### Introduction.

There has been in the last decades of the twentieth century an increased interest in student-oriented teaching. To many, R. L. Moore served as an exemplar of such, and he is now regarded as a prophet of modern methods in pedagogy. Moore began his teaching career as a Teaching Fellow at The University of Texas in 1901-02. He taught in high school in 1902-03 and in universities from 1905 until 1969. (See the chronology.) His teaching has been the subject of a number of articles (Jones, Whyburn, Wilder, Mahavier, Eyles), interviews, and the MAA film *Challenge in the Classroom* (1967). (See the bibliography.) A number of mathematicians influenced by his methods have adopted their own versions in their classes. The Third Annual Legacy of R. L. Moore Conference was held in April of 2000, under the auspices of the Educational Advancement Foundation. Its theme was "Modified Moore Methods as Used Today."

In this paper we list the Ph.D. students of R.L. Moore and some other students influenced by him, and we give some information about them, in the belief that a teacher is best known by means of the activities and accomplishments of his students.

**John R. Kline**(1891-1955), Ph.D., University of Pennsylvania, 1916.A.B. (1912) Muhlenburg; A.M. (1914) Pennsylvania. Thesis title: "Double elliptic geometry in terms of point and order", 19 pages in journal article.

Kline was Professor of Mathematics at Penn from 1920 (the year Moore returned to Texas) until his death in 1955. He was Chair of the Department from 1933 to 1954. He was a Guggenheim Fellow (Göttingen) in 1926-27. He was Associate Editor of the Transactions AMS, Bulletin AMS, American Journal of Mathematics at various times and served on the editorial board of the AMS Colloquium Publications. He was Associate Secretary of the AMS from 1933 to 1936 and Secretary (AMS) from 1936 to 1950. He directed the Ph.D. theses of thirteen students, including National Research Fellows W. L. Ayres, H. M. Gehman, N.E. Rutt, and Leo Zippin. Several of his students went to Austin in their Fellowship years. He also directed the Ph.D. theses of W.W. Dudley and W.W.S. Claytor, the second and third African American mathematicians to earn their Ph.D.'s. (In this instance, the sins of the father were not visited on the son). On his recommendation, Claytor went to the University of Michigan on a post-doctoral fellowship, where he worked with R. L. Wilder. After this, he was offered the opportunity for further study at Princeton but declined it in order to accept a position at Howard University, where he had a distinguished career. Kline was the thesis director of record for Lida K. Barrett, but she began her work with R.L. Moore at Texas; the thesis problems were suggested by Kline after her move to the University of Pennsylvania and essentially solved under his direction. After Kline suffered a heart attack the work was completed under the direction of R.D. Anderson, who had joined the faculty at Pennsylvania in 1948. Perhaps Kline's best known student was Leo Zippin, who wrote with Deane Montgomery the classic monograph

*Topological Transformation Groups.*The work described in this treatise, together with work done by Andrew Gleason, provides a solution to Hilbert's Fifth Problem. Throughout his career, Kline maintained regular contact with RLM; for example, he was instrumental in getting Mary-Elizabeth Hamstrom to go to Austin for her graduate studies. Kline wrote only one joint paper, and it was with RLM, who also wrote no other joint work. Title: "On the most general closed and bounded plane point set through which it is possible to pass an arc."**George H. Hallett**(1895-1985), Ph.D., University of Pennsylvania, 1918.A.B. (1915) Haverford College; A.M. (1916) Harvard. Thesis title: "Linear order in three dimensional Euclidean and double elliptical spaces".

Hallett did not enter academia; instead, he had a long career in public service, beginning with a post as Secretary of the Proportional Representation League. In 1937 he wrote

*Proportional Representation: The Key to Democracy.*He taught courses in government at several colleges in New York City: Brooklyn, Hunter, NYU, and CCNY. He was given special awards from LaGuardia Memorial Association (1963), New York City Club (1964), and Citizens Union (1947, 1969). During the Second World War he was active in the Committee for Civilian Defense. In his book*Creative Teaching: The Heritage of R.L. Moore,*D. R. Traylor quotes Hallett as saying, "... I think such success as I've had in the field of government probably has a good deal to do with that [ Moore's teaching] - because they don't catch me up very often in theories of logic in bills, or different parts of bills, that don't hang together."**Anna M. Mullikin**(1893-1975), Ph.D., University of Pennsylvania, 1922.A.B. (1915) Goucher College; A.M. (1919) U. Pennsylvania. Thesis title: "Certain theorems relating to plane connected point sets".

When Moore went to Texas in 1920, Mullikin went there, as well, to continue her work with him. Her thesis, "Certain theorems relating to plane connected point sets," received considerable attention. It overlapped somewhat with some of Janiszewski's work (which had been published in Polish). Mullikin went on to a long career as a high school teacher in Pennsylvania; Mary-Elizabeth Hamstrom was one of her students. During the Second World War she was active in Civilian Defense. In 1952, Goucher College, where she received a B.A. in 1915, honored her with an alumnae achievement citation.

**Raymond L. Wilder**(1896-1982 ), Ph.D., University of Texas, 1923.B.Phil. (1920) Brown; M.Sc. (1921) Brown. Thesis title: "Concerning continuous curves", (62 pages). Signers: R.L. Moore, H.J. Ettlinger, A.A. Bennett, H.Y. Benedict, M.B. Porter.

