# Morse, Marston

# MORSE, MARSTON

(*b*. Waterville, Maine, 24 March 1892; *d*. 22 June 1977, Princeton, New Jersey),

*mathematics, differential geometry, calculus of variations, functions of several variables*.

An American mathematician, Morse was an early and influential member of the Institute for Advanced Study at Princeton, New Jersey. He is particularly associated with the theory of functions of several variables on manifolds and on the calculus of variations.

**Early Life and Career** . Morse was the son of Howard Calvin Morse, a farmer and realtor, and Ella Phoebe Marston. He was a student at Coburn Classical Institute in Waterville, Maine. At age eighteen he entered Colby College, also in Waterville. Morse graduated from Colby College in 1914 and took his PhD from Harvard University in 1917. He then joined the American Expeditionary Force in World War I and was awarded the Croix de Guerre with Silver Star for bravery under fire. On his return to the United States he taught at Cornell University in Ithaca, New York, and then briefly at Brown University in Providence, Rhode Island, before joining the faculty at Harvard University at Cambridge, Massachusetts, in 1926. He became a professor at the newly established Institute for Advanced Study in Princeton in 1935 and retired from there in 1962 as professor emeritus, but he continued to conduct research for the rest of his life. His war work in World War II on terminal ballistics earned him a Meritorious Service Award, which he received from President Franklin Roosevelt in 1944.

After the war Morse was active in the creation of the National Science Foundation and served on its first board from 1950 to 1954. A deeply religious man, he converted to Catholicism and was the Vatican’s representative at the Atoms for Peace Conference of the United Nations in 1952. He received honorary degrees from more than twenty universities, and among other distinctions he was awarded the associate membership of the French Academy of Sciences and the corresponding membership of the Italian National Academy of the Lincei. His strong attraction to these countries, deepened by his religious convictions, made these awards particularly pleasing to him. He was also a fine pianist with a particular, but by no means exclusive, affinity for the works of Johann Sebastian Bach.

**The Morse Inequalities** . Morse’s long and productive mathematical career drew particular impetus from the work on Henri Poincaré. Therefore, he contributed to the growing subject of topology always with an eye to its origins and uses in dynamics, analysis, and differential geometry and in later years was somewhat opposed to the elaborate algebraic structures created by the new generations of topologists. As he put it in a lecture in 1949: “Always the foundation and never the Cathedral.” In an early work, “Recurrent Geodesics on a Surface of Constant Negative Curvature,” he took up a topic raised by Poincaré and Jacques Hadamard and studied surfaces of constant negative curvature bounded by closed geodesics. The question he answered was: what geodesics lie entirely in the surface (and do not end on one of the boundaries)? His answer, in more modern terminology, was given as a word in the generators of the fundamental group of the surface, and Morse’s later work with Gustav Hedlund in 1944, “Unending Chess, Symbolic Dynamics and a Problem in Semigroups,” marks the earliest occurrence of what has come to be called symbolic dynamics. Using this technique, Morse was able to characterize the geodesics that lie entirely in the surface as a limit of recurrent but nonperiodic geodesics and to show that there are uncountably many of them.

In 1925 Morse published his first paper, “Relations between the Critical Points of a Real Function of *n* Independent Variables,” on the distribution of the critical points of a function on a manifold. (These are the points where all its first derivatives vanish.) The simplest example is the height function on a torus. One can think of the torus as an inner tube of a bicycle wheel in the vertical position. The height function has critical points at the top and bottom of the torus and at the highest and lowest points where the torus (the inner tube) meets the wheel. At these points the derivatives vanish, but the matrix, *H*, of their second derivatives is nondegenerate. Morse called such functions nondegenerate, and his first result was that a nondegenerate function on a manifold has a finite number of critical points. Furthermore, the critical points are of various kinds. In the case of the torus, the first distinction is between the points at the top and bottom, where the surface is locally bowl shaped, and the other two points, where the surface is locally saddle shaped. A more refined distinction, also valid on a manifold of any dimension, depends on the number of negative eigenvalues of the matrix *H* (which varies from critical point to critical point). Morse called the number of negative eigenvalues of the matrix *H* at a critical point the index of the point and defined the number m_{k} to be the total number of critical points with *k* negative eigenvalues, and he showed that these numbers were related to the dimensions of the homology groups of the manifold by a series of inequalities (today called the Morse inequalities). In the case of the torus, the Morse numbers are *m _{0} = 1, m_{1} = 2, m_{2} = 1* (working from the lowest to the highest critical point). In this case the Morse numbers coincide exactly with the dimensions of the homology groups.

