( ) Year of initial appointment at Binghamton

Arcones, Miguel, Assistant Professor, PhD, 1991, City University of New York: Mathematical statistics, probability theory. (1998)

Brewster, Benjamin C., Professor, PhD, 1970, University of Kentucky: Algebra, group theory. (1970)

Brin, Matthew G., Professor, PhD, 1977, University of Wisconsin at Madison: Geometric topology. (1978)

Farrell, F. Thomas, Distinguished Professor, PhD, 1967, Yale University: Topology and differential geometry. (1990)

Feingold, Alex J., Professor, PhD, 1977, Yale University: Algebra, Lie algebras, conformal field theory. (1979)

Geoghegan, Ross, Professor, PhD, 1970, Cornell University: Topology, geometric group theory. (1972)

Guzman, Fernando, Associate Professor, PhD, 1985, Syracuse University: Algebra, algebraic logic, theoretical computer science. (1985)

Hanson, David L., Professor, PhD, 1960, Indiana University: Probability, mathematical statistics. (1973)

Head, Thomas, Professor, PhD, 1962, University of Kansas: Theoretical computer science, algebra, automata. (1988)

Hilton, Peter J., Distinguished Professor Emeritus, PhD, 1949, Oxford University: Algebraic topology, algebra. (1982)

Houghton, Charles J., Associate Professor Emeritus, PhD, 1964, Ohio State University: Computer science. (1964)

Kappe, Luise-Charlotte, Professor, Dr. rer nat, 1962, University of Freiburg, Germany: Group theory, number theory. (1968)

Kappe, Wolfgang P., Professor, Dr. phil nat, 1961, University of Frankfurt, Germany: Algebra, group theory. (1968)

Karagueuzian, Dikran, Assistant Professor, PhD, 1995, Stanford University: Algebraic topology, group theory. (1999)

Klimko, Eugene M., Associate Professor, PhD, 1967, Ohio State University: Probability and statistics. (1973)

Kronk, Hudson V., Associate Professor, PhD, 1964, Michigan State University: Graph theory. (1964)

Lercher, Bruce L., Associate Professor Emeritus, PhD, 1963, Pennsylvania State University: Mathematical logic. (1962)

McAuley, Louis F., Professor, PhD, 1954, University of North Carolina: Topology. (1969)

McAuley, Patricia T., Associate Professor and Director of the
MAT/MST Program, PhD, 1962, University of Wisconsin:

Algebraic topology. (1969)

Pedersen, Erik, Professor and Chair, PhD, 1974, University of Chicago: Algebraic and geometric topology. (1989)

Pixton, Dennis G., Associate Professor, PhD, 1974, University of California at Berkeley: Dynamical systems, formal languages. (1977)

Schick, Anton, Associate Professor, PhD, 1983, Michigan State University: Statistics, probability. (1984)

Sterling, Nicholas J., Associate Professor Emeritus, PhD, 1966, Syracuse University: Mathematical education. (1966)

Yu, Qiqing, Associate Professor, PhD, 1986, University of California at Los Angeles: Statistics. (1995)

Zacks, Shelemyahu, Professor, PhD, 1962, Columbia University: Statistics. (1980)

Zaslavsky, Thomas, Professor, PhD, 1974, Massachusetts Institute of Technology: Combinatorics, graph theory. (1985)

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Mathematics belongs both to the liberal arts and to the sciences. Not only is it the language of science (including social science), but it is also studied for its own beauty. It is therefore one of the most vital and lively subjects in the university curriculum. In the technology-oriented climate of today, the department’s graduates have excellent employment opportunities.

The Mathematical Sciences Department has programs leading to BA and BS degrees and MA, MAT/MST and PhD degrees. The challenging BS degree program is excellent preparation for graduate work at any university. Students considering a BS degree should seek advice as early as possible and plan their schedules carefully to meet the demanding requirements.

Three actuarial science seminars (MATH 324, 325 and 425) are offered for students interested in this profession.

The department serves other disciplines by providing instruction in various mathematical skills. For example, the department offers MATH 220, Calculus for Management Decisions, to students in the School of Management. Traditional mathematical preparation for the hard sciences (biology, chemistry, physics) is provided by MATH 221, 222, 304, 323, 371, 375, 471, 478, 479 and other courses.

Statistics has long been a fundamental tool in a variety of fields. MATH 147 does not demand the prior knowledge of calculus required by the more rigorous (but still basic) probability and statistics two-course sequence MATH 447-448.

1. A grade of C– or better is necessary for a math course to count toward the major.

2. A grade of C or better is necessary for a math course to serve as a prerequisite to another math course.

3. A Pass grade (P) does not count toward the major or as a prerequisite (unless the only grade available is Pass/Fail; in this case, consent of the department is required).

4. A grade-point average of 2.0 or higher in major courses is required for satisfactory completion of the major.

5. If you have received credit for a course, you may not take one of its prerequisites for credit at a later time.

The BA program is highly flexible and allows each student to fashion a course of study to meet his or her individual needs and interests.

A student must complete a minimum of 10 courses as follows:

1. Calculus-Linear Algebra: MATH 221, 222, 323 and 304.

2. Introduction to Higher Mathematics: MATH 330.

3. A pairing of two courses to be selected according to the student’s interests from the following: MATH 401-402, 401-404, 401-407, 351-451, 478-479, 375-478, 478-461, 461-462, 371-471, 357-358, 447-448, 381-386; CS 333-375, 333-350, 333-432 and 471-472.