Wilder was a member of the National Academy of Sciences. He served as President of the American Mathematical Society in 1955-1956 and as President of the Mathematical Association of America in 1965-1966. The bulk of his academic career was at the University of Michigan, where he directed twenty-five Ph.D. students. In his honor the University of Michigan established the Raymond L. Wilder Professorship of Mathematics. Before going to Michigan he taught at Ohio State for two years, and after retiring from Michigan he taught at UC Santa Barbara. He was the AMS Colloquium Lecturer in 1943 (

*The Topology of Manifolds*was published in the Colloquium series), and the Gibbs Lecturer in 1969. He wrote*Introduction to the Foundations of Mathematics,**The Evolution of Mathematical Concepts; an Elementary Study,*and*Mathematics as a Cultural System.*He was a recipient of the Distinguished Service Award of the MAA and the Lester R. Ford Award. He has observed that he went from Brown to Texas to study actuarial mathematics. He recalled that Moore did not at first want him as a student ( "I was a Yankee") and did not really accept him as a member of the class until he had proved the arcwise connectedness theorem. Wilder was one of the developers of algebraic topology. He maintained close contacts with Moore and also with Lefschetz at Princeton. Wilder was largely responsible for recognition and development of the abilities of Norman Steenrod (See the extensive correspondence between them at the Center for American History at the University of Texas). He also had influence on E.G. Begle and mathematics education reform.**Renke G. Lubben**(1898-1980), Ph.D., University of Texas, 1925.B.A. (1921) Texas. Thesis title: "The double-elliptic case of the Lie-Riemann-Helmholtz-Hilbert problem of the foundations of geometry", (102 pages). Signers: R.L. Moore, M.B. Porter, H.J. Ettlinger, G. Watts Cunningham, Edward L. Dodd, A. P. Brogan, Albert A. Bennett.

Lubben's Ph.D. thesis gave the solution to the then last remaining problem in the foundations of geometry. He was an independent discoverer of maximal compactifications of completely regular spaces, although priority in publication is assigned to Stone and Cech. He was a National Research Fellow (G&#ouml;ttingen) in 1926-27. He served on the Texas mathematics faculty until his retirement.

**Gordon T. Whyburn**(1904-1969), Ph.D., University of Texas, 1927.A.B. (1925) Texas; M.A. (Chemistry) (1926), Texas. Thesis title: "Concerning continua in the plane", (57 pages). Signers: R.L. Moore, M.B. Porter, H.J. Ettlinger, Edward L. Dodd, J.R. Bailey, H.L. Lochte, Arnold Romberg.

Whyburn was a member of the National Academy of Sciences. He was President of the American Mathematical Society in 1953-54. After leaving Texas, he taught at Johns Hopkins, but his principal academic appointment was at the University of Virginia, where he and E. J. McShane were brought in to develop a graduate program in mathematics. He served as Chair of the department from 1934 until 1966. He was the AMS Colloquium Lecturer in 1940; his Colloquium book

*Analytic Topology*resulted. He also published*Topological Analysis.*He received the Chauvenet Prize of the MAA. He directed thirty-two Ph.D. students, many of whom had distinguished academic careers. Whyburn studied chemistry as an undergraduate and in graduate school, obtaining an M.A. in Chemistry before deciding to concentrate on mathematics.**John H. Roberts**(1906-1997), Ph.D., University of Texas, 1929.A.B. (1927) Texas. Thesis title: "Concerning non-dense plane continua", (47 pages). Signers: R.L. Moore, M.B. Porter, H.S. Vandiver, H.J. Ettlinger, Edward L. Dodd, J.W. Calhoun, A.E. Cooper.

On leaving Texas, Roberts went to the University of Pennsylvania for a year, where he worked with J.R. Kline. He went to Duke University in 1931, and he remained there until his retirement. During the Second World War he served in the United States Navy. He directed twenty-four Ph.D. students, a number of whom are prominent mathematicians. He was Director of Graduate Studies in the Department from 1948 until 1960 and managing editor of the Duke Mathematical Journal from 1951 until 1960. He also served as Chair of the Department. His research centered at first on continua, then on dimension theory.

**Clark M. Cleveland**(1892-1969), Ph.D., University of Texas, 1930.B.S.in C.E. (1917) Mississippi. Thesis title: "On the existence of acyclic curves satisfying certain conditions with respect to a given continuous curve", Signers: R.L. Moore, M.B. Porter, Edward L. Dodd, H.J. Ettlinger, J.M. Kuehner, Arnold Romberg.

Cleveland joined the Department of Applied Mathematics and Astronomy at Texas and remained there throughout his academic career. He became Chair of the Department of Mathematics in 1953 when the Department of Applied Mathematics and Astronomy and the Department of Pure Mathematics were merged.

**Joe L. Dorroh**(1904-1989), Ph.D., University of Texas, 1930.B.A. (1926) Texas; M.A. (1927) Texas. Thesis title: "Some metric properties of descriptive planes", (38 pages). Signers: R.L. Moore, H.S. Vandiver, M.B. Porter, H.J. Ettlinger, J.M. Kuehner, S. Leroy Broun.

Dorroh was a National Research Fellow at Cal Tech in 1930-31 and at Princeton in 1931-32. He taught at LSU from 1942 to 1946 and at Illinois Tech in 1946-47. He then went to Texas A&I until his retirement in 1966. He was Chair there from 1952 until his retirement.

**Charles W. Vickery**(1906-1982), Ph.D., University of Texas, 1932.B.A. (1928) Texas. Thesis title: "Vickery: Spaces in which there exist uncountable convergent sequences of points", (37 pages). Signers: R.L. Moore, M.B. Porter, H.S. Vandiver, Edward L. Dodd, E.T. Mitchell, A.P. Brogan.

He worked as a statistician and economist for the State of Texas and the U.S. Government. Following the Second World War he taught at LSU but returned to work for the government and in the aircraft industry. He published in Econometrica, Bull. AMS, and the Amer. Math. Monthly. He was a Fellow of the Royal Statistical Society.

**Edmund C. Klipple**(1906-1992), Ph.D., University of Texas, 1932.B.A. (1926) Texas. Thesis title: "Spaces in which there exist contiguous points", (42 pages). Signers: R.L. Moore, H.S. Vandiver, M.B. Porter, Edward L. Dodd, H.J. Ettlinger, Homer V. Craig.