The Morse inequalities are extremely useful in determining the homology of a manifold and therefore, in certain cases, the manifold itself. For example, only a sphere can have just two critical points, a fact crucial to John W. Milnor’s remarkable discovery that the seven-dimensional sphere can have nonstandard differentiable structures.

Morse’s inspiration for this work was George D. Birk-hoff’s paper “Dynamical Systems with Two Degrees of Freedom” of 1917. Morse’s proof of his theorems was based on the idea of deforming the manifold by letting it flow along the lines of steepest descent for the given function. As is easily seen in the torus case, this concentrates the surface more and more at its critical points. A separate argument then establishes what can happen in homological terms in a neighborhood of each critical point. This is done nowadays with the machinery of algebraic topology; Morse gave more intuitive arguments of considerable profundity using the language of topology as established by Oswald Veblen at the time.

**The Calculus of Variations** . Remarkable as these results are, Morse intended them as results on the way to a global theory of the calculus of variations. His paper, “The Foundations of a Theory of the Calculus of Variations in the Large in *m*-Space,” published in the *Transactions of the American Mathematical Society* in 1929, and his book, *The Calculus of Variations in the Large*, published in 1934, go a considerable way in this direction, but the much greater difficulties of the subject (by comparison with Morse theory) give his results a less conclusive feel that is appropriate to the opening up of a new subject. He found important results regarding the index of an extremal curve, but this concept of an index, which generalizes the concept of an index of a critical point of a function, now leads to an infinite sequence of numbers. This is indicative of a number of difficulties in the general theory that arise because it is intrinsically infinite dimensional.

The continuing importance of Morse’s work was demonstrated by the fact he that gave two invited addresses at International Congresses of Mathematicians. One was at Zürich, Switzerland, in 1932, titled “Calculus of Variations in the Large.” The other was at Cambridge, Massachusetts, in 1950, titled “Recent Advances in Variational Theory in the Large.”

## BIBLIOGRAPHY

*Both Morse's* Selected Papers, *edited by Raoul Bott (New York: Springer-Verlag, 1981), and his* Collected Papers *(Singapore: World Scientific, 1987), carry a complete bibliography of his work*.

### WORKS BY MORSE

“Recurrent Geodesics on a Surface of Constant Negative Curvature.” *Transactions of the American Mathematical Society* 22, no.1 (1921): 18–100. Also in *Selected Papers*, pp. 21–37.

“Relations between the Critical Points of a Real Function of *n* Independent Variables.” *Transactions of the American Mathematical Society* 27, no. 3 (1925): 345–396.

“The Foundations of a Theory of the Calculus of Variations in the Large in *m*-Space.” *Transactions of the American Mathematical Society* 31 (1929): 379–404.

*The Calculus of Variations in the Large*. New York: American Mathematical Society, 1934.

With Gustav Hedlund. “Unending Chess, Symbolic Dynamics and a Problem in Semigroups.” *Duke Mathematical Journal* 11, no. 1 (1944): 1–7. Also in *Selected Papers*, pp. 583–589.

With Stewart S. Cairns. *Critical Point Theory in Global Analysis and Differential Topology*. New York: Academic Press, 1969.

*Selected Papers*. Edited by Raoul Bott. New York: Springer-Verlag, 1981.

*Collected Papers*. Singapore: World Scientific, 1987.

### OTHER SOURCES

Birkhoff, George D. “Dynamical Systems with Two Degrees of Freedom.” *Transactions of the American Mathematical Society* 18 (1917): 199–300.

Bott, Raoul. “Marston Morse and His Mathematical Works.” *Bulletin of the American Mathematical Society*, n.s., 3 (1980): 907–950.

Pitcher, Everett. “Marston Morse.” *Biographical Memoirs of the National Academy of Sciences* 65 (1994): 222–260. Also available from http://books.nap.edu/openbook.php?record_id=4548&page=222.

*Jeremy Gray*

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