(presentation to Mathematics faculty)

**September 23, 1997**

Our department has a long tradition of excellence in mathematical logic going back to Haskell Curry, one of the first Evan Pugh professors, who was also one of the leading American logicians through the 40's, 50's and 60's. Dick Mansfield arrived here in the early 70's, and Tom Jech and I were hired in 1975. Many students came to Penn State specifically in order to study logic. Many outstanding logicians were trained at Penn State. I myself have supervised 12 PhD students, most of whom continue to teach and perform research in this field; see the appendix below. Tom Jech has also had a distinguished record of accomplishment in supervising PhD students; see the appendix below. Dick Mansfield has also supervised several PhD students.

During the period 1975-1990, logic students had the possibility of taking a logic qualifying exam based on what was then known as the first-year graduate logic sequence, Math 557-558. This sequence was very successful in focusing their studies and teaching them basic concepts and methods of mathematical logic. At the same these students took other courses and qualifying exams, principally algebra and analysis. They then went on to advanced courses and seminars in other branches of mathematics as well as logic and set theory. This system produced many excellent logic PhDs.

In 1990, the present system of qualifying exams in algebra, analysis and topology was introduced. As part of this reform, the logic qualifying exam was abolished, and the 557-558 sequence was decoupled into two independent elective courses. Since 1990, few if any students have taken these courses in sequence. Because of the new situation created in 1990, it has been extraordinarily difficult to develop new PhD students in this area. The biggest problem is that potential logic students must spend most of their first two years preparing for qualifying exams in algebra, analysis, and topology. This preparation is time-consuming, and much of the material in those exams is irrelevant to their future specialty. During those two years, the logic students must put logic on hold for the most part. The effect is to severely impede their studies and postpone the time when they can begin thesis research. The only way around this problem is for a student to arrive at Penn State having already mastered a substantial amount of the graduate-level material that is covered in the qualifying exams. All recent PhDs in logic were people who either entered under the pre-1990 system, or who entered at a postgraduate level in terms of previous training and background.

The proposed logic option is an attempt to restore our ability to recruit and develop logic PhD students. The logic option would not in any way threaten the existing system of qualifying exams in algebra, analysis, and topology. The only purpose of the logic option is to improve opportunities for motivated logic students to succeed in the PhD program, as they did prior to 1990. The logic option will not take students away from any other field, because it will be of interest only to students who arrive at Penn State already committed to specializing in logic. Such students will be required to petition for the logic option almost immediately upon arrival at Penn State.

Our proposal is very narrow and modest, but it would make an enormous difference to us logicians. I urge you to consider it carefully. I welcome your comments or suggestions for improving the proposal.

Appendix: Stephen Simpson's Ph.D. students

Here is a list of my Ph.D. students, 13 in all.

- John Steel,
*Determinateness and Subsystems of Analysis*, Berkeley, 1977.(Steel is a tenured full professor at UCLA.)

[ Although Steel's official thesis adviser was Professor John Addison of Berkeley, the following is a quotation from the acknowledgements page of Steel's thesis. ``I owe a great debt to Stephen Simpson, who guided me expertly in the perilous transition from study to research. The results of Chapters 1 and 2, together with less tangible aspects of my research, are a product of Simpson's influence.'' The thesis consists of three chapters. ]

- Rick L Smith,
*Theory of Profinite Groups with Effective Presentations*, Pennsylvania State University, 1979.(Smith is a tenured associate professor at the University of Florida.)

- Galen Weitkamp,
*Kleene Recursion over the Continuum*, Pennsylvania State University, 1980.(Weitkamp is a tenured professor at the Western Illinois University.)

- Peter Pappas,
*The Model Theoretic Structure of Group Rings*, Pennsylvania State University, 1982.(Pappas is a professor at Vassar College.)

- Stephen H Brackin,
*On Ramsey-type Theorems and their Provability in Weak Formal Systems*, Pennsylvania State University, 1984.(Brackin is a mathematician at Odyssey Research Associates.)

