Hassler Whitney) (1907+March 23, 0907-1March10,0989), an American mathematician, specializes in differential geometry and studied graph theory in his early years. The wolf prize in mathematics winner is in 1982. He has published nearly 80 papers and 3 monographs, namely Geometric Integral Theory (1957), Variant of Complex Analysis (1972) and Mathematical Activities (1974).

Chinese name: hassler whitney.

Hassler Whitney

Date of birth:1March 23, 907

Date of death:1March 989 10.

Occupation: Mathematician

Representative works: geometric integral theory, etc.

The life of the character

school days

Whitney's grandfather is a linguist, his grandfather is a famous astronomer S. newcomb (newcomb, 1897- 1898 is the president of the American Mathematical Society), and his father is a judge. When he was young, he liked making mechanical toys and had no preference for mathematics. According to himself, the only thing related to his career as a mathematician is to think about the formula that numbers can be divisible by 9 at the age of 9. On this basis, the formula of the number divisible by 1 1 is derived. I only learned a little math during my primary and secondary school. 192 1- 1923 went to Switzerland to study. Besides German, I also studied French for a year and climbed mountains for a year. 5438+0924, he went to Yale University to study physics. I'll forget it when I'm done. After obtaining a bachelor's degree in physics at 1928, he continued to specialize in music and obtained a bachelor's degree in music at 1929. He loves music all his life, and has a high musical talent. He can play the piano, violin, viola, oboe and other musical instruments. He was the chief violinist of Princeton Symphony Orchestra. He also likes climbing mountains. The complete collection contains photos of him standing at the top of the steep Swiss Alps at the age of 65,438+04. After graduating from college, he went to Harvard University to take the doctoral exam of G.D. boekhoff because he was interested in the four-color problem. But he failed the exam the first time, which made boekhoff extremely angry. However, boekhoff took in this student who never felt inferior and gave him guidance in areas where he was not good at. Whitney's papers came out one after another. 10 when he got his doctorate in 932, he wrote nearly10 papers, all about graph theory. His doctoral thesis is entitled "Chromatic number graph", in which "chromatic number" is defined and calculated.

life

Because of his outstanding work, he was a researcher of the National Research Council of the United States from193 1933, a lecturer in the Department of Mathematics of Harvard University from1933, and a professor from 1946. At this time his direction changed from graph theory to topology. 1September, 935, participated in the international topology conference held in Moscow, Soviet Union. This meeting became a milestone in the history of topology. The title of his last paper was Moscow 1935: Topology Transferred to America. The article writes that at the meeting, H. hopf became his favorite topographer. At that time, all the adults were gone, and the face of topology was also changing: four people introduced homology in unison, and homotopy theory also formally appeared. The application of vector field problem leads to the appearance of the concept of fiber bundle, which is closely related to Whitney's work and also determines Whitney's work direction in the next 65,438+00 years.

During World War II, he participated in wartime research.

1943- 1945, the applied mathematics group of the national defense research Committee of the research and development bureau is engaged in research.

After the war, he gave a speech entitled "Topology of Smooth Manifolds" at the annual meeting of American Mathematical Society 1946.

1948- 1950 served as the vice chairman of the American Mathematical Society.

From 1944 to 1949, he is the editor of American mathematical magazine.

The editing of mathematical reviews ranges from 1949 to 1954.

Member of the Program Committee of the International Congress of Mathematicians held in Harvard from 65438 to 0950, who gave a report on "R-dimensional Integration in N-dimensional Space".

1952 was appointed professor at Princeton Institute for Advanced Studies, 1977 retired. During this period, he was the first chairman of the Mathematics Group of the National Science Foundation of the United States.

1966- 1967 National Research Council supports members of the Mathematical Science Committee.

