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The 1963 Booklet Were for the First Four Years of a Seven-Year Programleading to the Phd DOCUMENT RESUME ED 022 695 SE 005 082 PREPARATION FOR GRADUATE STUDY IN MATHEMATICS. Committee on the Undergraduate Program in Mathematics, Berkeley, Calif. Spons Agency-National Science Foundation, Washington, D.C. Pub Date Nov 65 Note 24p. EDRS Price 1F-S025 HC-$1.04 Desciiiptors-*ALGEBRA, *COLLEGE MATI-EMATICS, COURSE DESCRIPTIONS, CURRICULUMDEVELOPMENT, *CLERICULUM PLANNING, GRADUATE STWY, MATHEMATICS, RESEARCH identifiers-Committee on the Undergraduate Program in Mathematics, National Science Foundation The purpose of this booklet is to provide interim guidance for those collegesthat cannotimmediatelyeffecttherecommendationsoftheCommitteeonthe Undergraduate Program in Mathematics (CUPM) 1963 report. The recommendations in the 1963 booklet were for the first four years of a seven-year programleading to the PhD. and a career in mathematical research. Considered in this report are the undergraduate curricula in Abstract Algebra and Real Analysis. (RP) a _ I PREPARATION FOR .e._ -- GRADUATE STUDY IN MATHEMATICS - - A progvam in partial fulfillment offhe goals set mit in fhe Panel's earHer pamphlet, Pregraduate Preparation of Research Mathe- maticians. U.S. DEPARTMENT OF HEALTH, EDUCATION & WELFARE OFFICE OF EDUCATION THIS DOCUMENT HAS BEEN REPRODUCED EXACTLY AS RECEIVED FROM THE PERSON OR ORGANIZATION ORIGINATING IT,POINTS OF VIEW OR OPINIONS STAlED DO NOT NECESSARILY REPRESENT OFFICIAL OFFICE OF EDUCATION POSITION OR POLICY. NOVEMBER 1965 - :41 The Committee on the Undergraduate Program in Mathematics, supported by the National Science Foundation and appointed by the Mathematical Association of America, is charged with making recom- mendations for the improvement of college and university mathematics curricula at all levels and in all educational areas. Moreover, the Committee devotes its energies to reasonable efforts in realizing these recommendations. The present membership of the Committee is as follows: William L. Duren, Jr. University of Virginia Chairman Richard D. Anderson Robert C. James Louisiana State University Harvey Mudd College Ralph P. Boas Lowell J. Paige Northwestern University University of California, Los Angeles R. Creighton Buck University of Wisconsin Henry 0. Pollak Mathematics and Statistics C. E. Burgess Research Center University of Utah Bell Telephone Laboratories, Inc. Dan E. Christie Robert M . Thrall Bowdoin College University of Michigan M. L. Curtis A. W. Tucker Rice University Princeton University Charles R. De Prima R. J. Walker California Institute of Cornell University Technology Gail S. Young, Jr. Samuel Eilenberg Tulane University Columbia University E. G. Begle - Ex officio Leonard Gillman School Mathematics Study University of Rochester Group Samuel Goldberg Stanford University Oberlin College R. L. Wilder - Ex officio M. Gweneth Humphreys President, Mathematical Randolph-Macon Woman's Association of America College B. E. Rhoades Robert H. McDowell Executive Director Associate Director _ To reflect one of itsmajor concerns, the a Panel on Pregraduate Training, Committee has established with current membershipas follows: Ralph P. Boas Department of Mathematics Northwestern University Chairman Richard D. Anderson Samuel Eilenberg . Department of Mathematics Department of Mathematics Louisiana StateUniversity C olumbia University Louis Auslander Leonard Gillman Graduate Studies Division Department of Mathematics The City University ofNew York University of Rochester Paul T. Bateman t George E. Hay t Department of Mathematics Department of Mathematics University of Illinois University of Michigan D. W. Bushaw Leon A. Henkin Department of Mathematics Department of Mathematics Washington State University University of California, Berkeley Dan E. Christie Department of Mathematics Victor L. Klee Bowdoin College Department of Mathematics University of Washington Leon W. Cohen Department of Mathematics Herman Meyer University of Maryland Department of Mathematics University of Miami Leslie A. Dwight Department of Mathematics A. B. Willcox Southeastern StateCollege. Department of Mathematics Durant, Oklahoma Amherst College \ \ c / ......W.....