EPADEL:A Semisesquicentennial History, 1926-2000

We end the chapter with a brief analysis of the mathematical classification of the lectures sponsored by the Philadelphia Section at annual meetings from 1956 to 1978. Although the leading choice of categories by the speakers during this period continued to be analysis, topology emerged as the leading contender. The present period also saw seven talks on applications and five on algebra. Because strict classification is impossible, the interested reader is urged to consult the Monthly reports listed in Table 6.1.1.

No longer was the Penn School of Analysis responsible for the continued dominance of talks on analysis. In fact, Penn faculty gave only three of the 14 talks in this category. I. J. Schoenberg gave two – “Mass distributions on the circle and convex conformal maps” (1957) and “On spline interpolation” (1963). N. J. Fine gave the other, “Integrability of continuous functions” (1963). The most popular topic in analysis was differential equations, which was chosen by Solomon Lefschetz (in 1960), William Feller (1964), J. P. Diaz (1968), and Louis Nirenberg (1971).

Local speakers whose lectures mainly dealt with topics in analysis were Albert Wilansky, “On the Cauchy criterion for the convergence of an infinite series” (1956), G. A. Stengle, “Some asymptotic problems in analysis” (1962), Frederick Cunningham, “Arzela’s theorem” (1963) and “In search of a modern understanding of differentials” (1974), and James England, “Bernoulli processes after the isomorphism theorem” (1974).

Curiously the 14 talks on analysis dealt neither with complex variables nor with numerical analysis.

What caused the sudden upsurge of interest in topology? Two of the section’s leaders were perhaps responsible – William Pervin and Albert Wilansky. Wilansky spoke twice (in 1963 and 1970), Pervin once (1967). Besides, three MAA officers, all prominent members of the R. L. Moore School of Topology at Texas, accepted invitations address the section: R. L. Bing, Raymond Wilder, and Gail Young. None dealt with the logical relations among axioms popularized by their famous mentor. Instead Bing’s title was “Homogeneity,” Wilder’s “Intuition” (probably in algebraic geometry), and Young’s “Topological methods in analysis”. The section also heard Edwin Moise speak twice on knot theory. William Thurston delivered a notable lecture on topology titled simply “Symmetry”.

Herman Gluck gave yet another notable lecture on topology in 1974, when he stated that Cauchy’s rigidity problem was still open. Just four years later Gluck’s Ph.D. student, Robert Connelly, then at Cornell, announced his solution of the problem. The paper containing the counterexample Connelly constructed was subsequently published in the Journal of I.H.E.S.

Herman R. Gluck had obtained his Ph.D. in 1961 at Princeton under the legendary Ralph Fox for a dissertation on embedding 2-spheres in 4-spheres. Recall that Fox had spoken to the section at annual meetings in 1945 and 1953. As of this year Gluck has supervised seven doctoral dissertations at Penn, where he has taught since 1966.

Five lectures on geometry combined with the 12 on topology make the geometry/topology classification the most popular. We already mentioned geometry talks by Samuel Gulden and Doris Schattschneider. Famous mathematicians delivered the remaining three lectures. A. S. Besicovitch, known for his negative answer to the Kakeya conjecture, spoke on a related topic in his 1959 talk, “Some extremal problems in geometry”. Initially he had selected a different subject, as suggested by the title he first submitted, “Sharpening Lebesque theorems on differentiation”, but he must have changed his mind in the days leading up to the meeting. It does not seem to be well known that Besicovitch directed a Ph.D. dissertation at Penn in 1962. In 1970 Washington’s Victor Klee discussed “Some unsolved problems from intuitive geometry,” and one year later Rafael Artzy viewed “Analytic geometry stripped of all but incidence”. Artzy was then at Temple University, but in 1976 he became the department head at the University of Haifa in Israel.