Since 1967, his interest has completely turned to mathematics education, especially primary and secondary education. He personally went deep into the classroom to understand the students' thoughts and feelings and found many problems in mathematics teaching. He pointed out that children's intuitive way is very close to that of mathematicians. At that time, the school's teaching objectives were narrow and the language was poor. When students encounter problems, they only substitute formulas instead of learning to think. Teaching is to instill inexplicable concepts and cope with standardized exams. Students can only passively accept it. To this end, he made a teacher training plan and compiled teaching materials for teachers. In the United States, Britain, Belgium, Brazil and other countries as a mathematics teaching consultant. From 65438 to 65438, he served as the chairman of the Center of the International Mathematics Education Committee.

Personal glory

Because of his extraordinary contribution, he won many honors. 1945 was elected as an academician of the National Academy of Sciences, 1976 won the national medal of science award, 1982 won the Wolf Prize, and 1985 won the Steele Prize of the American Mathematical Society, which was a lifelong achievement.

Mathematical achievement

Whitney has published nearly 80 papers and 3 monographs, namely Geometry Integral Theory (1957), Variant of Complex Analysis (1972) and Mathematical Activities (1974).

graph theory

Whitney has been interested in four-color problems all her life. His earliest and last math papers were all about four colors. He gave the equivalent proposition of the four-color problem and studied the reducibility problem. Starting with the four-color problem, he studied the general graph theory, especially the conditions of homeomorphism of two graphs: for example, G and G' are two connected graphs, and neither of them contains ab, ac and ad arcs. If there is any one-to-one correspondence between two arcs with common vertices and two arcs with common vertices of another graph, then the two graphs are homeomorphic. He defined the connectivity of graphs. The necessary and sufficient conditions for n-reconnection are given (the so-called n-reconnection refers to a graph with at least n+ 1 vertices, which cannot be broken by removing n- 1 or fewer vertices and connecting them with arcs. If the graph Gn is reconnected, but not n+ 1, then its connectivity is called n). He also defined the dual G' of graph G and proved that graph G can be disconnected.

His doctoral thesis is about the coloring of graphs, in which the formula of M(λ) is proved and calculated, where M(λ) is the number of different coloring methods of a graph with λ colors. He introduced a set of numbers mij, which can be used not only to calculate M(λ), but also to define the topological invariants of graph G;

Where r is the rank of graph G and n is the zero degree of G, he used these invariants to study the classification of graphs.

Whitney's greatest achievement in combinatorial theory is that he introduced matroid theory, which is an abstract linear correlation theory, including not only graph theory as a special case, but also network theory, comprehensive geometry and crosscutting theory. His starting point is very simple, considering the columns C 1, C2, Cn of matrix M, and the subsets of these columns are either linearly independent or linearly related, so

Any subset of the (1) independent set is also independent;

(2) If Np and Np+ 1 are independent sets of P column and p+ 1 column, Np plus one of Np+ 1 constitutes an independent set of p+ 1.

He called the system satisfying these two conditions matroid, and extended the properties of many graphs to matroid.

Differentiable mapping and singularity theory

Analytical extension of (1) differentiable function Whitney's main contribution to topology is the establishment of differential topology. Therefore, the continuous mapping considered by topology must be extended to differentiable cases. Whitney laid the foundation for this in his early work (1932- 1942).

1925 The Soviet mathematician улысон(uryson) proved that if A is a closed set (bounded or unbounded) in N-dimensional Euclidean space E and f(x) is a continuous function defined in A, then F can be extended to the whole E. If f(x) belongs to A.

(2) Singularity theory Singularity theory is one of Whitney's most important creations, which comes from the problem of differential embedding and immersion. Singularity is the generalization of critical point. In 20942, for the first time,

In this paper, the singularity of differEntial mapping F from N-dimensional Euclidean space en to E2n- 1 is studied. It is found that if you change f slightly, you can get f*. Its singularity is an arc singularity, which can be transformed into a standard form:

yi=xi(i=2，，n)，

ym+i- 1=xixi(i=2，，n)。

In 1955, he first classified the types of singularities in plane E2 to E; There are only two kinds of results, one is fold and the other is Cusp, and its standard is.