4. Three additional MATH courses numbered above 330. CS 333 may be substituted for one of these three additional courses if the sequence in 3 is not a CS sequence.

No more than three transfer and independent study courses may be used to satisfy the requirements listed under 2, 3 and 4.

The 10-course requirement should be considered a strict minimum. Students are encouraged to take some additional mathematics courses numbered above MATH 330.

The flexibility of the BA program makes it especially important for the student to get early and regular advice from the faculty adviser. See further comments under the headings "Departmental Advising" and "Mathematics and Computer Science."

This degree affords excellent preparation for graduate study in mathematics or the teaching of mathematics. A student must complete the following courses: MATH 221, 222, 304, 323, 330, 375, 401, 402 or 404, 461, 478 and 479.

In addition, students must complete five additional departmental courses numbered above 330 (including graduate courses).

Courses above the introductory level in the Division of Science and Mathematics and in the Department of Computer Science of the Thomas J. Watson School of Engineering and Applied Science may be substituted for any of these five courses if they have mathematical content comparable to that of an upper-level undergraduate mathematics course and are approved on a case-by-case basis by the student’s adviser and the Undergraduate Committee. Students electing this substitution are strongly advised to obtain this approval by petitioning the Undergraduate Committee before taking the course(s) in question.

Transfer and independent study may not be used for more than five courses numbered above MATH 330.

Exceptions to the requirements for the BS degree may, in rare cases, be allowed. They must be approved by the department.

The honors program in mathematics is designed for students who have a serious interest in advanced mathematics, particularly in research.

One requirement for the honors program is strong and broad coursework in mathematics. The student must complete, by graduation, with a grade-point average of at least B, the following: MATH 375; 401; 402 or 404; 478 and 479; and 461, or 447 and 448. Courses on the same subjects at the same or higher level may be substituted upon approval of the Mathematics Undergraduate Committee.

The additional requirements for the honors program are individually designed by the student in consultation with a faculty sponsor. A proposal for this extra work must be presented to the Mathematics Undergraduate Committee during the student’s junior year, with the support of the faculty sponsor. Such a proposal typically involves extra coursework at the graduate level and/or independent research leading to a thesis. If independent study is required in the proposal, the student may register for MATH 498 under the direction of the faculty sponsor.

In cases of unusual merit, the Undergraduate Committee may award honors to a student who is judged to have met the above standards, even though no program sponsored by a faculty member was submitted.

The Mathematics Undergraduate Committee has final authority for accepting a student into the honors program (based on the merits of the proposal) and for granting graduation with honors (based on the student’s success in fulfilling the goals of the honors proposal).

More details, including sample proposals, are available from the Department of Mathematical Sciences.

Students considering a major in mathematical sciences should seek advice from the faculty as early as possible. Every declared major should be assigned to a faculty adviser, and should meet regularly with the adviser to discuss course selection and career goals.

Mathematicians and statisticians are in demand, not only in mathematics teaching and research, and in the traditional fields of physics, chemistry, computer science and engineering, but also, and increasingly, in business, economics, environmental sciences, geology, biology and the health sciences, among others. Students interested in applications of mathematics should consider a minor in another discipline or even a double major, and consult the faculty in the relevant departments.

A basic knowledge of computer programming is useful for most mathematics majors.

Actuaries analyze and solve complex business and social problems related to insurance and pension plans. They are employed by federal and state agencies, consulting firms and universities, as well as insurance companies. Professional advancement results from passing a series of examinations administered by the actuarial societies. A strong background in mathematics is essential to success.

Students interested in an actuarial career should include MATH 221, 222, 304, 323, 447 and 448 in their programs, as well as the actuarial seminar MATH 324, which is a preparation for the first exam. They should have knowledge of computer programming equivalent to CS 140, and also take courses in accounting, economics, insurance marketing and other areas of business administration.

The actuarial profession has instituted some major changes in its professional requirements as of 2000. Students with an actuarial bent should receive advice from the actuarial adviser.

Binghamton University is a site for administering the first two actuarial exams (Course 1 and 2).

The Computer Science Department in the Watson School of Engineering and Applied Science offers a minor program that can be combined with a BA in mathematics to provide a strong background leading to careers in computer science. The BA in mathematics is designed to facilitate this combination by allowing two computer science courses to be included in the degree program. Students interested in mathematics and computer science should also consult with the Computer Science Department.

A minor in mathematical sciences requires the student to complete, with a
grade higher than D, at least six departmental courses numbered above MATH 300,
of which at least three are numbered above MATH 330. Transfer and independent
study credit may not be used for more than one of the latter three courses.
Students interested in pursuing a mathematics minor should consult with the
undergraduate director. Note that Harpur College mandates that at least four of
the courses for the minor must be in addition to those counted toward
fulfillment of the student’s major.

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The department is committed to the idea that pure and applied mathematics are two faces of the same subject. The research of the faculty and the training of the students cover a wide variety of topics in pure mathematics, as well as statistics and computer science. The department offers a lively research atmosphere. Students are encouraged to take a broad range of courses. Teaching assistants are given varied assignments intended to increase their experience and employability. The distinguished research faculty offers considerable personal attention to graduate students.