Klipple joined the Texas A&M mathematics faculty in 1935, and he stayed there until his retirement in 1971. He was Chair of the department for many years; he resigned as Chair in 1966 when asked by the Dean to rank his faculty in order, 1-40. In 1968 he was given a Faculty Distinguished Achievement Award. He nurtured a good many students who later became well known mathematicians, including Peter Lax, Efraim Armendariz, and William T. Guy.

**Robert E. Basye**(1908-2000), Ph.D., University of Texas, 1933.B.A. (1929) Missouri; M.A. (1931) Princeton. Thesis title: "Simply connected sets", (26 pages). Signers: R.L. Moore, M.B. Porter, H.S. Vandiver, P.M. Batchelder, H.J. Ettlinger, R.V. Haskell.

Basye's principal academic appointment was at Texas A&M University, from 1940 until his retirement in 1968. After retirement, he devoted his full time to rose research, becoming a renowned genetic hybridizer and grower. Among his many achievements was Basye's Purple Rose. He served on active duty in the U.S. Naval Reserve during the Second World War.

**F. Burton Jones**(1910-1999), Ph.D., University of Texas, 1935.B.A. (Chemistry) (1932) Texas. Thesis title: "Concerning R.L. Moore's Axiom 5-1", (80 pages). Signers: R.L. Moore, M.B. Porter, H.J. Ettlinger, Edward L. Dodd, E.P. Tchoch, W.A. Felsing.

As was the case with Whyburn before him, Jones was a chemistry student who changed to work with Moore. He stayed on the Texas faculty until 1950, serving as chair. He later served as Chair at the University of North Carolina (Chapel Hill) and at the University of California, Riverside. During the war years he worked for the U.S. Navy in Cambridge, on methods for locating and identifying submarines. He directed fifteen Ph.D. students, many of whom became well known mathematicians. At Texas he helped develop a number of students; in fact, Mary Ellen Rudin has described him as the mathematician who had the greatest influence on her development. After leaving Texas he continued to provide counsel and support and guidance to younger mathematicians trained in the Texas tradition. He worked in continua and in abstract spaces. He originated the famous normal Moore space problem. In 1975 he received a Fulbright-Hays Fellowship to visit Canterbury University in Christchurch, New Zealand. He was twice a fellow at the Institute for Advanced Study and spent two summers in Europe on the National Academy of Sciences Exchange Program.

**Robert L. Swain**(1913-1962), Ph.D., University of Texas, 1941.B.A. (1934) Reed College. Thesis title: "I. Proper and reductive transformations. II. Continua obtained from sequences of simple chains of point sets. III. Distance axioms in Moore spaces. IV. Linear metric space. V. A space in which there may exist uncountable convergent sequences of points", (101 pages). Signers: R.L. Moore, H.S. Vandiver, M.B. Porter, P.M. Batchelder, Edward L. Dodd, H.J. Ettlinger.

Swain's major academic appointments were at the University of Wisconsin (Madison), Teacher's College at New Paltz, and Rutgers University. In 1955-56 he held a Ford Foundation Faculty Fellowship.

**Robert H. Sorgenfrey**(1915-1996), Ph.D., University of Texas, 1941.B.A. (1937) UCLA. Thesis title: "Concerning triodic continua", (56 pages). Signers: R.L. Moore, F. Burton Jones, H.S. Vandiver, P.M. Batchelder, Edward L. Dodd, H.J. Ettlinger.

W.M. Whyburn, who was Department Head at UCLA at the time of Sorgenfrey's undergraduate work, arranged for him to study with Moore. Sorgenfrey was on the faculty at UCLA from 1942 until his retirement in January of 1979. In 1963 he received the UCLA Distinguished Teaching Award, the first mathematician to do so. He is known for his work in general topology, especially in the production of counter-examples. He directed four Ph.D. students. After retirement he wrote several successful high school mathematics textbooks.

**Harlan C. Miller**(1896-1981), Ph.D., University of Texas, 1941.B.A. (1916) Wellesley; M.A. (1930) Columbia. Thesis title: "On compact unicoherent continua", (72 pages). Signers: Miller: R.L. Moore, R.G. Lubben, H.S. Vandiver, P.M. Batchelder, Edward L. Dodd, H.J. Ettlinger.

Prior to her graduate program, Miller taught for a number of years at the Hockaday school in Dallas. After her Ph.D., she taught at Winthrop College and North Texas for one year each before joining the faculty at Texas Woman's University, where she spent the rest of her academic career. She was active in University administration there, serving as Director of Mathematics. She helped direct Lida K. Barrett toward studying with RLM. She advised J.R. Boyd in his graduate program. He later developed the Moore-style mathematics program at Guilford College. Texas Woman's University has an annual Harlan Miller Lecture Series.

**Gail S. Young**(1915-1999), Ph.D., University of Texas, 1942.B.A. (1939) Texas. Thesis title: "Concerning the outer boundaries of certain connected domains", (70 pages). Signers: R.L. Moore, F. Burton Jones, R.G. Lubben, Edward L. Dodd, H.S. Vandiver, N. Coburn, H.J. Ettlinger.