Through this paper, the singularity theory was founded. In 1956, he classified some cases of singular points of differential mapping of En→Em, and obtained standard forms, including n ≥ m)=(4 2,3 and (n, m) = (4,4), (5,5), (5,4), (n, 2n. Little was known at that time. This basic problem of singularity classification, together with other problems, has become a hot topic in singularity theory. In the same year, R. Thorm made a breakthrough by using his own transverse theory and universal folding theory, which became the basis of his catastrophe theory. Later, J. Mather established the stability theory in 1968- 197 1 year. Since 1967, the Soviet school headed by Soviet mathematician B. ирнолъв has made brilliant achievements in theory and application.

1948 also published the Ideal of Differentiable Functions, which opened up another new direction of singularity theory. Later, B. Malgrange and others made great breakthroughs in this respect, including proving the "preparatory theorem".

(3) Hierarchy Theory Hierarchy theory is Whitney's last theory, and in a sense, it is also a natural continuation of singularity theory. Euclidean space and manifold usually studied have good homogeneous structure (local structure is the same), but this is not satisfied even for algebraic clusters. In particular, there are singularities in real algebraic clusters inherited from analytic geometry. From 1957 to 1965, Whitney studied the topology of real algebraic clusters and discussed the decomposition from clusters to manifolds. In 1957, Whitney introduced the concept of hierarchy, which decomposed algebraic clusters and analytic clusters into layers, and was later developed into hierarchical set theory by Tom. In 1965, S. Schaevitz proved that any semi-analytic set has Whitney stratification. In 1965, Whitney defined the concepts of tangent vector, tangent plane family and tangent cone for analytic clusters, and considered the harmony of tangent sets.

Differential popular topology

Although Poincare and even Riemann have studied the topology of differential manifolds, Whitney really founded the topology and differential topology of differential manifolds due to the lack of tools. In 1936, he laid the theoretical foundation of differential manifold. He gave the inherent definition of differential manifold. Define the Cr structure (1≤r≤∞) above. He proved that all Cr structures of Cr manifold contain C∞ coordinate system, and their C∞ structures are unique. This C∞ structure is called manifold microstructure or differential structure or smooth structure, and the corresponding manifold is called Hui manifold or differential manifold or smooth manifold. There are essential differences between differential manifolds and topological manifolds. That is, a topological manifold may not allow any differential structure or multiple differential structures, but any differential structure allows real analytic structure and Riemann metric, which is also proved by Whitney. In this paper, he proved some basic theorems, especially the embedding and immersion theorems: any N-dimensional differential manifold can be differentially embedded in R2n+ 1(2n+ 1 dimensional Euclidean space). In 1944, he improved that n-dimensional differential manifolds can be embedded in R2n and immersed in R2n- 1. For some manifolds, these results have reached perfection. This work has opened up an important field of differential manifolds, and many topologists such as Wu Wenjun have made contributions since then.

Fiber bundle and indicator class

Whitney first defined the real "fiber space" in 1935, then he called it "ball space", and in 1940 he changed it to "ball cluster". In 1937 and 194 1, he made two reports on this, including many fundamental results, and he intends to do the same. It was never finished. His interest has always focused on the "typical class". 1936, he and Swiss mathematician E. Stiefel independently defined this kind of feature class in 1935, and later called it Stiefel-Whitney feature class. His purpose is to study the topology of differential manifolds with characteristic classes. Fiber bundle is just a tool, so its definition is not clear in every detail, but it is very general. 1940- 1950, fiber bundles became the main tool to study many topological problems (especially homotopy, homotopy and differential geometry problems). The publication of N.E. Steenrod's monograph Topologyoffi-berbundles marked the maturity of fiber bundle theory, among which Whitney's contribution was particularly prominent.

(1) classification from the beginning, Whitney mainly studied the classification of fiber bundles. In 1937, he obtained the classification space of the ball bundle, that is, the Glassman manifold Gn, r, and asserted that the isomorphic class of the ball bundle with base space b and rank r is [B, Gn, R], that is, the homotopy class (nr) of the mapping from b to Gn, r.