- Mark Stephen Legrand,
*Coanalytic Sets in the Absence of Analytic Determinacy*, Pennsylvania State University, 1985.(Legrand is an assistant professor at Auburn University.)

- Douglas K Brown,
*Functional Analysis in Weak Subsystems of Second Order Arithmetic*, Pennsylvania State University, 1987.(Brown is an associate professor at the Altoona Campus of Penn State.)

- Jeffry L Hirst,
*Combinatorics in Subsystems of Second Order Arithmetic*, Pennsylvania State University, 1987.(Hirst is an associate professor at Appalachian State University in North Carolina.)

- Xiaokang Yu,
*Measure Theory in Weak Subsystems of Second Order Arithmetic*, Pennsylvania State University, 1987.(Miss Yu is an associate professor at the Altoona Campus of Penn State.)

- Fernando Ferreira,
*Polynomial Time Computable Arithmetic and Conservative Extensions*, Pennsylvania State University, l988.(Ferreira is a professor at the University of Lisbon.)

- Kostas Hatzikiriakou,
*Commutative Algebra in Subsystems of Second Order Arithmetic*, Pennsylvania State University, l989.(Hatzikiriakou is a professor at the University of Crete.)

- Alberto Marcone,
*Foundations of BQO Theory and Subsystems of Second Order Arithmetic*, Pennsylvania State University, 1992.(Marcone is a professor at the University of Torino.)

- A James Humphreys,
*On the Necessary Use of Strong Set Existence Axioms in Analysis and Functional Analysis*, Pennsylvania State University, 1996.(Humphreys is an instructor at Penn State.)

Appendix: Thomas Jech's Ph.D. students

Tom Jech's Ph.D. students at Penn State:

- Robert Mignone (1979) College of Charleston
- Vladimir Zadrozny (1981) Bell Labs
- Carlos Alves (1985) U. of Trenton
- Qi Feng (1987) National U., Singapore
- Tomasz Weiss (1989)
- Wenzhi Sun (1991) Salem College
- Chaz Schlindwein (1993) Lander College
- Jiri Witzany (1994) Charles U., Prague
- Jindrich Zapletal (1995) Cal Tech

I gathered information on math PhD qualifying exams at the following peer institutions: Illinois, Indiana, Michigan, Michigan State, Minnesota, Northwestern, Ohio State, Purdue, Iowa, Wisconsin. Five of the ten have an option for a logic qualifying exam, and all ten have qualifying exam systems that are much more flexible than ours.

This data seems to support the view that we ought to have a more flexible system, or an option for a logic qualifying exam, or both.

Comprehensive Examinations - written examinations - three examination units; basic algebra, basic analysis, and a third area which the student chooses subject to the approval of the Director of Graduate Studies. Each unit examination will take no longer than three hours. The areas, and the corresponding courses on which the examinations are based, are

- Algebra: 401 - 402
- Analysis: 440 - 441
- Combinatorics & Optimization: any two of 417 418 473 474
- Differential Equations: 443 - 444
- Geometry: 423 - 424
- Logic: 410 and 411 or 412 or 413
- Number Theory: 405 - 453
- Probability: 451 - 452
- Topology: 431- 435

Qualifying Examination - three fields of mathematics - Each exam is a written one of three hours' duration. Associated with each examination are certain 500-level courses. The level of sophistication of the examinations is the same as these courses.

- Pure Mathematics
The student must take examinations in subjects from at least two of the following groups:

- Algebra, logic
- Real analysis, complex analysis, probability, statistics, applied mathematics, numerical analysis
- Topology, differential geometry

- Applied Mathematics
A student must take the examinations in applied mathematics, analysis (real or complex, but not both), and any other subject from the above list.

- Statistics
A student opting for statistics must take the qualifying examinations in probability, statistics, and real analysis.