Eight addresses in the category of algebra/number theory were delivered, and all six in algebra dealt with mainstream topics. We already mentioned Samuel Plotkin’s 1976 talk on applications of group theory to music and Stephen Shatz’s 1977 talk on algebraic curves. The four other talks on algebra were M. O. Rabin (1956), “Impossibility of computational algorithms for group-theoretic problems”; J. C. Moore (1966), “Some aspects of homological algebra – Background and recent developments”; James Brooks (1967), “Equivalence of matrices and modules over Dedekind domains”; and Charles Curtis (1968), “Characters of finite groups”. The two talks on number theory were Peter Scherk (1956), “Integers,” and Hans Rademacher (1961), “Gaussian polynomials and pentagonal numbers”. Rademacher’s talk was the last of four that he gave to the section.

Chapter 5 noted that many of the lectures on applications from 1942-1955 were delivered during World War II. In the period 1956-1978, however, there was a much wider range of applications, and not one was war-related. The first, titled “Mathematical models in the biological sciences,” was given in 1971 by Willard Baxter, who played a key role in developing the excellent program in applied mathematics at the University of Delaware. Two years later A. J. Goldman, Chief of Operations Research at the National Bureau of Standards (now NIST, the National Institute of Standards and Technology), spoke on “Some mathematical operations research in government”. The 1976 meeting featured two talks on applications, Nelson Max, “Catastrophe theory and its applications,” and Jane Cronin, whose lecture carried the intriguing title, “Mathematical aspects of periodic catatonic schizophrenia”. The following year Steve Rohde, of General Motors Research Laboratories, drove home several points in his lecture, “Some mathematical aspects in the design of automotive components”. In 1977 Rohde received the Newkirk Award from American Society of Mechanical Engineers for his contributions to tribology (the study of friction, wear, and lubrication). We already mentioned Dorothy Bernstein’s 1978 lecture, “The role of applications in pure mathematics”. That same year Chris Rorres presented a lecture titled “The application of linear programming to the optimal harvesting of a renewable resource”.

A new interest in the section surfaced during this period – combinatorics/ graph theory. Not surprisingly, Herbert Wilf fanned this interest with his 1967 lecture “Counting finite graphs”. Wilf delivered another talk from this field eight years later, “How to choose k out of n”. In 1972 Roger Entringer traveled from New Mexico to discuss “Open problems in combinatorial analysis and graph theory”. Earlier we mentioned the two remaining talks in graph theory – John Koch (1976), “The proof of the four color theorem,” and Thomas Saaty (1978), “Priorities, hierarchies, and behavioral systems”.

As usual, only a few talks were dedicated to probability or statistics. Three were given in this period: Stuart Hunter (1957), “Experimental statistics – Some of the concepts and mathematical requirements,” R. D. Luce (1959), “Probabilistic models in psychology for the study of choice behavior,” and Bennett Eisenberg (1975), “Uniformly distributed sequences, stationary processes and the ergodic theorem”.

Five of the lectures do not fit into any category yet are deserving of mention because they celebrated mathematics for its own sake. Three were delivered at the 1963 meeting: C. K. Brown, “The search for delightful results,” Marguerite Lehr, “A little mathematics of the multiplication table variety,” and Pincus Schub, “Some mathematical crumbs”. Lehr’s lecture was the fourth she gave to the section, 31 years after her initial talk in 1932. She also spoke in 1944 and 1954. We already mentioned Mordell’s 1969 talk titled “Reminiscences”. We end with the delightful title of a lecture delivered in 1971 by longtime MAA Executive Director Alfred B. Willcox, “England was lost on the playing fields of Eton: A parable for mathematics”.

Overall, this discussion shows that most of the activities that took place at annual meetings from 1956 to 1978 centered on educational issues, as evidenced by both panel discussions and invited lectures. Among the invited lectures, the categories of analysis and topology/geometry garnered the greatest number of speakers, although an increasing number were concerned with algebra, applications, and combinatorics/graph theory. Talks in algebra considered mainstream topics for the first time, while the applications were varied and unrelated to war projects.