- TPBLE OF CONTENTS Foreword: The Goal 4 The Present Recommendations 5 The lower division 5 The upper division 6 Introducing the program 6 Objectives of the program 6 Identifying students 7 Lower division courses 8 Upper division courses 9 Course Outlines Abstract Algebra, Outline A 10 Abstract Algebra, Outline B 14 Real Analysis 18 FOREWORD: THE GOAL The task of the Panel on Pregraduate Training isto recommend programs for undergraduates who intend to go on to graduate workin mathematics. The first stage of the Panel's work,to consider an ideal undergraduate program for the futureresearch mathematician, was complred in 1963, and the conclusions have been publishedin a booklet. The recommendations in the 1963 bookletare for the first four years of a seven-year program leading to a solid Ph.D. anda career in mathematical research. The booklet contains guidelines and several detailed sample course outlines.Note, however, that in forming these recommendations the Panel madeno allowance for the student's possible deficiencies in preparationor tardiness in selecting a goal, for inadequacies of staff, forlack of suitable textbooks, etc. Furthermore, the program was designed for the very gifted undergraduate working under idealcircumstances.Some schools already offer programs compatible withthe 1963 recom-- mendations. At many others, however, therecommendations cannot be put into effect verysoon. For them, the 1963 booklet provides a goal. The present booklet provides interimguidance. *Preliminary Recommendations forPregraduate Preparation of Research Mathematicians, May 1963. Reprinted with minor modificationunder the title Pregraduate Preparation of ResearchMathematicians, 1965. Distributed through CUPM, P. 0. Box 1024, Berkeley,California 94701. -4- THE PRESENT RECOMMENDATIONS With the goal clearly in mind,the Panel has turned to theurgent practical task of recommendingspecific underg aduate curriculaat those schools where it is unlikelyto be achieved quickly. The present recommendations are firststeps toward the goal. The lower division (first two years).Another subcommittee of CUPM, the Subcommitteeon a General Curriculum in Mathematics for Colleges (GCMC), has recentlydrawn up a program in mathematics designed for almost all students whorequire t-aining in mathematics. This program was developed with fullrealization of the great diversity of problems in the various schools.It includes the standard basic subjects, but the order of topics issometimes unconventional. Further- more, the GCMC committee recommends attitudestoward the subject matter that are more consistent withcurrent developments in mathematics than those that many students acquiretoday. The Panel feels that the first part of the GCMCprogram (Mathematics 1, 2, 3, 4, 5)can also serve well for the pregraduate trainingof mathematicians, preferably with some increase in rigor anddepth. The Panel makes the following observations: 1. The first two years of the GCMCprogram include a thoughtful study of calculus through partial derivatives andmultiple integrals, and of linear algebra through the elementarytheory of vector spaces and linear transformations. 2.If all mathematics students follow the firsttwo years of the GCMC program, those who decide relativelylate on graduate work in mathematics will not losevery much, even though early work in the spirit of the 1963 recommendations isvery desirable for the future professional mathematician. 3. The first two years of the GCMCprogram are within the reach of all schools, even if they have to offera pre-calculus course as preparation. The lower division of the GCMCprogram and its relation to pregraduate training are discussed inmore detail later on. *A General Curriculum in Mathematics for Colleges, 1965. Distributed through CUPM, P. 0. Box 1024, Berkeley, California94701. -5- The upper division (last two years).Because of the heterogeneity of the mathematical world, the Panelrecognizes that no single curriculum will work for all schools.We therefore recommend the following priorities. 1. A minimal upper divisionprogram for mathematics majors who intend to continue the study of mathematicsin graduate school appears in the GCMG report (page 19). Any collegenot already offering a comparable program should takeimmediate steps to do so. 2.If, however, the collegecan also supply courses designed especially for the pregraduate student,then it should provideone, or if possible both, of the one-year sequences (analysis and algebra) described below. The choice, if onlyone can be given, should of course be determined by the staff's capabilities. Theone-semester analysis course (Mathematics 11) of theGCMG recommendations is itself aimed primarily at the pregraduatestudent; it could be replaced by the one-year course (Mathematics11-12). On the other hand, the one-semester algebra course (Mathematics6) of the GCMC serves many purposes; it should be retained forthese purposes and the one-year algebra course described belowshould be added for the pregraduate student. The Panel has consulted many outstandinggraduate mathematics departments; it turns out that a solid groundingin algebra and analysis is what they most want from incoming students.If time and resources permit, it is of course desirable to introducethe pregraduate student to a broader range of material, but notat the expense of depth in algebra and analyis. Introducing the program. Many departmentsof
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