CiNii Books - Collected papers

We present here the mathematical papers of Hassler Whitney. This collection contains all the published papers, with the exception of some short announcements that Whitney did not wish to be included. We also include the introduction to his book Geometric Integration Theory, and one previously unpublished manuscript on the four-color problem. The papers are presented under some broad categories: graphs* and combinatorics, differentiable functions and singularities, analytic spaces, manifolds, bundles and characteristic classes, topology and algebraic topology, geometric integration theory. Whitney intended to write an introduction to this collection. Unfortunately he left us no manuscript at the time of his death, May 10, 1989. We had discussed the possibility of using his paper "Moscow 1935 - Topology moving toward America," written for the Centennial of the American Mathematical Society, as part of his introduction to this collection, an idea which he much liked. We therefore include this paper, which contains personal information as well as mathematical reflections, as Whitney's own introduction to these volumes. Whitney's mathematical style, like his personal style, was that of an explorer and pioneer. One of the pictures included in these volumes shows him as a mountain climber. In mathematics, he preferred to work on undeveloped areas: break new ground and build foundations. During the last twenty years of his life he concentrated his efforts on developing an educational system that builds on the natural tendency in children to be explorers.

Table of Contents

- Volume 1.- [82] Moscow 1935: Topology Moving Toward America.- 1 Graphs and Combinatorics.- [3] A theorem on graphs, Annals of Math. (2) v. 32, 1931,378-390.- [5] Non-separable and planar graphs, AMS Transac. v. 34, 1932, 339-362.- [6] Congruent graphs and the connectivity of graphs, Am. Jour. Math. v. 54, 1932, 150-168.- [10] The coloring of graphs, Annals of Math., (2) v. 33, 1932, 688-718.- [12] A set of topological invariants for graphs, Am. Jour. Math., v. 55, 1933, 231-235.- [13] On the classification of graphs, Am. Jour. Math., v. 55, 1933, 236-244.- [14] 2-Isomorphic graphs, Am. Jour. Math., v. 55, 1933, 245-254.- [17] Planar graphs, Fundamenta Math., V. 21, 1933, 73-84.- [23] On the abstract properties of linear dependence, Am. Jour. Math., v. 57, 1935, 509-533.- [37] A numerical equivalent of the four color problem, Monatshefte fur Math, un Phys. 3, 1937-207-213.- [77] On reducibility in the four color problem, unpublished manuscript, 1971.- [78] (With W. T. Tutte) Kempe chains and the four colour problem, Utilitas Mathematica 2(1972), 241-281.- 2 Differentiable Functions and Singularities.- [18] Analytic extensions of differentiable functions defined in closed sets, AMS Transac., v. 36, 1934, 63-89.- [19] Derivatives, difference quotients and Taylor's formula, AMS Bull., v. 40, 1934, 89-94.- [20] Differentiable functions defined in closed sets I, AMS Transac., v. 36, 1934, 369-387.- [21] Derivatives, difference quotients and Taylor's formula II, Annals of Math. (2) v. 35, 1934, 476-481.- [22] Functions differentiable on the boundaries of regions, Annals of Math. (2) v. 35, 1934, 482-485.- [26] A function not constant on a connected set of critical points, Duke Math. J., v. 1, 1935, 514-517.- [27] Differentiable functions defined in arbitrary subsets of Euclidean space, AMS Transac., v. 40, 1936, 309-317.- [45] Differentiability of the remainder term in Taylor's formula, Duke Math. J., 10, 1943, 153-158.- [46] Differentiable even functions, Duke Math. J., 10, 1943, 159-160.- [47] The general type of singularity of a set of 2n ? 1 smooth functions of n variables, Duke Math. J., 10, 1943, 161-172.- [49] On the extension of differentiable functions, AMS Bull., 50, 1944, 76-81.- [55] On ideals of differentiable functions, Am. Jour. Math. 70, 1948, 635-658.- [61] On totally differentiable and smooth functions, Pacific J. Math. 1, 1951, 143-159.- [63] On singularities of mappings of Euclidean spaces, I. Mappings of the plane into the plane, Annals of Math. (2)62, 1955, 374-410.- [64] On functions with bounded n-th differences, J. de Maths. Pures et Appl. 36, 1957, 67-95.- [67] Singularities of mappings of Euclidean spaces, Symposium Internacional de Topologia Algebraica, Mexico, 1956, 285-301, Mexico, La Universidad Nacional Autonoma, 1958.- [70] On bounded functions with bounded n-th differences, AMS Proc. 10, 1959, 480-481.- 3 Analytic Spaces.- [66] Elementary structure of real algebraic varieties, Annals of Math. (2) 66, 1957, 545-556.- [68] (With F. Bruhat) Quelques proprietes fondamentales des ensembles analytiques-reels, Comm. Math. Helv. 33, 1959, 132-160.- [73] Local properties of analytic varieties, in: differential and combinatorial topology (Symposium in Honor of Marston Morse), Princeton, NJ, Princeton University Press, 1965, 205-244.- [74] Tangents to an analytic variety, Annals of Math (2) 81, 1965, 496-549.- Permissions.

