
• Clark Barwick et al. a .: Daniel M. Kan . Notices AMS, 2015, No. 9

Individual evidence

1. ↑ Clark Barwick, Michael Hopkins , Haynes Miller , Ieke Moerdijk : Daniel M. Kan (1927-2013) . In: Notices of the American Mathematical Society . tape 62 , no. 9 , October 2015, p. 1042-1045 ( ams.org [PDF; accessed May 18, 2016]).
2. ^ CRF Maunder : A short proof of a theorem of Kan and Thurston . In: Bulletin of the London Mathematical Society . tape 13 , no. 4 , 1981, p. 325–327 , doi : 10.1112 / blms / 13.4.325 .