Young held appointments at Purdue, Michigan, Tulane, Rochester, Case-Western Reserve, Wyoming, and Columbia. He was Chair at at least two of these. He served as President of the Mathematical Association of America in 1970-71. He won the Distinguished Service Award of the MAA in 1987.He worked with the School Mathematics Study Group and with the Committee on the Undergraduate Program in Mathematics. He directed fourteen Ph.D. students, with one of whom (John Hocking) he wrote the successful textbook

*Topology*. Another was Beauregard Stubblefield, who has engagingly described in an interview conducted by Albert C. Lewis for the Center for American History how he became aware of Moore's principles of teaching through R.L. Wilder, Gail Young, and E.E. Moise, three of Moore's students at Michigan at the time of Stubblefield's graduate program.**R.H. Bing**(1914-1986), Ph.D., University of Texas, 1945.B.S. (1935) Southwest Texas; M.Ed. (1938) Texas. Thesis title: "Concerning simple plane webs", (34 pages). Signers: R.L. Moore, Edwin Ford Beckenbach, H.J. Ettlinger, H.S. Vandiver, P. M. Batchelder, C.T. Gray, J.G. Umstattd, Hob Gray.

Bing's principal academic appointments were at Wisconsin (1943-1973) and Texas (1973-1978). He was a member of the National Academy of Sciences. He served as President of the American Mathematical Society in 1977-78 and President of the Mathematical Association of America in 1963-64. His work was centered at first on continuum theory, then on 3-manifolds. He is also well known for the Bing metrization theorem. He was the Colloquium Lecturer in 1970, resulting in his Colloquium book

*Topology of 3-manifolds.*The American Mathematical Society published, in two volumes, the Collected Papers of R.H. Bing. He directed thirty-eight Ph.D. students, many of whom developed substantial reputations. His background was that of a high school teacher and football coach. When F.B. Jones was asked whether RLM at first did not recognize Bing's talent, Jones replied, with a twinkle in his eye, that "In later years Moore didn't remember it that way." Bing served as Chair of the Wisconsin and of the Texas Mathematics Departments. He was responsible for the MAA's film Challenge in the Classroom, which was about Moore's teaching methods.**Edwin E. Moise**(1919-1998), Ph.D., University of Texas, 1947.B.A. (1940) Tulane. Thesis title: "An indecomposable continuum which is homeomorphic to each of its nondegenerate subcontinua", (24 pages). Signers: R.L. Moore, H.S. Wall, F. Burton Jones, H.J. Ettlinger.

Moise's dissertation involved the pseudo-arc, a term he coined. It was used to solve an old problem of Knaster. He held academic appointments at Michigan, Harvard, and Queen's College, CUNY. It was at Michigan that he began his most important work on 3-manifolds, culminating in his proof, completed at the Institute for Advanced Study, that every 3-manifold can be triangulated. He went to Harvard as James B. Conant Professor of Mathematics and Education. From 1955 to 1958 he was a member of the Executive Committee of the International Comission on Mathematical Instruction. He served as Vice-President of the American Mathematical Society in 1973-74 and as President of the Mathematical Association of America in 1967-1968. He wrote a number of successful textbooks, and a treatise on

*Geometric Topology in Dimensions 2 and 3.*He directed three Ph.D. students. In his last years he devoted his attention to 19th century English poetry. During the Second World War he served in the U.S. Navy as a Japanese translator.**Richard D. Anderson**(1922,--), Ph.D., University of Texas, 1948.B.A. (1941) Minnesota. Thesis title: "Concerning upper semi-continuous collections of continua", (22 pages). Signers: R.L. Moore, F. Burton Jones, H.S. Wall, H.J. Ettlinger, H.S. Vandiver.

Anderson was recruited by Moore to do graduate work in the Fall of 1941. His graduate program was interrupted by a ture of duty in the U.S. Navy, where he served at sea. His principal academic appointments were at Pennsylvania and LSU. As a number of Moore's other students, including Bing, Jones, Moise, and Burgess, he held appointments at the Institute for Advanced Study in Princeton. His work at first centered around the geometric topology of continua. He subsequently was largely responsible, along with his students, for developing infinite-dimensional topology. He directed ten Ph.D. students at LSU and, as noted earlier, contributed to the direction of Lida Barrett at Pennsylvania. A number of his students have had distinguished careers. He served as Vice-President of the American Mathematical Society in 1972-73 and as President of the Mathematical Association of America in 1981-82. He received the Distinguished Service Award of the MAA. He has in more recent years devoted his major efforts to reform in mathematics education, more generally, in science education. He is currently Senior Consultant to the NSF sponsored Louisiana Systems Initiatives Program.

**Mary Ellen (Estill) Rudin**(1924,--), Ph.D., University of Texas, 1949.B.A. (1944) Texas. Thesis title: "Concerning abstract spaces", (27 pages). Signers: R.L. Moore, H.J. Ettlinger, H.S. Wall, F. Burton Jones, E.F. Mitchell, David L. Miller.

Rudin's principal academic appointments were at Duke, where she met Walter Rudin, Rochester, and Wisconsin, from which she retired as Grace Chisolm Young Professor of Mathematics. She served as Vice-President of the American Mathematical Society in 1980-81 and has been very active in AMS affairs and committee work. Her research has been in set-theoretic topology, especially using axiomatic set theory. She has, as C.E. Aull has noted, ushered in the Rudin Era in general topology. She has directed sixteen Ph.D. students and is largely responsible for directing the Ph.D. research of several others, including Judy Roitman and William Fleissner at UC Berkeley.

Note: In their article "By their fruits shall ye know them: some remarks on the interaction of general topology with other areas of mathematics", appearing in

*History of Topology,*edited by I.M. James, Elsevier, 1999, Teun Koetsier and Jan van Mill write:In that period general topology rather unexpectedly succeeded in solving several difficult problems outside its own area of research, in functional analysis and in geometric and algebraic topology. ...There were in that period at least two major developments in general topology that revolutionized the field: the creations of

*infinite-dimensional topology*and*set theoretic topology.*It was mainly due to the efforts of Dick Anderson and Mary Ellen Rudin that these fields have played such a dominant role in general topology ever since.It is interesting to note that Anderson and Rudin comprised a two person class under Moore in the immediate post-war years.