Whitney also knows that the cluster space of ball clusters based on B only depends on the homotopy type of B, which was proved by J. Feldbau in 1939. On the other hand, as early as 1935, Whitney constructed a new fiber bundle g*(ξ) for fiber bundle ξ and continuous mapping G: b' → B.

(2) The characteristic class Stiefel only considers the characteristic class of tangent bundle of differential manifold, while Whitney considers it much wider. He thinks that the bottom space b of any ball cluster (e, b, p) can also be an arbitrarily locally finite simple complex. He defined the characteristic class as the integer coefficient homology class of Stiefel manifold Sn, m. He pointed out the homology group of Sn and m.

In 1937, he defined the sex class with homology. In 1940, he pointed out that for continuous mapping,

g:B'0→B，

If E'=g*(E) is the callback of e, then

Wr(E')=g*(Wr(E))。

At the same time, he gave Whitney's summation formula: define the income of two ball clusters e' and e "on the same bottom space.

Among them, the area of ∪ curve, he pointed out that when r≥4, it is "extremely difficult" to prove. In 194 1, he only gave the proof that E and E' are both line bundles. The first published proof was given by Wu Wenjun in 1948. He also replaced the ball bundle with a vector bundle. In the same year, Chen Shengshen also published another proof.

Whitney also gave the concept of demonstrative class or even formal power series of demonstrative class. At this point, the theoretical basis of Steefel-Whitney demonstration class was formally established. Later, based on the four theorems put forward by Whitney, Milnor opened the theory of deixis and other deixis, especially Pontryagin (понт).

(3) The application of demonstrative classes plays an extremely important role in topology and geometry. Whitney herself mainly uses indicative courses to learn immersion. For example, he proved that the 8-dimensional real projective space P8(R) can not be immersed in R 14, but can be immersed in R 15. His theory was later developed by Wu Wenjun and others.

algebraic topology

1935 is a turning point in algebraic topology, which is mainly marked by the establishment of homology theory and homology ethics. Forty years after Poincare introduced the concept of homology, four mathematicians introduced the concept of homology almost simultaneously and independently. They are J.W. Alexander, Whitney, E. Cech (Cech) and A.H. André Andrey Kolmogorov (колмогоров). The other three were at 660.

In the same ethics theory, in 1937, Whitney expressed hopf-Hurewicz criterion in the same tone. If x is an n-dimensional locally finite cell complex and y is an n-dimensional (n- 1) connected space, then f, G: X→ Y is homotopy if and only if.

HN(Y； z)→Hn(X； z)。

Infer from this

〔X，x0； y，y0】úHn(X； πn(Y))

There is a one-to-one correspondence. These conditions are not necessarily true for the mapping of different dimensions. Whitney gave the homotopy algebraic condition of two-dimensional complex mapping to two-dimensional or three-dimensional projective space in 1936, but it was not published. In 194 1, H.E. Robbins extended the homotopy classification of mappings from two-dimensional complex to arbitrary space. Later, Olum simplified and popularized it on a large scale. For the three-dimensional complex, Pontryagin considered the homotopy classification mapped to S2 in 194 1, in which the newly emerging upper product was applied for the first time. In fact, Whitney got the corresponding results as early as 1936. He studies simple connectivity in 1948. On this basis, the necessary and sufficient conditions for homotopy of two continuous mappings in three-dimensional complex K to R and the obstacle classes of mapping expansion are given. It should also be pointed out that Whitney introduced the concept of tensor product of Abel group in 1938, which is an essential tool for algebraic topology and homology algebra.

Geometric integral theory

During the period of 1946- 1957, Whitney established geometric integration theory, which is a more general integration theory, such as R-dimensional integration in N-dimensional space. From this, he gave an analytical explanation of winding and upper closed chain, such as geometric chain is a function on the singular chain of "general position" He used Lipschitz condition to replace the differentiable condition in E Cartan's and G deRham's external differential form theory, and the obtained integral theory is equivalent to algebraic homology theory, which is also true for more general Lipschitz spaces, including polyhedron and absolute neighborhood contraction kernel as its special cases, especially extending Stokes theorem to Lipschitz spaces. His theory is summarized in ""