The department offers the MA, MAT, MST and PhD degrees. Research areas of faculty expertise include algebra, combinatorics, dynamical systems, geometry, graph theory, probability, statistics, theoretical computer science and topology.

The MA program is intended to give the student a solid
professional basis either for proceeding to the PhD program or for work in
government, industry or teaching at the community college level. The PhD degree
prepares a student for university or college teaching and for higher-level
employment in government and industry. The MAT and MST degrees are preparation
for careers in high school teaching. Entering students having substantial
graduate level training may enter the PhD program, skipping

the MA.

The department is noted for its method of graduate education. In first-year courses, the emphasis is on training the student to do mathematics in depth. Many students report that these courses are the formative experiences of their professional lives.

Teaching assistantships are available. They provide not only financial support but also valuable experience, either in teaching a variety of courses or assisting faculty in special courses. The aim is to enhance students’ training with actual experience helpful in obtaining employment.

Department members assist students in obtaining suitable employment and offer advice for career development.

Although there is no official requirement of a minor, the
department supports the concept of suitable study outside the area of primary
emphasis, particularly for doctoral students. Doctoral students in pure
mathematics are encouraged to obtain expertise in an area of applied mathematics
sufficient for competency in instruction in that area at the undergraduate
level. Students in statistics and other applied areas naturally obtain
appropriate training in pure mathematics in the regular course of study.

**
Note:** A departmental graduate student handbook is available on request.

For admission to regular standing, a student should have a bachelor’s degree and have completed (with an average of at least 3.0) a set of mathematics courses approximately equivalent to those required for a bachelor’s degree at Harpur College with a specialization in mathematics. The department encourages submission of Graduate Record Examination scores for the aptitude and advanced tests that are useful in evaluating applicants.

The official requirement for a master’s degree is a minimum of 30 credit hours at the graduate level. This requirement can technically be satisfied in three semesters. However, the 30-hour requirement is regarded as minimal, and most students take four semesters to complete the master’s degree. Each student’s program is worked out in consultation with an adviser, under the general supervision of the graduate committee. While it is possible for a student to fulfill up to eight hours of course requirements by writing a master’s essay, only in certain circumstances is this encouraged. Students writing a master’s essay must pass an oral examination covering the subject matter of the essay.

MA students who do not write master’s essays must pass an oral examination in the last semester of their MA program. For this purpose, a committee of three or more faculty members is appointed. Usually these are faculty members who have taught the student. The examination syllabus is arranged by the committee in consultation with the student; in general it covers 30 hours of the student’s coursework.

in Teaching

The Department of Mathematical Sciences offers jointly with the Division of Education the MAT (master of arts in teaching) and the MST (master of science in teaching) degrees.

The MAT degree program is for those with no pre-service teacher preparation at the undergraduate level. The MST degree program is for those already provisionally certified to teach in New York state. Requirements for these degrees are listed elsewhere in this Bulletin. Mathematics courses specifically designed for these programs are indicated by MAT/MST following the course title.

Inquiries about these programs should be directed to the MAT/MST adviser, Mathematical Sciences Department, Binghamton University, Binghamton, New York 13902-6000.

A minimum of 14 courses at the graduate level (including those counted for the MA) is required. A total of five or six years of full-time graduate study (including study toward the MA) is normally required to complete the doctorate.

Admission to PhD candidacy begins with informal discussions among the student, the adviser and other members of the department on whether it is wise for the student to consider pursuing a doctorate. Such discussions generally take place early in the student’s fourth semester of graduate study, near the end of the master’s program, when department members are able to assess the student’s abilities. Then, or later, the student finds a prospective dissertation adviser who is an active and established researcher and who is willing (at least provisionally) to supervise the student’s doctoral dissertation.

At an appropriate time, the adviser presents to the Graduate Committee a format for the "admission to candidacy" examination of the student. This format, worked out in consultation with the student, may be one or several examinations, written or oral, in several areas; an oral presentation of research papers; or a combination of these. The adviser provides syllabi for the areas to be covered on the examination. The Graduate Committee either accepts the adviser’s recommendation or suggests alternatives; it then appoints an examining committee to carry out its instructions. The examining committee reports the results of its examination and its recommendations to the Graduate Committee. The Graduate Committee makes the final decision on the student’s admission to candidacy. A detailed explanation of this procedure is available in the department.

It is department policy that the PhD candidate have working experience in at least two foreign languages. The chair of the candidate’s thesis guidance committee is responsible for implementing this policy.

The student must, of course, do research and write a dissertation. It is the student’s responsibility to find an adviser willing to supervise the research and guide the student in writing the thesis. The Graduate Committee then appoints a guidance committee for the student. The dissertation must be defended in an oral examination.

Within the MA or PhD program there are two components or areas of emphasis. The flavor of these components can be indicated as follows:

The department is committed to the idea that the student whose primary interest is pure mathematics should also be acquainted with some applications. Thus, even students pursuing the PhD degree in mathematics are encouraged to take some courses in computer science and/or statistics. The department has special emphasis in algebra, combinatorics, dynamical systems, functional analysis, geometry, graph theory, probability, statistics, theoretical computer science and topology. The department has a tradition of developing intellectual independence in its graduate students. Much time is given to the education of graduate students, both individually and in small classes.

The statistics component gives broad training. The master’s degree prepares students for jobs as statisticians and data analysts in government and industry. The PhD degree prepares students for university teaching and research, as well as consulting and research roles in industry and government. Students are given training in many diverse statistical methods used to analyze data, as well as the mathematical, statistical and probabilistic foundation.