Note that of the preceding five students, four became President of the MAA and four became vice-president of the AMS. One might wonder whether this is duplicated by any other successive group of five students by any one thesis advisor.

**Cecil E. Burgess**(1920-2004), Ph.D., University of Texas, 1951.B.S. (1941) West Texas. Thesis title: "Concerning continua and their complementary domains in the plane", (38 pages). Signers: R.L. Moore, H.S. Wall, H.S. Vandiver, H.J. Ettlinger.

Burgess's graduate program was also interrupted by service in the U.S. Navy. After leaving Texas, he went to the University of Utah, where he remained throughout his career, except for leaves, which he usually spent working with R. H. Bing. Most of his work, and that of his students, has been centered on Bing-style topology. He directed ten Ph.D. students, and some of them are quite prominent. He served for a number of years as Chair of the Department.

**B.J. Ball**(1925-1996), Ph.D., University of Texas, 1952.B.A. (1948) Texas. Thesis title: "Concerning continuous and equicontinuous collections of arcs", (36 pages). Signers: R.L. Moore, H.S. Wall, R.G. Lubben, H.J. Ettlinger, Homer V. Craig.

Ball entered the Navy before getting his BA. On his return in 1946 he moved into Moore's graduate program. His major academic appointments were at Virginia and Georgia. He served as Chair at Georgia for a number of years. His work was in continuum theory, general topology, and, in later years, shape theory. He directed eight Ph.D. students and contributed to the direction of many others.

**Eldon Dyer**(1929-1993), Ph.D., University of Texas, 1952.B.A. (1947) Texas; B.S. (1947) Texas. Thesis title: "Certain conditions under which the sum of the elements of a continuous collection of continua is an arc", (14 pages). Signers: R.L. Moore, H.S. Wall, R.G. Lubben, H.J. Ettlinger, Homer V. Craig.

Dyer's academic appointments included Georgia, Johns Hopkins, Chicago, Rice, and CUNY, from which he retired as Distinguished Professor in 1991. He chaired the Department of Mathematics at CUNY, Center for Graduate Studies 1967-1970. He is best known for his work in algebraic topology and for his six Ph.D. students, among whom is Robion Kirby. As were Wilder and Bing, he was a consulting editor for the Encyclopedia Britannica. He held a Sloan Fellowship in 1960-62 and an NSF Post-Doctoral Fellowship in 1955-56. On two occasions he was a visiting member of the Institute for Advanced Study in Princeton. He served as Editor of the Proceedings AMS 1960-65 and as Associate Editor Transactions AMS.

**Mary-Elizabeth Hamstrom**(1927–2009), Ph.D., University of Texas, 1952.B.A. (1948) Pennsylvania. Thesis title: "Concerning webs in the plane", (35 pages). Signers: R.L. Moore, H.S. Wall, R.G. Lubben, H.J. Ettlinger, Homer V. Craig.

Hamstrom's principal academic appointment was at the University of Illinois. Her research was mostly in geometric topology. She directed nine Ph.D. students. Before going to Austin Hamstrom had been taught by two of Moore's earlier students: Anna Mullikin in high school and J.R. Kline at Penn. As were many of the students in Austin in the 1940s and early 1950s, she was strongly influenced by F.B. Jones.

**John M. Slye**(1923,--), Ph.D., University of Texas, 1953.B.S. (1945) California Institute of Technology. Thesis title: "Flat spaces for which the Jordan Curve Theorem holds true", (19 pages). Signers: R.L. Moore, H.S. Wall, H.J. Ettlinger, D.S. Hughes, R.N. Little, Jr.

Slye's academic appointments were at the Universities of Minnesota and Houston. His work was in geometric topology. He directed two Ph.D. students. During the Second World War he served in the U.S. Navy.

**John T. Mohat**(1924-1993), Ph.D., University of Texas, 1955.B.A. (1950) Texas Western. Thesis title: "Concerning spirals in the plane", (76 pages). Signers: R.L. Moore, H.S.Wall, H.J. Ettlinger, Homer V. Craig.

Mohat spent his academic career at the University of North Texas. He served in the U.S. Army during the Second World War.

**Bennie J. Pearson**(1929,--), Ph.D., University of Texas, 1955.B.A. (1950) Texas. Thesis title: "A connected point set in the plane that spirals down on each of its points", (18 pages). Signers: R.L. Moore, H.S.Wall, H.J. Ettlinger, Homer V. Craig.

Pearson's major academic appointment was at the University of Missouri- Kansas City. He served as Chair of the Department for six years. He directed three Ph.D. students.

**Steve Armentrout**(1930,--), Ph.D., University of Texas, 1956.B.A. (1951) Texas. Thesis title: "On spirals in the plane", (34 pages). Signers: R.L. Moore, H.J. Ettlinger, D.S. Hughes, C.W. Horton, H.S. Wall.

Armentrout taught at the University of Iowa for a number of years and then at Penn State. He has been active in AMS committee work and served as Treasurer of the AMS. He has worked in geometric topology and differential topology. He has directed twelve Ph.D. students, many of whom are very active.

**William S. Mahavier**(1930–2010), Ph.D., University of Texas, 1957.B.S. (1951) Texas. Thesis title: "A theorem on spirals in the plane", (26 pages). Signers: R.L. Moore, H.S. Wall, H.J. Ettlinger, C.W. Horton.

Mahavier was a physics major; in fact, his only degree in mathematics was the Ph.D. Mahavier's academic appointments included Illinois Institute of Technology, University of Tennessee, and Emory University. His work has centered on continuum theory. He has directed eight Ph.D. students.

**L. Bruce Treybig**(1931–2008), Ph.D., University of Texas, 1958.B.S. (1953) Texas. Thesis title: "Concerning locally peripherally separable spaces", (21 pages). Signers: R.L. Moore, H.J. Ettlinger, F.A. Matsen?, Homer V. Craig.

