(presentation to Graduate Council)

**March 18, 1998**

- I.
- New programs, options, and minors.
The Department of Mathematics proposes to institute a Logic and Foundations Option within its Ph.D. program. This option was approved by the mathematics faculty on September 23, 1997.

- [A.] Objectives.
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.

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 Logic and Foundations Option would allow such students to pursue a Ph.D. in mathematics with a concentration in these areas.

The design of the new option takes into account the success of the current mathematics Ph.D. 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.

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.

- [B.] New courses.
None. We are not proposing any new courses. The Department of Mathematics currently offers several courses in logic and foundations; see appendix 4. The resources needed to offer the Logic and Foundations option are already in place.

- [C.] Program statement.
The new program is called the Logic and Foundations Option. It is an option within the existing Ph.D. program in the department of mathematics.

Students who have been admitted to the mathematics Ph.D. program will be allowed to select the Logic and Foundations Option by filing a petition with the departmental graduate studies committee at any time during the interval between admission to the Ph.D. program and the add/drop deadline for the student's first semester in the Ph.D. program. Degree requirements and procedures for the Logic and Foundations Option will be exactly the same as for the standard mathematics Ph.D. program (see section 6.2 in appendix 6). The only difference is that candidates under the Logic and Foundations Option will take a written qualifying examination in Logic/Foundations, instead of Geometry/Topology as in the standard mathematics Ph.D. program. A first-year graduate sequence of two 3-credit courses, Math 557-558, will prepare students for the Logic/Foundations qualifying examination. The content of these courses will be as currently offered and listed in the graduate catalog (see appendix 5). The written qualifying examinations in Algebra and Analysis and all other requirements will be administered under the standard mathematics Ph.D. program. Candidates under the Logic and Foundations Option will graduate with the same title of Ph.D. in Mathematics as those in the standard mathematics Ph.D. program.

- [D.] Justification.
It is anticipated that the Logic and Foundations Option will have a large positive 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 departments of the University.

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. (See appendix 1.) For many years, Penn State was considered one of the best places in the world to study these subjects. Unfortunately, our ability to recruit and develop students in these areas has been negatively impacted by the system of Ph.D. qualifying examinations which the Department of Mathematics instituted in 1990. The Logic and Foundations Option would help to restore our ability to develop Ph.D. students in these area.

- [E.] Statements of affected departments.
No other departments are affected.

- [F.] Graduate faculty.
The mathematics department graduate faculty currently numbers about 60 including three specialists in mathematical logic: Thomas Jech, Richard Mansfield, and Stephen G. Simpson. Professors Jech, Mansfield, and Simpson have made substantial contributions to the graduate program, by supervising Ph.D. students in mathematical logic and set theory and in other ways. (See appendices 2 and 3.) Therefore, we anticipate that the current faculty in mathematical logic will be able to sustain the Logic and Foundations Option for at least the next five years. Although additional faculty would be desirable, this is not a requirement for restarting the Logic and Foundations Option. As noted in item B above, the resources needed to offer the Logic and Foundations Option are already in place.

- [A.] Objectives.
- II.
- Changes in programs, options, and minors.
The Logic and Foundations Option has been narrowly drawn so that it will have no effect on the existing standard Ph.D. program in mathematics. (See appendices 5, 6 and 7.) It will also have no effect on any other program, option, or minor.

- III.
- Dropping of programs, options, and minors.
The Logic and Foundations Option will not require the dropping of any program, option, or minor.

**Appendices**

- History of logic at Penn State
- Jech's former Ph.D. students
- Simpson's former Ph.D. students
- Graduate course offerings in logic

- Graduate degree requirements in Mathematics

- Side by side comparison

History of logic at Penn State

Penn State 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. Richard Mansfield arrived here in the early 70's, and Thomas Jech and Stephen Simpson were hired in 1975. Many students came to Penn State specifically in order to study logic. Many outstanding logicians were trained at Penn State. Simpson has supervised 12 PhD students, most of whom continue to teach and perform research in this field; see appendix 3 below. Jech also has a distinguished record of accomplishment in supervising PhD students; see appendix 2 below. 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 time 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.

Jech's former Ph.D. students

- 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

Simpson's former Ph.D. students

- 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.)