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Undergraduate

**MATH 101. BASIC MATHEMATICS 2 cr.
**Ratios and percents, geometric concepts and measurement; introduction to
algebra. Credit given only to students with deficiencies in the mathematics
admission requirement. Does not fulfill all-college distribution requirements.
Not open to students who have credit for any higher-numbered mathematics course.

**MATH 102. BASIC ALGEBRA every sem., 2 cr.
**Polynomials and rational fractions. Solving equations and inequalities.
Functions and graphing. Roots and exponents. College credit given only to
students with deficiencies in the mathematics admissions requirement. May not be
used to satisfy major requirements or all-college distribution requirements. Not
open to students who have credit for any higher-numbered mathematics course.
Prerequisite: MATH 101 or equivalent with a grade of C or better.

**MATH 103. BASIC ALGEBRA every sem., 2 cr.
**Continuation of MATH 102. The same restrictions apply. Prerequisite: MATH
102 or equivalent with a grade of C or better.

**MATH 104. INTRODUCTION TO FUNCTIONS every sem., 2 cr.
**The concepts of functions and their graphs. Logarithm and exponential
functions. Right triangle trigonometry. This course is preparation for MATH 108.
Credit given only to students with deficiencies in the mathematics admissions
requirement. May not be used to satisfy major requirements or all-college
distribution requirements. Not open to students who have credit for any
higher-numbered mathematics course. Prerequisite: MATH 103 or equivalent with a
grade of C or better.

**MATH 107. BASIC INTEGRATED MATHEMATICS
**Development of basic algebraic skills with some geometry. The course is
designed as a bridge between high school mathematics and elementary statistics.
It is not adequate preparation for calculus. Prerequisite: grade of C or higher
in MATH 104 or two years of high school math.

**MATH 108. ALGEBRA AND TRIGONOMETRY every sem.
**Topics essential for study of calculus, including elements of trigonometry,
complex numbers, logarithms and basic algebra. Skill development in algebraic
and trigonometric manipulations.

**MATH 147. ELEMENTARY STATISTICS every sem.
**Classification of data, frequency distributions, probability and the normal
curve, elementary sampling theory. Not open to students who have credit for any
other course in statistics. Prerequisite: MATH 108 or equivalent with a grade of
C or better.

**MATH 220. CALCULUS FOR MANAGEMENT DECISIONS every sem.
**Elements of calculus; emphasis on maximum and minimum problems. Primarily
for School of Management students, who may satisfy their mathematics requirement
with either MATH 220 or 221, and for economics majors in the BS specialization
in financial economics. Not equivalent to MATH 221 as prerequisite for MATH 222.
Prerequisite: MATH 108 or equivalent with a grade of C or better.

**MATH 221. CALCULUS I every sem.
**Differentiation and integration of elementary functions. Prerequisite: MATH
108 or equivalent with a grade of C or better.

**MATH 222. CALCULUS II every sem.
**Techniques and application of integration. Sequences and series.
Prerequisite: MATH 221 with a grade of C or better.

**MATH 304. LINEAR ALGEBRA every sem.
**Vector spaces, linear transformations, determinants, characteristic values.
Prerequisite: MATH 221 with a grade of C or better.

**MATH 314. DISCRETE MATHEMATICS every sem.
**
Logic, sets, relations, functions. Induction, recursion, counting methods.
Graphs, trees. Some abstract algebra. Prerequisite: MATH 221 with a grade of C
or better.

**MATH 323. CALCULUS III every sem.
**Calculus of functions of several variables. Prerequisite: MATH 222 with a
grade of C or better.

**MATH 324. SEMINAR IN ACTUARIAL SCIENCE 2 cr.
**Advanced problem solving seminar for students interested in careers as
actuaries. Prepares for Course 1. Does not satisfy major requirements.
Prerequisites or corequisites: MATH 323 and either 447 or 341. P/F only.

**MATH 330. INTRODUCTION TO HIGHER MATHEMATICS every sem.
**How to write mathematically correct proofs. Preparation for upper-level math
courses through learning a body of mathematics in a rigorous and careful way.
The student not only solves problems but learns to express the solutions
according to the universally accepted norms of mathematics. Content includes
induction and recursion, key properties of the real numbers, sets, functions and
countability. Other topics at instructor’s discretion. Methods taught include
direct proof, proof by contradiction and, in general, logical presentation of
mathematical thought. Prerequisite: MATH 222 with a grade of C or better.

**MATH 335. MATHEMATICAL LOGIC
**Development of predicate calculus. Introduction to metatheory of
propositional and predicate calculus: completeness, consistency, decidability.
Axiomatics. Prerequisite: MATH 314 or 330 with grades of C or better, or consent
of department.

**MATH 339. PROBLEM SOLVING SEMINAR 1 cr.
**Techniques of problem solving. Focus on hard problems not usually addressed
in ordinary coursework. Problems chosen from a variety of mathematical topics
and levels. Prerequisite: consent of department. P/F only.

**MATH 341. PROBABILITY WITH STATISTICAL METHODS 3 cr.
**Development of probabilistic concepts in discrete and absolutely continuous
cases. Classical combinatorial methods, independence, random variables,
distributions, moments, transformations, conditioning, confidence intervals,
estimation. Open only to students in the Watson School. Does not serve as a
prerequisite for MATH 448. Prerequisite: MATH 222 with a grade of C or better,
or consent of department.