Treybig has taught at Tulane University and at Texas A&M University. His work has been primarily in general topology and continuum theory. He has directed seven Ph.D. students.

**James N. Younglove**(1927,--), Ph.D., University of Texas, 1958.B.A. (1951) Texas. Thesis title: "Concerning dense metric subspaces of certain non-metric spaces", (22 pages). Signers: R.L. Moore, R.G. Lubben, R.N. Little, H.S. Wall.

After graduating from high school, Younglove served two years in the U.S. Navy before entering Texas as a freshman. Younglove's academic appointments were at University of Missouri (Columbia), and at University of Houston, where he served as Chair for a number of years. His work was primarily in general topology, especially, metrication theory. He directed one Ph.D. student.

**George W. Henderson**(1936,--), Ph.D., University of Texas, 1959.B.A. (1958) Texas. Thesis title: "Proof that every compact continuum which is topologically equivalent to each of its nondegenerate subcontinua is an arc", (19 pages). Signers: R.L. Moore, H.P. Hanson, R.E. Lane, H.S. Wall.

Henderson's thesis was a proof that every decomposable hereditarily equivalent continuum is an arc. (The word "decomposable" was inadvertently omitted from title pages of final copies of his thesis.) He taught at the University of North Carolina, University of Virginia, Rutgers, and the University of Wisconsin, Milwaukee, where he directed one Ph.D. student.

**John M. Worrell**(1933,--), Ph.D., University of Texas, 1961.B.A. (1954) Texas; M.D. (1957) Texas. Thesis title: "Concerning scattered point sets", (41 pages). Signers: R.L. Moore, H.S.Wall, H.J. Ettlinger.

Before entering graduate school, Worrell obtained an M.D. degree. After he received his Ph.D. in mathematics, he worked at Sandia for a number of years, on his own research and on problems of interest to the space program. His mathematical work has centered on general topology, often in collaboration with Howard Wicke. He later taught at Ohio University and is now in private (medical) practice. While at Ohio University he created and developed, with the assistance of George M. Reed, the Institute for Medicine and Mathematics. He held an NSF post-doctoral fellowship in 1961-62.

**Howard Cook**(1933--2010), Ph.D., University of Texas, 1962.B.S. (1956) Clemson. Thesis title: "On the most general closed and bounded plane point set through which it is possible to pass a pseudo-arc", (34 pages). Signers: R.L. Moore, H.J. Ettlinger, R.E. Lane, H.S. Wall.

Cook's academic appointments have been at Auburn, North Carolina, Georgia, Tasmania, and Houston, mostly at Houston. In his thesis he characterized those compact sets in the plane which can be embedded in pseudo-arcs, an analogue of the Moore-Kline characterization of those which are subsets of arcs. His work has been in continuum theory and in general topology (Moore spaces). He has directed five Ph.D. students.

**James L. Cornette**(1935,--), Ph.D., University of Texas, 1962.B.S. (1956) West Texas; M.A. (1959) Texas. Thesis title: "Cornette: Continuumwise accessibility", (52 pages). Signers: R.L. Moore, H.J. Ettlinger, R.E. Lane, H.S. Wall.

Cornette's principal position has been at Iowa State University. His earlier work was in continuum theory; he has turned to biomathematics in more recent years. He has directed seven Ph.D. students. He is currently University Professor of Mathematics and Director of the Center for Bioinfomatics and Biological Statistics at Iowa State. In 1985 he began a collaborative program of research with three other scientists which has resulted in twenty-one journal articles, ten review and expository articles, and three patents.

**Dennis K. Reed**(1933-1986), Ph.D., University of Texas, 1965.B.S. (1959) Texas. Thesis title: "Concerning upper semi-continuous collections of finite point sets", (41 pages). Signers: R.L. Moore, H.S. Wall, Homer V. Craig, Patrick L. Odell.

Reed's academic career was at the University of Utah. He won the University's Distinguished Teaching Award in 1973.

**Harvy L. Baker**(1938,--), Ph.D., University of Texas, 1965.B.A. (1960) Texas. Thesis title: "Complete amonotonic collections", (35 pages). Signers: R.L. Moore, Homer V. Craig, R.G. Lubben, H.S. Wall.

Baker has taught at the University of Nebraska, where he directed two Ph.D. students, and at The University of Texas at Arlington.

**Blanche Joanne**(Monger) Baker (1934,--), Ph.D., University of Texas, 1965.B.A. (1956) Lamar State; M.A. (1958) Texas. Thesis title: "Concerning uncountable collections of triods", (45 pages). Signers: R.L. Moore, R.G. Lubben, Homer V. Craig, H.S. Wall.

Baker has taught at the University of Nebraska and at Lamar University.

**Roy D. Davis**(1938,--), Ph.D., University of Texas, 1966.B.A. (1961) Texas; M.A. (1964) Texas. Thesis title: "Concerning the sides from which certain sequences of arcs converge to a compact irreducible continuum", (35 pages). Signers: R.L. Moore, H.J. Ettlinger, R.G. Lubben, H.S. Wall.

After leaving Texas, Davis worked in the aerospace industry in Southern California.

**Jack W. Rogers**(1943,--), Ph.D., University of Texas, 1966.B.A. (1963) Texas; M.A. (1965) Texas. Thesis title: "A space whose regions are the simple domains of another space", (34 pages). Signers: R.L. Moore, H.J. Ettlinger, R.G. Lubben, W.E. Millett, H.S. Wall.

Rogers has taught at Emory University and at Auburn University, where he is currently Professor of Mathematics and Director of the Auburn University Honors College. His early work was in continuum theory; he later changed to applied mathematics and computational linear algebra. He has directed three Ph.D. students. Rogers first encountered Moore as a 10th grade high school student. He took the summer geometry course that year.