The Ph.D. program requires at least 72 semester hours of graduate course work and is designed to be completed within approximately five years by those with a bachelor's degree. Degree candidates must pass written comprehensive examinations in three areas (to be chosen from algebra, analysis, logic, topology, or partial differential equations) and demonstrate reading proficiency in one language among French, German or Russian.

Qualifying Review - demonstration of competence in the four core areas, algebra, analysis, applied analysis, and geometry/topology. This will include passing a written exam in at least two of these areas and passing certain courses or exams in the other areas with satisfactory grades. ... Each student must also successfully complete during Stage I at least one course outside of the core areas.

Preparation for the exams is provided by the following course sequences.

- Algebra: 593, 594
- Analysis: 596, 597
- Applied Analysis: 556, 572
- Geometry/Topology: 591, 592

One of the exams must be passed by the beginning of the student's fourth term in the program. The entire review should be completed by the end of the fourth term and must be completed by the beginning of the sixth term.

Written qualifying examinations are given in four areas: (1) Algebra, (2) Geometry/Topology, (3) Differential Equations, and (4) Real and Complex Analysis. Parallel to these exams the department offers four ``core'' sequences: (1) Algebra -MTH 818-819, (2) Geometry/Topology - MTH 868-869, (3) Differential Equations - MTH 848-849, and (4) Real and Complex Analysis - MTH 828-829. ....

The qualifying examination requirements can be satisfied in two ways: (1) Pass exams in three areas; (2) Pass examinations in two areas and complete a core sequence in a third area with a grade of at least 3.5 each semester. These requirements must be completed within four semesters of entering the doctoral program. ...

During the first year most of your time will be spent on course work. The standard full-time course load is three courses per quarter. ... Ordinarily, you will take the written preliminary examination in the September preceding your second year. ... This examination covers three subjects, chosen from among the following topics:

- algebra
- algebraic topology
- complex analysis
- dynamical systems
- geometry of manifolds
- probability
- real analysis

We encourage you to choose algebra and real analysis plus one other subject from the list.

(telephone call to the Director of Graduate Studies, Don Kahn)

There are written qualifying exams in four areas: (1) real analysis, (2) complex analysis, (3) algebra, (4) topology and manifolds. To stay in good standing, a student must pass all four courses, plus a written qualifying exam in either real or complex analysis, and a written qualifying exam in either algebra or topology/manifolds, by the end of two years of graduate study.

(telephone call to the Graduate Secretary, Patrick Bonace)

By the end of the second year of graduate study, students must pass written qualifying exams in (1) real analysis, (2) algebra. In addition, there is a breadth requirement: students must pass 2-course sequences in three other areas. Among the allowable areas are logic, differential geometry, numerical analysis, etc. etc.

Qualifying Examinations - The student must pass five examinations in the following areas, based on material that is covered in the courses listed and on material from undergraduate prerequisites. Credit for passing a similar examination at another university cannot be transferred.

- Complex Analysis (MA 530)
- Real Analysis (MA 544)
- Abstract Analysis (MA 553)
- Linear Algebra (MA 554)
- One exam selected from
- Probability (MA 519)
- Topology (MA 571)
- Logic (MA 585)
- Applied Math (MA 523)
- Numerical Analysis (CS 514)

All students must take the examinations by the end of two years of graduate work unless a written request for postponement has been submitted and approved by the Graduate Committee at least two months prior to the examination date.

Students in the PhD program must pass qualifying exams in two of five areas (algebra, analysis, applied mathematics, logic and topology), pass a specialty exam in the intended area of research, and under the guidance of a member of the department, complete a significant piece of original research. The PhD program also requires a minor area, reading knowledge of one foreign language (French, German or Russian) and 54 credits of graduate work.

of the Doctoral Program in Mathematics

The creation of a new Logic and Foundations Option within the current mathematics graduate program is aimed towards creating an environment where research and education in mathematical logic and foundations of mathematics can prosper and thrive beyond the current level.