**MATH 357. OPERATIONS RESEARCH
**Theory and applications of operations research, including linear
programming, mathematical programming and queueing theory. Prerequisites: MATH
222 and 304 with grades of C or better. No computer programming experience is
required.

**MATH 358. NUMERICAL ANALYSIS I
**Floating-point arithmetic, error analysis, root finding, interpolation and
approximation by polynomials, numerical integration and differentiation,
numerical differential equation methods, solutions of linear systems by Gaussian
elimination with pivoting, direct factorization of matrices. Prerequisites: MATH
222 and 304 with grades of C or better, and CS 140 or equivalent.

**MATH 371. MATHEMATICAL METHODS IN SCIENCE I
**Ordinary differential equations. Emphasis on applications to problems in
physics, chemistry, biology, economics, etc. Prerequisite: MATH 222 with a grade
of C or better.

**MATH 375. COMPLEX VARIABLES
**Analytic functions. Cauchy’s integral theorem, power series. Prerequisite:
MATH 323 with a grade of C or better.

**MATH 381. GRAPH THEORY
**Directed and undirected graphs, trees, connectivity, Eulerian and
Hamiltonian graphs, planar graphs, coloring of graphs, graph parameters,
optimization and graph algorithms. Prerequisite: MATH 304, and either MATH 314
or 330 with grades of C or better, or consent of department.

**MATH 386. COMBINATORICS
**Topics from among counting techniques, generating function and recurrence
relations, pigeonhole principle, Ramsey’s Theorem, Latin squares,
combinatorial designs. Prerequisite: MATH 304 and either MATH 314 or 330 with
grades of C or better, or consent of department.

**MATH 391. PRACTICUM IN COLLEGE TEACHING 1 cr.
**Independent study through teaching in particular mathematics course. Various
assignments closely directed by instructor in course, including development of
syllabi and other course materials; construction and reading of examinations;
lecturing and/or discussion leadership; laboratory supervision; academic
counseling of student. May be repeated for total of no more than eight credits.
Credits may not be earned in conjunction with course in which student is
currently enrolled. Does not satisfy major or all-colIege requirements.
Prerequisite: consent of instructor. P/F only.

**MATH 401. MODERN ALGEBRA I
**Groups, rings, integral domains, fields. Prerequisites: MATH 304 and 330
with grades of C or better, or consent of department.

**MATH 402. MODERN ALGEBRA II
**Further study of topics in MATH 401. Vector spaces, modules, lattices,
Galois theory. Prerequisite: MATH 401 with a grade of C or better.

**MATH 404. ADVANCED LINEAR ALGEBRA
**Modules, normal forms of linear transformations, quadratic forms.
Prerequisite: MATH 304 and 330 with grades of C or better, or consent of
department.

**MATH 407. INTRODUCTION TO THE THEORY OF NUMBERS
**Classical number theory. Divisibility, prime numbers, quadratic reciprocity,
Diophantine equations. Prerequisite: MATH 330 with a grade of C or better, or
consent of department.

**MATH 447. INTRODUCTION TO PROBABILITY AND STATISTICS I
**Development of probabilistic concepts, sampling distributions, estimation,
confidence intervals, tests of hypotheses. Prerequisite: MATH 323 with a grade
of C or better, or consent of department.

**MATH 448. INTRODUCTION TO PROBABILITY AND STATISTICS II
**Methods of probability applied to estimation and testing on hypotheses, both
parametric and non-parametric; random variables, limit theorems, Markov chains,
stochastic processes. Prerequisite: MATH 447 with a grade of C or better.

**MATH 461. TOPOLOGY I
**Study of topological spaces. Metric spaces, separation properties,
connectivity, compactness. Prerequisites: MATH 304, 323 and 330 with grades of C
or better, or consent of department.

**MATH 462. TOPOLOGY II
**Topology of the plane, introductory algebraic topology, local connectivity,
applications of topology to analysis. Prerequisite: MATH 461 with a grade of C
or better.

**MATH 465. FOUNDATIONS OF GEOMETRY
**Postulational treatment of geometric systems, including projective, affine
and non-Euclidean geometries. Prerequisites: MATH 304 and 330 with grades of C
or better, or consent of department.

**MATH 471. MATHEMATICAL METHODS IN SCIENCE II
**Vector calculus, Fourier series, partial differential equations, with
emphasis on applications. Prerequisite: MATH 371 with a grade of C or better.

**MATH 478. REAL ANALYSIS I
**Geometry and topology of Rn, functions and limits, calculus of functions on
Rn and higher dimensional spaces. Prerequisites: MATH 304, 323 and 330 with
grades of C or better, or consent of department.

**MATH 479. REAL ANALYSIS II
**Sequences and series of functions, more advanced study of differentiation
and integration. Prerequisite: MATH 478 with a grade of C or better.

**MATH 480. SEMINAR IN ALGEBRA var. cr.
**Current research. Prerequisites: MATH 401 with a grade of C or better and
consent of instructor. May be repeated for credit.

**MATH 488. TOPICS IN HIGHERMATHEMATICS as needed
**Some topic in higher mathematics not normally part of regular curriculum.
Prerequisite: consent of department. May be repeated for credit.