**Martin D. Secker**(1927,--), Ph.D., University of Texas, 1966.B.A. (1949) North Texas; M.A. (1950) North Texas; M.A. (1964) Texas. Thesis title: "Reversibly continuous bisensed transformations of an annulus into itself", (28 pages). Signers: R.L. Moore, H.J. Ettlinger, R.G. Lubben, H.S. Wall.

After leaving Austin, Secker taught first at Iowa State University, then at Branson School, a private college preparatory school in California, and at the College of Marin.

**David E. Cook**(1935,--), Ph.D., University of Texas, 1966.B.A. (1958) Texas; M.A. (1960) Texas. Thesis title: "Concerning compact point sets with noncompact closures", (39 pages). Signers: R.L. Moore, Homer V. Craig, W..T. Guy, Jr., Harold P Hanson, R.G. Lubben.

Cook's main academic appointment was at the University of Mississippi, where he directed three Ph.D. students.

**John W. Hinrichsen**(1940,--), Ph.D., University of Texas, 1967.B.A. (1961) Texas; M.A. (1964) Texas. Thesis title: "Certain web-like continua," (26 pages). Signers: R.L. Moore, W.T. Guy, Jr., R.G. Lubben, H.S. Wall, W.E. Millett.

Hinrichsen's academic career was spent at Auburn University. His research has been in continuum theory.

**Joel L. O'Connor**(1942,--), Ph.D., University of Texas, 1967.B.A. (1962) Texas. Thesis title: "Holes in two-dimensional space", (51 pages). Signers: R.L. Moore, Homer V. Craig, Robert E. Greenwood, W.E. Millett, H.S.Wall.

O'Connor taught at the University of Florida and then went into industry as an applied mathematician. He has consulted with the NSA and with the Vanderbilt University College of Medicine and, jointly with a medical physicist, founded Clinical Database Systems. He has consulted with other private firms and with state and local governments. O'Connor first encountered Moore as a high school student in Moore's summer geometry course.

**John W. Green**(1943,--), Ph.D., University of Texas, 1968.B.A. (1965) Texas; M.A. (1966) Texas. Thesis title: "Concerning the separation of certain plane-like spaces by compact dendrons", (85 pages). Signers: R.L. Moore, Homer V. Craig, R.G. Lubben, H.S. Wall.

Green was on the University of Oklahoma faculty for fifteen years. He then obtained a Ph.D. in mathematical statistics from Texas A&M University, then taught at the University of Delaware for five years. He has since been employed by E.I. DuPont as senior research biostatistician. He has said that his success in his present position is largely due to the training he obtained in Moore's classes. He has directed four Ph.D. students, two at Oklahoma in topology and two at Delaware in statistics.

**Michael H. Proffitt**(1942,--), Ph.D., University of Texas, 1968.B.A. (1964) Texas; M.A. (1966) Texas. Thesis title: "Concerning uncountable collections of mutually exclusive compact continua", (32 pages). Signers: R.L. Moore, Homer V. Craig, R.E. Greenwood, H.S. Wall.

After a post at SUNY New Paltz, Proffitt returned in 1972 to U.Texas, where he was a Robert A. Welch Fellow in chemistry and later in physics. In 1980 he moved to the University of Colorado, working in atmospheric research, more specifically, measurements of ozone. His measurements identified the cause of the ozone hole and its spread into other latitudes (published as articles in Nature and Science). He has written over 100 journal publications. He retired in 2004 as Senior Scientific Officer of the World Meteorological Organization and continues his scientific work in Buenos Aires, Argentina.

**Jesse A. Purifoy**(1938,--), Ph.D., University of Texas, 1969.B.A. (1963) Texas; M.A. (1965) Texas. Thesis title: "Some separation theorems", (50 pages). Signers: R.L. Moore, P.R. Meyer, D.S. Hymann, H.S. Wall, J.R. Whiteman.

After leaving Texas, Purifoy joined the faculty at Memphis State University, where he helped start a Ph.D. program in mathematics. While there he began consulting with the manufacture of programmable calculators and consulting with municipal bond companies and municipalities and other governmental agencies. He then moved to Houston and remained busy with computer hardware and software. He currently owns a software company, Purifoy Systems Analysis, Inc.

**Robert E. Jackson**(1943,--), Ph.D., University of Texas, 1969.B.A. (1964) Texas; M.A. (1966) Texas. Thesis title: "Concerning certain plane-like domains", (57 pages). Signers: R.L. Moore, Homer V. Craig, Robert E. Greenwood, W.T. Guy, Jr., H.S. Wall.

On leaving Texas, Jackson taught at Dickinson College, Carlisle, PA for several years and then went into industry, first with NCNB in North Carolina as a systems analyst, then with Diagnostic Laboratories, and then with BMC Software, where he is now a senior computer scientist.

**Nell Elizabeth (Stevenson) Kroeger**(1944,--), Ph.D., University of Texas, 1969.B.A. (1965, in microbiology) Texas; M.A. (1968) Texas. Thesis title: "Concerning indecomposable continua and upper semi-continuous collections of nondegenerate continua", (38 pages). Signers: R.L. Moore, Robert E. Greenwood, Homer V. Craig, W.T. Guy, Jr., H.S. Wall.

Stevenson held an academic position at SUNY Binghamton before going into the private sector. She currently is involved with computer software and has an intense interest in classical music.

Moore directed very few M.A. students; two were Lucille S. Whyburn, wife of Gordon T. Whyburn, and Martin Ettlinger, son of H.J. Ettlinger. Martin Ettlinger had an illustrious career in chemistry, being a professor at Rice and then at University of Copenhagen. In an interview, Ettlinger has remarked that the intellectual atmosphere in Moore's classes was never duplicated in his experience except when he was a Junior Fellow at Harvard.