The design of the new option takes into account the success of the current graduate program in providing an excellent doctoral education. However it also acknowledges its limitations in supporting some areas of graduate education in mathematical logic and foundations of mathematics. Within such a context, the establishment of the new Logic and Foundations Option is intended to augment and enrich the current graduate program rather than alter it.

This new option will have a major impact in research and education in various branches of mathematical logic, including recursive function theory, set theory, proof theory, and model theory. Since a major goal of such research is to obtain insight into the foundations of mathematics, such kinds of research are very unlikely to be pursued in other Science and Engineering Departments.

It is relevant to point out that Penn State Mathematics Department has a long history of research and graduate education in mathematical logic and foundations of mathematics, going back to key figures such as Haskell Curry. For many years, Penn State was considered one of the best places in the world to study these subjects.

For the Logic and Foundations Option to succeed, it is essential to ensure a stable supply of graduate students. Because of realistic constraints in personal resources and time, the present qualifying exam system has negatively impacted recruitment of candidates. Discussion with department faculty in all areas reveals that a reasonable solution to accommodate the Logic and Foundations Option can be found without jeopardizing the present one, while simultaneously ensuring and strengthening the quality level of the current system.

Typical candidates for the Logic and Foundations Option would be students with a strong undergraduate mathematics background who are interested in mathematical logic and foundations of mathematics.

The basic points of the proposal can be summarized as follows:

- Maintain the standard Analysis and Algebra exams for mathematics Ph.D. candidates.
- Allow students to petition for the option of replacing the current exam in Topology with an exam in Logic and Foundations.
- Graduate students planning to write a thesis in the area of mathematical logic and foundations of mathematics may still elect to follow the standard examination system (Analysis, Algebra, Topology).
- Time limits to complete the new set of exams will remain the same as in the present system.
- On the discretion of the advisor, Ph.D. candidates having completed the new set of exams may be required to take certain courses, a certain number of credits, and/or certain exams in areas of pure mathematics as a graduation requirement.
- Candidates under the new option will graduate with the same title of Ph.D. in Mathematics as those in other areas.
- The proposed option will be established in a trial basis for the next five years. During that period of time, fine tuning and adjustments will take place to ensure the best results. A committee will re-evaluate the program and recommend changes, if deemed appropriate. In the latter case, proposed changes will follow the guidelines of the ``Constitution, Bylaws, and Standing Rules of the University Faculty Senate''.

- The process of application to the mathematics graduate program and the guidelines for selection of candidates will continue as in the present, i.e., choosing the academically strongest candidates.
- No option should be selected in the application form to graduate school.
- Candidates to the Ph.D. program select the new Foundations Option by filing a petition with the Graduate Studies Committee anytime between admission to the Ph.D. program and the add/drop deadline for the student's semester.
- All students are entitled to a free trial of all the qualifying exams at entry, as in the current system.
- Students must complete any qualifying examinations by the end of the second year of graduate school as currently required. Candidates failing to pass the exams by the end of the two year period will be discontinued from the graduate program. In no way is the existence of a new option meant to give a second chance to re-enter the Ph.D. program after having failed the qualifying exams.

The second phase of the procedure will be to design specific curriculum recommendations for the new courses. This task should be completed during the Spring semester of 1998.

We propose to keep Math 557 and Math 558 as the two-semester sequence on Logic and Foundations, with some minor changes to the syllabus as necessary. These courses include the Gödel completeness and incompleteness theorems, basic computability theory, solvable and unsolvable problems, and an introduction to proof theory, model theory, and set theory.

The new Logic and Foundations exam should be available to the students recently accepted to the Ph.D. program as well as to the next group of incoming Ph.D. candidates. The Foundations Option should start offering exams by August of 1998 to those incoming students that wish to select the Foundations Option upon arrival.

(presentation to Mathematics faculty)

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