Graduate course offerings in logic

MATH 457. Introduction to Mathematical Logic (3)

MATH 459. Computability and Unsolvability (3)

MATH 557. Mathematical Logic (3) The predicate calculus. Completeness and compactness. Gödel's first and second incompleteness theorems. Introduction to model theory. Introduction to proof theory. Prerequisite: MATH 435 or 457 or equivalent.

MATH 558. Foundations of Mathematics I (3) Decidability of the real numbers. Computability. Undecidability of the natural numbers. Models of set theory. Axiom of choice. Continuum hypothesis. Prerequisite: any 400-level MATH course or equivalent.

MATH 559-560. Recursion Theory I, II (3 each) Recursive functions; degrees of unsolvability. Hyperarithmetic theory; applications to Borel combinatorics. Computational complexity. Combinatory logic and the lambda calculus. Prerequisite: MATH 459 or 557 or 558.

MATH 561-562. Set Theory I, II (3 each) Models of set theory. Inner models, forcing, large cardinals, determinacy. Descriptive set theory. Applications to analysis. Prerequisite: MATH 557 or 558.

MATH 563-564. Model Theory I, II (3 each) Interpolation and definability. Prime and saturated models. Stability. Additional topics. Applications to algebra. Prerequisite: MATH 557.

MATH 565. Foundations of Mathematics II (3) Subsystems of second order arithmetic. Set existence axioms. Reverse mathematics. Foundations of analysis and algebra. Prerequisite: MATH 557 and 558.

MATH 574. Topics in Logic and Foundations (3-6; may be taken repeatedly) Topics in mathematical logic and the foundations of mathematics. Prerequisite: MATH 558.

=10pt 0pt

JanuaryFebruaryMarchAprilMayJune
JulyAugustSeptemberOctoberNovemberDecember
, =60
=-60
:<10 0

Present catalog descriptions are in ordinary type.
Proposed revisions are in slanted type.

n=557-558 t=Mathematical Logic and Foundations of Mathematics I, II c=3 each d=First-order logic. Completeness and incompleteness theorems of Gödel. Introduction to model theory, axiomatic set theory, computability, and unsolvability. p=MATH 457 or consent of instructor (for MATH 557), MATH 557 (for MATH 558).

20pt20pt

*
n=557
t=Mathematical Logic
c=3
d=The predicate calculus. Completeness and compactness.
Gödel's first and second incompleteness theorems.
Introduction to model theory. Introduction to proof theory.
p=MATH 435 or 457 or equivalent.
*

*
n=558
t=Foundations of Mathematics I
c=3
d=Decidability of the real numbers.
Computability. Undecidability of the natural numbers.
Models of set theory. Axiom of choice. Continuum hypothesis.
p=any 400-level MATH course or equivalent.
*

*
n=5XX
t=Foundations of Mathematics II
c=3
d=Subsystems of second order arithmetic. Set existence axioms.
Reverse mathematics. Foundations of analysis and algebra.
p=MATH 557 and 558.
*

n=559 t=Recursion Theory I c=3 d=Recursive functions, enumeration theorem, recursion theorem, recursively enumerable sets, the jump operator, arithmetical hierarchy, subrecursive hierarchies, complexity theory, degrees of unsolvability. p=MATH 557 or MATH 558 or CMPSC 559.

n=560 t=Recursion Theory II c=3 d=Continuation of MATH 559. Recursively enumerable sets, degrees of unsolvability, hierarchy theory, inductive definitions, recursion in higher types. p=MATH 559.

20pt20pt

*
n=559-560
t=Recursion Theory I, II
c=3 each
d=Recursive functions; degrees of unsolvability.
Hyperarithmetic theory; applications to Borel combinatorics.
Computational complexity.
Combinatory logic and the lambda calculus.
p=MATH 459 or 557 or 558.
*

n=561 t=Set Theory I c=3 d=Models of set theory, constructible sets, forcing, large cardinals and elementary embeddings; introduction to descriptive set theory; introduction to infinitary combinatorics. p=MATH 558.

n=562 t=Set Theory II c=3 d=Continuation of MATH 561. Large cardinals, indiscernibles, iterated ultrapowers; forcing; infinitary combinatorics, trees; descriptive set theory, the axiom of determinacy. p=MATH 561.