Mathematicians who studied with Moore but who wrote Ph.D. theses under others include W. L. Ayres (Kline), Lida Barrett (Kline-Anderson), Robert Williams (G.T. Whyburn), Steven Jones and Gary Richter (R.H.Bing), D.R. Stocks and E. Hensley (Greenwood), D.R. Traylor (Fitzpatrick), W.T. Reid, W.M. Whyburn, O. H. Hamilton, J.H. Barrett, D.H. Tucker, B. Fitzpatrick, and E.I. Deaton (H.J. Ettlinger). Of the last group, Hamilton and Fitzpatrick subsequently worked mainly in the directions in which Moore had started them. In the 1966 film Challenge in the Classroom, Hamilton was the only student Moore mentioned by name. Many of H.S.Wall's Ph.D. students, including John S. Mac Nerney, Pasquale Porcelli, John Neuberger, Saul Drobnies, Sam Young, Coke S. Reed, Jack B. Brown, Robert Dorroh, and R. Daniel Mauldin were profoundly influenced by Moore. A special class of students are those who left Austin in the 1960s to study elsewhere; these include Raymond Houston and George Golightly (Houston), Michel Smith and Tom Jacob (Emory), Kenneth Van Doren, Kermit Smith, Douglas Moreman, John Bales, and Nick Williams (Auburn), and Don Fox (Riverside).

A number of women who were or later became wives of students of Moore took courses from him. These include Jean Mahavier, Katherine Cook, June Treybig, Janet Rogers, and Gayle Ball.

Finally, there are those persons who did not become mathematicians at all, but were very successful in other endeavors and who attribute their success, to greater or lesser extent, to the training they got from Moore. These include James Wm. McClendon, Distinguished Scholar-in-Residence at the Fuller Theological Seminary in California, and Patricia Pound, Secretary of the Governor's Committee (for the State of Texas) for Persons with Handicaps. McClendon has written a number of books on philosophy and theology; he is currently completing a three volume systematic theology. He was long (1971-1990) a professor at the Graduate Theological Union, Berkeley, California. Also, there are Robert Boyer, Professor of Computer Science and Philosophy at The University of Texas, Harry Lucas, Jr., a successful businessman, Joel Finegold, a free-lance detergent chemist, and Margaret Ball, a successful writer. Lorene Rogers, a distinguished chemist and a President Emeritus of the University of Texas at Austin, has said that she had a severe case of mathematics anxiety, especially at the prospect of having to take calculus, until she took solid geometry and then calculus under Moore. That she was successful in calculus is evinced by Moore's having tried to recruit her into a career in mathematics.

Reference has been made above to several of Moore's students who did not become mathematicians and who commented very favorably on their training under him. For the sake of completeness, and for balance, it should be noted that some of his students who did become mathematicians with successful academic careers viewed their training as a mixed blessing, and expressed their wish that they had had a wider mathematical education, specifically including the learning of algebraic methods in topology. Most of his students who directed Ph.D. students made sure that their students did learn some algebra.

At least two other mathematicians, H.S. Wall and L.E. Dickson, directed more Ph.D. students than did Moore. It is very doubtful that anyone else directed students over a longer period of time, from 1916 to 1969. This is the more remarkable in that his first student, J.R. Kline, did not graduate until eleven years after Moore's own Ph.D. was awarded in 1905.

Of the first half of Moore's students, more than half first encountered him in their graduate program. Of the remainder, a substantial majority took courses from him as undergraduates, and at least two studied with him while still in high school. More precisely, of his first twenty-seven students, seventeen already had Bachelor's degrees before taking a course from him, Dyer took two courses from him the summer he received his B.A. and B. S., and Young came as a senior and started directly in Moore's graduate course. Of the other twenty-three, all save four took lower level courses either from him or from his colleagues.

In looking over the Ph.D. students' dissertation titles, we note that four of them, some of the early ones, are in geometry. Many of the later ones deal with continuum theory; not as many are in abstract spaces, and several are on spirals in the plane, a subject that has not stirred much interest outside Moore's school. The same is true of webs. Bing once remarked that if anyone wanted a reprint of the journal article based on his dissertation, he still had forty-eight of the fifty copies provided him.

The list of signers of dissertations helps to provide an overview of the mathematics program at Texas from 1920 until 1969.

Moore's calculus classes are the subject of one completed study (Eyles) and one ongoing study (H. Cook). There are many essays and articles available on-line at the above URL. Moreover, over the last several years a number of interviews of persons with knowledge of Moore's methods have been conducted. Transcripts of these interviews are available through the Center for American History at The University of Texas at Austin. I particularly mention those with R.D. Anderson, Lida K. Barrett, John W. Neuberger (this has a full discussion of Moore's calculus course), John M. Worrell, Mrs. B.J. Ball, Mrs. R.H. Bing, Coke S. Reed, John W. Green.

For a complete inventory of transcripts of such interviews, one may contact info@edu-adv-foundation.org.

Acknowledgements. In addition to the sources cited, a number of sources were used in the formulation of this listing, including various publications of the American Mathematical Society and the Mathematical Association of America. The Archives of the University of Pennsylvania and of the University of Texas also provided much helpful information. The Center for American History at The University of Texas and its staff, specifically including Ralph Elder and Christopher Bourell, have been very helpful. The Educational Advancement Foundation and Harry Lucas, Jr. have also provided a great deal of assistance. I especially thank Sherry White, Connie Lang, Laurie Schmid, and Kate Walden of the EAF for their help. A number of individuals, including some of Moore's students, kindly provided information that could not have been obtained by other means.

The author, Ben Fitzpatrick, Jr. (1932-2000), was a professor of mathematics at Auburn University through most of his career.

[This article has been revised to bring some biographical details up to date. -- Albert C. Lewis, Nov. 2011.]