Marston Morse

Harold Calvin Marston Morse, or shortly Marston Morse, (born 24 March 1892 in Waterville, Maine, † June 22, 1977 in Princeton, New Jersey) was an American mathematician who worked in the field of variational calculus and differential geometry.

Life

He graduated from Colby College in Waterville and then went to Harvard University, where he earned his degree in 1915 and his doctorate in 1917 at George David Birkhoff with a work on geodesics on surfaces of negative curvature. In World War I he served with distinction in an ambulance unit in France. Then he taught from 1919 at Harvard, Cornell University ( 1920-1925 ), Brown University, 1925/26, and then again at Harvard, before he went to the Institute for Advanced Study at Princeton in 1935. In 1962 he retired.

His field was predominantly the calculus of variations " in the large" (global analysis). Here he created his own theory of the maxima and minima of functions, Morse theory, which has applications in many areas of mathematical physics, the theory of differential equations to differential topology has ( book by John Milnor, " Morse theory", there is the application given on the proof of Raoul Bott's Periodiziätstheorem in the theory of homotopy groups of spheres ). Bott himself was interested as a student of Birkhoff also to applications in dynamic systems (yes, he also tried the topology in quantum mechanics apply ).

Preliminary work for Morse theory were already in the 19th century by James Clerk Maxwell ("On hills and dales ", 1870) and Arthur Cayley (On contour and slope line 1859) made ​​- the " mountaineer's equation ": Number of peaks plus the number of valleys minus number of passes equal to two (the proof can be done by elementary counting on rising or falling sea level). One variety is a height function f assigned ( Morse function) and it is considered the critical points where the gradient ( derivative) of f vanishes. This can be maxima, minima or saddle points. Each critical point is assigned a Morse - index corresponding to the number of independent directions in which the function f is decreased (that is, in surfaces at the peaks 2, minima 0, saddle points 1). The whole can be formalized: critical points in the 1st derivative vanishes. If the matrix of 2.Ableitungen ( Hessian matrix ) is non-singular ( determinant equal to zero), so the critical points are non- degenerate and geometrically isolated points. The number of negative eigenvalues ​​of the Hessian matrix are the index. A Morse function is a function with only nondegenerate critical points ( "almost all" functions on manifolds are such Morse functions). After the Morse Lemma can the function near the critical point represent a quadratic form, where r in n dimensions at index time it is a positive sign of the squares r times negative, (nr ).

In classical Morse theory are then drawn from the behavior of the Morse function and its critical points to draw conclusions about the topology of the manifold. The homotopy type only changes in appearance of critical points, in such a way that a cell (in the sense of algebraic topology ) "added" to the dimension of the index of the critical point is.

Morse inequalities exist between the alternating sum of the numbers of critical points of index n and the alternating sum of the rank of homology groups. As a special case, the Euler -Poincare characteristic is measured as the alternating sum of the numbers of critical points.

Already Morse was noticed that the degenerate points for transitions in dynamical systems of meaning, which was expanded in Rene Thom's " catastrophe theory ".

In 1933 he received the Bôcher Memorial Prize for his work in analysis. In 1950 he gave a plenary lecture at the International Congress of Mathematicians in Cambridge (Massachusetts ) ( Recent Advances in Variational Theory in the Large) and also in Zurich 1932 ( The calculus of variations in the large ).

Since 1922 he was married to Celeste Phelps, with whom he had a son and a daughter. The marriage ended in 1930 and there was a minor scandal in 1932 at Harvard, as his ex- wife the math professor William Fogg Osgood married colleague, who was then 28 years older than Morse and had to withdraw it.

He always wanted to be addressed by their first name Marston, his mother's maiden name.