20pt20pt

*
n=561-562
t=Set Theory I, II
c=3 each
d=Models of set theory. Inner models, forcing, large cardinals,
determinacy. Descriptive set theory. Applications to analysis.
p=MATH 557 or 558 or consent of instructor.*

n=563 t=Model Theory I c=3 d=Compactness and upward Löwenheim-Skolem theorems, interpolation and definability; element types, saturation, indiscernibles, omitting types theorem, applications to algebra. p=MATH 558.

n=564
t=Model Theory II
c=3
d=Continuation of MATH 563. Ultrapowers, categoricity, infinitary logic,
stability and superstability; other topics; applications to algebra.
p=MATH 563.

20pt20pt

*
n=563-564
t=Model Theory I, II
c=3 each
d=Interpolation and definability. Prime and saturated models.
Stability. Additional topics. Applications to algebra.
p=MATH 557.*

n=574 t=Topics in mathematical Logic and the Foundations of Mathematics c=3-6 d= p=.

20pt20pt

*
n=574
t=Topics in Logic and Foundations
c=3-6; may be taken repeatedly
d=Topics in mathematical logic and the foundations of mathematics.
p=MATH 557 or 558.*

Graduate degree requirements in Mathematics

The following requirements apply to all degree options.

- No credit will be given for any course in which a grade of less than B is received.
- A minimum grade point average of 3.0 is required for graduation for all advanced degrees.

The Ph.D. program

(1) **Qualifying Examinations**
(Departmental requirement). All doctoral students are required to take
three written qualifying examinations. Two of these examinations must
be completed prior to the beginning of the student's second year of
graduate study, and third prior to the beginning of their third year.
The written qualifying examinations are in the areas of Analysis,
Algebra, and Topology/Geometry.

**Students who elect the Logic and Foundations Option will take
a written qualifying examination in Logic/Foundations instead of
Topology/Geometry. ]**

Students who fail a qualifying examination twice after enrolling may not continue in the Ph.D. program. Students who do not pass all three examinations by the beginning of their third year may not continue in the Ph.D. program.

The examinations are offered starting approximately 10 days before the beginning of the fall semester and at the end of the spring semester each academic year. Basic one-year sequences in each subject are offered annually to help prepare the student for the examinations. Typically, an entering Ph.D. student takes two of the basic sequences in the first year and the third basic sequence in the second year of study, and takes the qualifying examinations in the spring after completing the corresponding courses.

However, entering Ph.D. students may take any of the examinations on arrival in August without penalty. If a pre-entrance examination is failed, the student still has two more opportunities to pass it. Entering Ph.D. students are advised to take at least two basic sequences (in the subjects they did not pass qualifying examinations on arrival) and the subsequent qualifying examinations in the first year of graduate study.

Some qualifying exam problems from previous years are available, along with a selection of problems from some first-year graduate courses. There are also outlines of the first-year courses on which the qualifying examinations are based.

(2) **Course Requirements** (Departmental requirement). Students
must receive a minimum grade of B in at least 11 3-credit 500-level mathematics
courses.

(3) **Language Requirement** (Departmental requirement). Before scheduling
their comprehensive examination, students must pass a reading examination
in one language, chosen from French, German, and Russian, which is not
their native language. Doctoral students must pass the language exam before
the end of their third year. The candidate must translate (the equivalent
of) four typewritten pages of mathematics (selected from a journal or textbook)
into English with the aid of a dictionary. The translation must be accomplished
in two hours or less. The material to be translated is usually selected
by the student's advisor. It is subject to approval by the Director of
Graduate Studies.

(4) **Ph.D. Candidacy** (Graduate School requirement). The Graduate
School will usually recommend candidacy after a student has passed 2 of
the 3 qualifying examinations and demonstrated satisfactory progress toward
(2) and (3) above. Admission to candidacy is conferred by the Graduate
School.

(5) **Residence Requirement **(Graduate School requirement). After
being admitted to candidacy, the student must be a full-time graduate student
as defined by the Graduate Bulletin for two consecutive semesters (excluding
summers) before comprehensive examinations can be scheduled.