**MATH 497. INDEPENDENT WORK var. cr.
**Individual study under direct supervision of faculty member. Prerequisite:
consent of department. May be repeated for credit with maximum of eight credit
hours of MATH 497 allowed toward major requirements.

**MATH 498. HONORS STUDY IN MATHEMATICS
**Independent studies/research open only to students who have been accepted in
the mathematics honors program. May be repeated for credit, with maximum of four
credit hours of MATH 498 allowed toward major requirements. Prerequisite:
consent of department.

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Graduate

It should be noted that a substantial number of the department’s advanced graduate courses are offered under the "Topics’’ number 590, and are therefore not described. This allows for flexibility and the offering of once-only courses on topics of current research interest. Recent topics have included: geometric topology, differential geometry, dynamical systems, geometric methods in group theory, Lie algebras, group theory, recent developments in knot theory, theoretical computer science, homological algebra, algebraic K-theory, stochastic differential equations, recursive estimation and control theory, sequential analysis, reliability theory, finite state structures and varieties of formal languages.

**MATH 501. PROBABILITY
**Basic probability notions, classical combinatorial methods, conditional
probabilities. Random variables and properties of distributions. Moments, moment
generating functions, covariances, correlations. Transformations, order
statistics. Convergence in probability and large sample properties.
Prerequisite: MATH 323 or equivalent.

**MATH 502. STATISTICAL INFERENCE
**Likelihood functions and sufficient statistics. Theory of estimation;
completeness and UMVU estimators. Blackwell-Rao theorem; information inequality.
MLEs and their asymptotic properties. Confidence intervals. Testing hypotheses
and Neyman-Pearson theory. Introduction to linear models and non-parametric
methods. Prerequisite: MATH 501 or equivalent.

**MATH 503-504. ALGEBRA
**Higher algebra, especially groups, rings, fields and modules. Prerequisites:
MATH 401 and 402, or consent of department.

**MATH 505-506. ANALYSIS
**Real analysis including theory of Lebesque measure, integration and
elementary theory of Banach and Hilbert spaces. Complex analysis. Prerequisites:
MATH 478 and 479, or consent of department.

**MATH 507. LINEAR ALGEBRA AND MATRIX THEORY
**Linear algebra over the complex numbers and finite fields, eigenvectors and
eigenvalues, quadratic forms, normal forms of matrices, selected topics in
matrix theory. Prerequisite: consent of department.

**MATH 508. COMPLEX ANALYSIS
**A rigorous introduction to complex analysis. Rational functions; conformal
maps; Cauchy’s Integral Theorem with applications; representations of analytic
functions as series, products, integrals; topics selected by instructor.
Prerequisite: MATH 479 or consent of department.

**MATH 509. GRADUATE COMPUTER SCIENCE FOR MATHEMATICIANS
**Graduate-level introduction to computer science from mathematician’s point
of view, models of computation, automata theory, programming languages, program
semantics, proof theory for programs. Prerequisite: undergraduate degree in
mathematical sciences.

**MATH 513-514. GENERAL TOPOLOGY
**Topological spaces, metric spaces, separation axioms, compactness,
connectedness, quotient spaces. Topics from geometric topology, including
fundamental group, complexes and homotopy.

**MATH 517-518. ALGEBRAIC TOPOLOGY
**Concept of homotopy, fundamental group, covering spaces, categories and
functors, simplicial complexes, simplicial homology and cohomology, singular
homology and cohomology, cup product structure, CW-complexes, higher homotopy
groups. Prerequisite: MATH 461, 513-514 or equivalent.

**MATH 519. THEORY OF FIBER SPACES
**Various types of fibrations (Serre, Hurewicz, Dold fibrations, fiber
bundles, covering spaces), applications of homotopy theory, topics from
classical theory of bundles, classification theorems, spectral sequences.
Prerequisites: MATH 513, 514, consent of department.

**MATH 520. HOMOLOGICAL ALGEBRA
**Modules, chain complexes, tensor products, derived functors, homology of
groups, other topics selected by the instructor. Prerequisite: MATH 504 or
consent of department.

**MATH 521-522. DIFFERENTIAL TOPOLOGY
**Differentiable manifolds, imbeddings and immersions, Whitney’s imbedding
theorem, tangent and cotangent bundles, Morse theory. Prerequisite: MATH
513-514.

**MATH 523-524. GROUP THEORY
**Properties of groups, extensions, transfer, generators, defining relations.
Prerequisite: MATH 503-504 or equivalent.

**MATH 525-526. RINGS AND ALGEBRAS
**Advanced study of rings and algebras; special topics selected from current
literature. Prerequisite: MATH 503-504 or equivalent.

**MATH 527. REPRESENTATION THEORY
**Representations of groups and rings by linear transformations, characters,
applications in structure theory of groups and rings. Prerequisite: consent of
department.

**MATH 532. ADVANCED NUMERICAL ANALYSIS
**Solution to non-linear equations, differential equations, eigenvalue
problems, finite element method, discretization error, iterative methods,
computer implementation. Prerequisites: undergraduate differential equations,
linear algebra, advanced calculus, some programming experience.

**MATH 537. ANALYSIS OF ALGORITHMS
**Time and space analysis of algorithms for applications such as sorting,
searching, graphics manipulation, pattern matching and algebraic calculation.
Statistical analysis. Empirical analysis of complex algorithms arising in
computer systems. Prerequisite: MATH 509.