Volume: v. 2 ISBN 9780817635596

Description

We present here the mathematical papers of Hassler Whitney. This collection contains all the published papers, with the exception of some short announcements that Whimey did not wish to be included. We also include the introduction to his book Geometric Integration Theory, and one previously unpublished manuscript on the four-color problem. The papers are presented under some broad categories: graphs and combinatorics, differentiable functions and singularities, analytic spaces, manifolds, bundles and characteristic classes, topology and algebraic topology, geometric integration theory. Whimey intended to write an introduction to this collection. Unfortunately he left us no manuscript at the time of his death, May 10, 1989. We had discussed the possibility of using his paper "Moscow 1935 - Topology moving toward America," written for the Centennial of the American Mathematical Society, as part of his introduction to this collection, an idea which he much liked. We therefore include this paper. which contains personal information as well as mathematical reflections. as Whimey's own introduction to these volumes. Whimey's mathematical style, like his personal style. was that of an explorer and pioneer. Ooe of the pictures included in these volumes shows him as a mountain climber. In mathematics. he preferred to work on undeveloped areas: break new ground and build foundations. During the last twenty years of his life he concentrated his efforts on developing an educational system that builds on the natural tendency in children to be explorers.

Table of Contents

- Volume 2.- 1 Manifolds.- [28] Differentiable Manifolds, Annals of Math.(2) v.37, 1936, 645-680.- [29] The imbedding of Manifolds in families of analytic manifolds, Annals of Math.(2) v. 37, 1926, 865-878.- [30] On regular closed curves in the plane, Compositio Math. 4, 1937, 276-284.- [35] Analytic coordinate systems and arcs in a manifold, Annals of Math.(2) 38, 1937, 809-818.- [50] The self-intersections of a smooth n-manifold in 2n-space, Annals of Math. (2), 45, 1944, 220-246.- [51] The singularities of a smooth n-manifold in (2n - 1)-space, Annals of Math.(2), 45, 1944, 247-293.- [72] The work of John W. Milnor, Proceedings ICM 1962, Institut Mittag-Leffler, Djursholm, Sweden, xlviii-1.- 2 Bundles and Characteristic Classes.- [36] Topological properties of differentiable manifolds, AMS Bull. 43, 1937, 785-805.- [42] On the theory of sphere-bundles, NAS Proc., 26, 1940, 148-153.- [44] On the topology of differentiable manifolds, Lectures in Topology, U. of Michigan Press, 1941, 101-141.- [69] (With A. Dold) Classification of oriented sphere bundles over a 4-complex, Annals of Math. (2) 69, 1959, 667-677.- 3 Topology and Algebraic Topology.- [11] A characterization of the closed 2-cell, AMS Transac., v. 35, 1933, 261-273.- [15] Regular families of curves, Annals, of Math.(2) v. 34, 1933, 244-270.- [31] On matrices of integers and combinatorial topology, Duke Math. J., 3, 1937, 35-45.- [32] On the maps of an n-sphere into another n-sphere, Duke Math. J., 3, 1937, 46-50.- [33] The maps of an n-complex into an n-sphere, Duke Math. J., 3, 1937, 51-55.- [38] Cross sections of curves in 3-space, Duke Math. J., 4, 1938, 222-226.- [39] On products in a complex, Annals of Math.(2) 39, 1938, 397-432.- [40] Tensor products of abelian groups, Duke Math. J., 4, 1938, 495-528.- [41] Some combinatorial properties of complexes, NAS Proc., 26, 143-148.- [43] On regular families of curves, AMS Bull., 47, 1941, 145-147.- [48] Topics in the theory of Abelian groups, I. Divisibility of Homomorphisms, AMS Bull., 50, 1944, 129-134.- [54] Complexes of manifolds, NAS Proc., v. 33, 1947, 10-11.- [56] Relations between the second and third homotopy groups of a simply-connected space, Annals of Math.(2) 50, 1949, 180-202.- [57] Classification of the mappings of a 3-complex into a simply-connected space, Annals of Math.(2) 50, 1949, 270-284.- [58] An extension theorem for mappings into simply-connected spaces, Annals of Math.(2) 50, 1949, 285-296.- 4 Geometric Integration Theory.- [52] Algebraic topology and integration theory, NAS Proc., v. 33, 1947, 1-6.- [53] Geometric methods in cohomology theory, NAS Proc., v. 33, 1947, 7-9.- [59] La topologie algebrique et la theorie de l'integration, Colloques Internationaux du CNRS XII, Topologie Algebrique, 1947, 107-113, published by CNRS, Paris, 1949.- [62] r-dimensional integration in n-space, Proc. Int. Cong. Math., 1950, vol. 1, 245-256, Amer. Math. Soc. 1952.- [65] Introduction to "Geometric Integration Theory", Princeton, NJ, Princeton University Press, 1957, pp. 3-31.- 5 Other Subjects.- [9] A logical expansion in mathematics, AMS Bull., v. 38, 1932, 572-579.- [16] Characteristic functions and the algebra of logic, Annals of Math.(2) v. 34, 1933, 405-414.- [60] (With L.H. Loomis) An inequality related to the isoperimetric inequality related to the isoperimetric inequality, AMS Bull., 55, 1949, 961-962.- [71] (With A.M. Gleason) The extension of linear functionals defined on H-infinity, Pacific J. Math. 12, 1962, 163-182.- [75] The mathematics of physical quantities. Part I, Mathematical models for measurement, Am. Math. Monthly 75(1968), 115-138, Part II, Quantity structures and dimensional analysis, ibid. 237-256.- [76] Logic fad or tool? nico 4, 1969, Revue per. du centre Beige de Pedagogie de la Mathematique, 2-14.- [81] Comment on the division of the plane by lines, Am. Math. Monthly 86(1979), p. 700.- Permissions.

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