(6) **Continuous Registration **(Graduate School requirement). After
a Ph.D. candidate has passed the comprehensive examination and has met
the two-semester full-time residence requirement (5) above, the student
must register continuously for each fall and spring semester (beginning
with the first semester after both of the above requirements have been
met) until the Ph.D. thesis is accepted and approved by the doctoral committee.

(7) **Advisers and Doctoral Committees **(Graduate School requirement)
Consultation or arrangement of the details of the student's semester-by-semester
schedule is the function of the adviser. General guidance of a doctoral
candidate is the responsibility of a doctoral committee consisting of four
or more active members of the Graduate Faculty, which normally includes
at least two faculty in the major field and is chaired by the student's
adviser. This committee is appointed by the Graduate Dean through the Office
of Graduate Programs, upon recommendation of the Director of Graduate Studies,
soon after the student is admitted to candidacy.

(8) **English Competency** (Graduate School and Departmental requirement)
A candidate for the degree of Doctor of Philosophy is required to demonstrate
high-level competence in the use of the English language, including reading,
writing, and speaking, as part of the language and communication requirements
for the Ph.D. It is Penn State's policy that students who are non-native
speakers of English be certified as competent to teach in English by the
Department of Speech and Communication. The oral competency of both native
and non-native (once certified) speakers of English is assessed by the
Graduate Studies Committee before the end of the second year of study based
mainly on faculty evaluations of the student's classroom performance while
teaching undergraduate mathematics courses, which are returned to the Associate
Chair of the Department every semester. In the cases when students have
not had the opportunity to teach before the end of their second year, oral
competence may be assessed based on a faculty report on student's presentations
made in seminars. Students whose oral competency is judged to be substandard
will be assigned a faculty mentor who will work with them to improve their
oral presentation skills. In particular, the student may be required to
enroll in the Graduate Student Seminar or to give a presentation in one
of the departmental seminars. The faculty mentor will certify in writing
to the Director of Graduate Studies when the student has attained the required
standard of oral competency. Non-native speakers of English may be required
to take and to pass with a B or better SPCOM 114G (Basic ESL).

To satisfy the written competency, a student should prepare a short expository paper (approximately 4 pages) on topics related to proposed dissertation research. The advisor and one other mathematics graduate faculty member evaluate the paper and report the result to the Director of Graduate Studies (in the case of disagreement the Director will have the paper evaluated by a third member of the graduate mathematics faculty). Students whose written competency is judged to be substandard will be required to take and to pass with a B or better one of the following courses: ENGL 202 (Effective writing), ENGL 418 (Advanced Technical Writing and Editing), ENGL 198G (Writing in the Disciplines) for native speakers of English, and SPCOM 116G (ESL: Reading and Writing) for non-native speakers of English.

Competence must be formally attested by the Graduate Studies Committee before the doctoral comprehensive examination is scheduled. (International students should note that passage of the minimal TOEFL requirement does not demonstrate the level of competence expected of a Ph.D. from Penn State.)

(9) **Comprehensive Examination** (Graduate School requirement).
The comprehensive examination is scheduled by the Graduate School after
the student has passed three qualifying examinations, has satisfied the
language requirement, has been admitted to candidacy, and agreed on a thesis
advisor and a research program. Doctoral students must pass their comprehensive
examination by the end of their seventh semester. A doctoral committee
chaired by the thesis advisor determines whether the proposed problem is
acceptable for the thesis and whether the student has the necessary background
to pursue the work proposed. The committee is at liberty to inquire into
any aspect of the student's preparation and progress.

(10) **Ph.D. Thesis** (Graduate School requirement). The ability
to do independent research and competence in scholarly exposition must
be demonstrated by the preparation of a thesis on some topic related to
the major subject. It should represent a significant contribution to knowledge,
be presented in a scholarly manner, reveal an ability on the part of the
candidate to do independent research of high quality, and indicate considerable
experience in using a variety of research techniques. The contents and
conclusions of the thesis must be defended at the time of the final oral
examination. A draft of the thesis must be submitted to the doctoral committee
a month before the final oral examination.

(11) **Final Oral Examination** (Graduate School requirement). The
final oral examination is primarily a defense of the thesis and is scheduled
by the Graduate School at least three months after the date of passing
the comprehensive examination and no more than seven years after admission
to candidacy.