**MATH 538. COMPILERS AND FORMAL LANGUAGES
**Formal description of syntax and semantics of computer languages. Transition
from formal description to implementation as compiler or interpreter. Various
languages compared as to their data structures, procedures and input-output.
Prerequisite: MATH 509.

**MATH 545. TOPOLOGICAL GROUPS
**Locally compact topological groups, open homomorphism and closed graph
theorems, measure and integration on locally compact topological groups.
Prerequisite: MATH 505-506, 513-514 or consent of department.

**MATH 547-548. DECOMPOSITION SPACES
**Upper and lower semi-continuous decompositions, properties inherited by
decomposition spaces, applications (in particular to manifolds). Prerequisites:
MATH 513-514 and consent of department.

**MATH 549. KNOT THEORY
**Knots and knot types, presentation of a knot group, combinatorial covering
spaces, absolute calculus, cubes with holes. Prerequisites: MATH 513-514 and
consent of department.

**MATH 551-552. POLYHEDRAL TOPOLOGY
**Regular neighborhood theory, general position, unknotting balls and spheres,
engulfing techniques, handlebody theory and s-cobordism. Prerequisites: MATH
513-514 and consent of department.

**MATH 553. NON-PARAMETRIC INFERENCE
**Order statistics and quantiles, non-parametric confidence intervals,
non-parametric measures of association, tests based on ranks, tests of
independence, symmetry, location differences, chi-square and Kolmogorov-Smirnov
goodness of fit tests, non-parametric regression, robustness, asymptotic
relative efficiency of tests, concepts of non-parametric density estimation.
Prerequisite: MATH 448 or 502.

**MATH 554. SAMPLING FROM FINITE POPULATIONS
**The classical model and sampling strategies. Sampling distributions of
estimators of population quantities. Simple random sampling, stratified
sampling, two-stage and multistage cluster sampling, optimal allocation of
resources, and other design aspects. Sampling inspection techniques for quality
control. Other topics as time permits. Prerequisite: MATH 447 or 501.

**MATH 555. LINEAR MODELS
**Inference in linear models based on the least squares approach: Point
estimation, confidence regions, hypothesis testing, model building and
verification, residual analysis, selection of best regression, influential
observations. Prerequisites: MATH 448 or 502 and MATH 404 or 507.

**MATH 556. DESIGN OF EXPERIMENTS
**The role and principles of DE in scientific research. Reference
distributions, ANOVA, multiple comparisons. Randomized complete block designs,
latin squares, Pn factorial design and the calculus of factorial experiments.
Balanced incomplete block designs, the recovery of intrablock information.
Exploration of response surfaces. Prerequisite: MATH 555.

**MATH 558. MULTIVARIATE STATISTICAL ANALYSIS
**Multivariate normal distributions, Wishart distributions, inferences on
means and covariances, Hotelling’s T2, multivariate linear models, regression,
ANOVA, tests of independence, discriminant analysis, principal components,
canonical correlations and variables, factor analysis. Prerequisite: MATH 555.

**MATH 559. TIME-SERIES ANALYSIS
**Trend analysis and smoothing. Estimation, testing, modeling, and forecasting
for ARMA and ARIMA models. Prerequisite: MATH 555.

**MATH 561. ALGEBRA SEMINAR 1-4 cr.
**Prerequisite: consent of department.

**MATH 564. PROBABILITY SEMINAR 1-4 cr.
**Prerequisite: consent of department.

**MATH 565. TOPOLOGY SEMINAR I 1-4 cr.
**Prerequisite: consent of department.

**MATH 567. TOPOLOGY SEMINAR II 1-4 cr.
**Prerequisite: consent of department.

**MATH 570. APPLIED MULTIVARIATE ANALYSIS
**Multivariate normal distributions, Wishart distributions, Hotelling’s T,
tests of independence, large sample distribution theory, multivariate linear
models, discriminant analysis, factor analysis, principal components and other
selected topics. Prerequisite: MATH 558.

**MATH 571. ADVANCED PROBABILITY THEORY 5 cr.
**Measure theoretic probability. Axiomatic foundations, random variables,
conditional probability and expectation, characteristic functions, infinite
divisibility and stable laws, types of convergence, law of large numbers,
central limit theorem, other topics as time permits. Prerequisite: MATH 447 or
501, and MATH 506 or consent of instructor.

**MATH 572. STOCHASTIC PROCESSES 5 cr.
**A continuation of the subject matter presented in MATH 571. Martingales and
Markov processes (if not covered in MATH 571), orthogonality, stationary
processes, other topics as time permits. Prerequisite: MATH 571.

**MATH 573. APPLIED PROBABILITY AND STOCHASTIC PROCESSES
**Introduction to Markov chains, Markov processes with emphasis on
applications. Classification of states, stationarity. Continuity, integration,
and differentiation of second order processes. Stochastic differential
equations. Prerequisite: MATH 501

**MATH 574. NUMBER THEORY (MAT/MST)
**Elementary number theory, divisibility, fundamental theorem of arithmetic,
prime numbers, quadratic reciprocity, Diophantine equations. Prerequisite:
consent of instructor.

**MATH 575. SPECIAL TOPICS FOR TEACHERS (MAT/MST) 1-4 cr.
**Special topics of interest to teachers. Prerequisite: consent of instructor.