*To be admitted to this program, the student must have three years
of teaching experience at the college level. Experience gained as a
teaching assistant may not be used to satisfy this requirement.*

(1) **Qualifying Examinations** (Departmental requirement).
Passage of the qualifying examinations in algebra and analysis. These
examinations are subject to the same conditions as discussed in
requirement (1) of the Ph.D. program. In particular, students who do
not pass both examinations before the beginning of the third year of
study may not continue in the program. Entering D.Ed. students who do
not pass at least one pre-entrance examination must take at least one
of the basic sequences in algebra or analysis.

(2) ** Course Requirements (**Graduate School and Departmental requirement).
90 course and research credits of which at least 30 must be earned at
the University Park Campus (see requirement 5 below). 15 credits in
educational foundation courses approved by the College of Education
(see requirement 7 below). 45 credits of approved mathematics courses,
including 24 credits of 500-series courses and 15 credits of
600-series research credits. In order to receive credit for a course,
the student must receive a grade of B or better.

(3) ** Language Requirement **(Departmental requirement). Before scheduling
their comprehensive examination, students must pass a reading
examination in one language, chosen from French, German, and Russian,
which is not their native language. Doctoral students must pass the
language exam before the end of their third year. The candidate must
translate (the equivalent of) four typewritten pages of mathematics
(selected from a journal or textbook) into English with the aid of a
dictionary. The translation must be accomplished in two hours or less.
The material to be translated is usually selected by the student's
advisor. It is subject to approval by the Director of Graduate
Studies.

(4) **Candidacy** (Departmental requirement). The
Department will recommend candidacy after a student has passed 1 of
the 2 qualifying examinations and demonstrated satisfactory progress
toward (2) above. Admission to candidacy is conferred by the Graduate
School.

(5) **Residence Requirement **(Graduate School
requirement). A minimum of six semesters of full-time graduate study
and research (15 credits per semester), or their equivalent in credits
(90 credits), of which at least 30 credits must be earned in residence
at University Park Campus is required for the D.Ed. dgree. A candidate
may register for a maximum of 30 credits of research in absentia, but
none of these may count toward the minimum of 30 credits that must be
earned at University Park Campus. It is expected that students will
register for a minimum of 15 credits of thesis research. The maixmum
credit load permitted a student who is employed full time is 6 credits
per semester.

(6) **Advisers and Doctoral Committees**
(Graduate School requirement) Consultation or arrangement of the
details of the student's semester-by-semester schedule is the function
of the adviser. General guidance of a doctoral candidate is the
responsibility of a doctoral committee consisting of four or more
active members of the Graduate Faculty, which normally includes at
least two faculty in the major field and is chaired by the student's
adviser. This committee is appointed by the Graduate Dean through the
Office of Graduate Programs, upon recommendation of the Director of
Graduate Studies, soon after the student is admitted to candidacy.

(7) **Major Program and Minor Field** (Graduate School
requirement) The program of study includes a major and either a minor
or a group of general studies. A majority of the courses offered in
fulfillment of the requirements must be in the major program. A
candidate choosing a major outside the fields of professional
education (such as mathematics) shall have a minor consisting of no
fewer than 15 graduate credits in professional education, as
recommended to the dean of the Graduate School early in the major
program with the approval of a faculty adviser from the minor
area.

(8) **Comprehensive Examination **(Graduate School
and Departmental requirement). The comprehensive examination is
scheduled by the Graduate School after the student has passed two
qualifying examinations, has satisfied the language requirement, has
been admitted to candidacy, and agreed on a thesis advisor and a
research program. The comprehensive examination will cover three areas
to be chosen by the student from those listed below. It will be based
on the contents of the following 400-level series courses in these
areas: applied analysis (MATH 411), geometry (MATH 427), logic (MATH
457), number theory (MATH 465), numerical analysis (MATH 455-456),
probability and statistics (MATH 414) and topology (MATH 429).