**MATH 576. COMPUTER APPLICATIONS IN MATHEMATICS EDUCATION (MAT/MST)
**Computer usage in education from historical point of view, evaluation of
various levels of computer usage in learning situation (low-key approach,
interactive CAI approach, artificial intelligence approach). Prerequisite:
consent of instructor.

**MATH 577. RECREATIONAL MATHEMATICS (MAT/MST)
**Sources of recreational mathematics, magic squares, dissection problems, map
coloring problems, traversing of mazes, chessboard recreations, instant
insanity, arithmetical and geometrical fallacies. Prerequisite: consent of
instructor.

**MATH 578. COMBINATORICS (MAT/MST)
**Combinations and permutations, enumeration techniques, recursion, sum and
difference sequences, partitions, applications to pre-college mathematics.
Prerequisite: consent of instructor.

**MATH 579. ADVANCED STATISTICAL INFERENCE
**Weak convergence of probability measures on Euclidean spaces. Interval
estimation, point estimation, and hypothesis testing. General decision theory
including the minimax theorem, the complete class theorem, the abstract Rao-Blackwell
theorem, the theorem of Hunt and Stein, and Bayes methods. Asymptotic decision
theory. Prerequisite: MATH 571 and 502.

**MATH 580. TOPICS IN COMBINATORIAL ANALYSIS
**Variable subject matter chosen from field of combinatorial analysis.
Prerequisite: MATH 401. May be repeated for credit with consent of department.

**MATH 581. TOPICS IN GRAPH THEORY
**Theoretical and applied graph theory. Applications including personnel
assignment problem, construction of reliable communications networks, chromatic
polynomials. Prerequisite: MATH 401 or consent of instructor. May be repeated
for credit with consent of department.

**MATH 582. ALGEBRA (MAT/MST)
**Classical theory of equations, algebraic systems (including groups, rings,
fields, modules) and their properties. Prerequisite: consent of instructor.

**MATH 583. METRIC AND AFFINE GEOMETRY (MAT/MST)
**Affine and metric geometry from transformational point of view. Finite and
infinite geometries, Euclidean geometry, applications to pre-college
mathematics. Prerequisite: consent of instructor.

**MATH 584. EUCLIDEAN AND NON-EUCLIDEAN GEOMETRY (MAT/MST)
**Algebraic (analytic) approach to classical geometries (Euclidean,
hyperbolic, projective). Prerequisite: consent of instructor.

**MATH 588. PROBABILITY AND STATISTICS (MAT/MST)
**Finite probability and probability-related statistical problems. Mixture of
formal development and problem solving with applications to pre-college
mathematics. Prerequisite: consent of instructor.

**MATH 589. HISTORY AND CONCEPTUAL DEVELOPMENT OF THE CALCULUS
(MAT/MST)
**Historical and conceptual development of mathematical ideas underlying
modern calculus, including problems of infinity and of continuity as treated in
ancient and modern times. Applications to pre-college mathematics wherever
appropriate.

**MATH 590. TOPICS IN MODERN MATHEMATICS 1-4 cr.
**Study (at graduate level) of some topic in mathematics not a part of regular
graduate curriculum. Content changes from term to term. With consent of
department, students may repeat course for credit. Prerequisite: consent of
department.

**MATH 591. THE TEACHING OF COLLEGE MATHEMATICS 1-4 cr.
**Required for teaching assistants, suggested for graduate assistants
interested in college teaching. Does not count toward required number of courses
for MA or PhD.

**MATH 597. INDEPENDENT WORK 1-4 cr.
**Reading and research on special topic, under direction of adviser. May be
repeated for credit with consent of department. Commonly taught topics under
Independent Work include but are not limited to the following:

• MATH 597A. STUDIES IN MODERN ALGEBRA I

• MATH 597B. STUDIES IN MODERN ALGEBRA II

• MATH 597C. STUDIES IN REAL ANALYSIS I

• MATH 597D. STUDIES IN REAL ANALYSIS II

**MATH 599. THESIS 1-4 cr.**

**MATH 601. TOPICS IN TOPOLOGY
**Variable subject matter chosen from field of topology. May be repeated for
credit with consent of department.

**MATH 603. TOPICS IN ALGEBRA 1-4 cr.
**Variable subject matter chosen from field of algebra. May be repeated for
credit with consent of department.

**MATH 604. ADVANCED TOPICS IN THE THEORY OF GROUPS
**Topics selected from current research. May be repeated for credit with
consent of department.

**MATH 605. SEMINAR IN STATISTICS 1-4 cr.
**Variable subject matter chosen from field of statistics. Topics selected
from current research. May be repeated for credit with consent of department.

**MATH 698. PRE-DISSERTATION RESEARCH 1-9 cr./sem.
**Independent reading and/or research in preparation for comprehensive
examinations for admission to PhD candidacy and/or preparation of dissertation
prospectus. Graded on S/U basis only.

**MATH 699. DISSERTATION 1-12 cr./sem.
**Research for and preparation of the dissertation.

**MATH 700. CONTINUOUS REGISTRATION 1 cr./sem.
**Required for maintenance of matriculated status in graduate program. No
credit toward graduate degree requirements.

**MATH 707. RESEARCH SKILLS 1-4 cr.
**Development of research skills required within graduate programs. May not be
applied toward course credits for any graduate degree. Prerequisite: approval of
relevant graduate program directors or department chairs.

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