(9) **Thesis **(Graduate School requirement). Evidence of a
high degree of scholarship, competence in scholarly exposition, and
ability to select, organize, and apply knowledge must be presented by
the candidate in the form of a written thesis. The candidate must
demonstrate a capacity for independent thought, as well as ability and
originality in the application of educational principles or in the
development of a new generalization under scientific controls. A
thesis may be based upon a product or project of a professional
nature, provided scholarly research is involved. In order to be an
acceptable thesis, the professional project must be accompanied by a
written discourse demonstrating the nature of the research and
including such theories, experiments, and other rational processes as
were used in effecting the final result. The topic and outline of the
proposed thesis must have the approval of the doctoral committee. A
draft of the thesis must be submitted to the doctoral committee a
month before the final oral examination.

(10) **Final
Oral Examination** (Graduate School requirement). The final oral
examination is primarily a defense of the thesis and is scheduled by
the Graduate School at least three months after the date of passing
the comprehensive examination and no more than seven years after after
admission to candidacy.

The following requirement applies to both the thesis and non-thesis options:

(1) MATH 405, 441, 470, and 471 are not approved for the Master of Arts degree. MATH 401 and 435 cannot both be counted towards the degree.

* M.A. Degree, Thesis Option: Both Major and Minor in Mathematics* (30
course and thesis credits).

(1) above.

(2) 12 credits of 500-series mathematics courses.

(3) 6-9 thesis credits (600-series).

(4) Remainder of course credits in approved 400- and 500-series courses.

(5) Final (oral) examination based on thesis and general course work.

(6) Time limitation is six years or a period spanning seven consecutive summers.

*M.A. degree, Non-thesis Option: Both Major and Minor in Mathematics*
(30 course credits).

(1) above.

(2) 18 credits of 500-series mathematics courses in an approved program (3 of these credits may be taken in a related area provided the student receives permission from the Graduate Studies Committee in advance).

(3) Remainder of courses in approved 400- or 500-series courses. With the approval of the Graduate Studies Committee, the student may substitute credits in a related area. A total of 6 credits at the 400 and 500 level in another area is the maximum allowed.

(4) A paper on an approved topic in mathematics (no credit).

(5) Time limitation is six years or a period spanning seven consecutive summers.

*M.A. Degree, Non-thesis Option: Major in Mathematics and Minor in
Applied Mathematics* (30 course credits).

(1) above.

(2) A minimum of 21 credits in mathematics (including at least four courses at the 500 level) and a minimum of 9 credits (with at least 6 credits at the 500 level) in the minor area are required.

(3) The possible minors are computer science, operations research, physics, and statistics. The courses required and recommended for each of the areas can be obtained from the department.

(4) A paper on an approved topic in mathematics (no credit).

(5) Time limitation is six years or a period spanning seven consecutive summers.

*The M.Ed. Degree*

(1) 18 credits of mathematics courses. (6 credits may be taken in a related area if approved by the Graduate Studies Committee).

(2) 6 credits of science or additional mathematics courses.

(3) 6 credits of education courses approved by the College of Education.

(4) A paper on an approved topic in mathematics (no credit).

(5) Time limitation is six years or a period spanning seven consecutive summers.

Side by side comparison

As noted in appendix 6, a requirement for the Mathematics Ph.D. program is to take 11 500-level mathematics courses. Although no specific courses are required, the majority of Mathematics Ph.D. students take a common core of 500-level courses in Analysis, Algebra, and Topology/Geometry. This pattern will be unaffected by the Logic and Foundations Option.

For example, a student interested in logic and foundations might fulfill the 11-course requirement by taking: Math 501-502 (Real and Complex Analysis), Math 535-536 (Algebra), Math 527-528 (Geometry/Topology), Math 557-558 (Logic/Foundations), Math 561 (Set Theory), Math 565 (Foundations II), Math 574 (Topics in Logic and Foundations). Under the Logic and Foundations Option, such a student would take qualifying examinations in Analysis, Algebra, and Logic/Foundations.

By way of comparison, a student interested in, say, differential geometry might fulfill the 11-course requirement by taking, for example: Math 501-502 (Real and Complex Analysis), Math 535-536 (Algebra), Math 527-528 (Geometry/Topology), Math 530-531 (Differential Geometry and Topology), Math 533-534 (Lie Theory), Math 580 (Special Topics in Geometry). Such a student would take qualifying examinations in Analysis, Algebra, and Topology/Geometry.

(presentation to Graduate Council)

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