The Department of Mathematics provides an environment of research and comprehensive instruction in mathematics and applied mathematics at both undergraduate and graduate levels. Degrees available in mathematics include two baccalaureate degrees, the Bachelor of Arts and the Bachelor of Science (the Bachelor of Science is also available in applied mathematics); and two postgraduate degrees, the Master of Science and the Doctor of Philosophy.

The requirements for a degree in mathematics or in applied mathematics express the educational intent of the Department of Mathematics; they are drawn with an eye toward the cumulative character of an education based in mathematics, the present emerging state of mathematics, and the scholarly and professional prerequisites of an academic career in mathematics.

Requirements for the bachelor's degree look to the advancement of students' general education in modern mathematics and their knowledge of its relation with the other sciences (Bachelor of Science) or with the other arts (Bachelor of Arts).

Descriptions of the detailed requirements that give meaning to these educational intentions follow. Students should understand that any particular degree requirement can be modified if persuasive reasons are presented to the department; petitions to modify degree requirements are submitted in person to the director of undergraduate studies or the departmental counselor.

Placement. At what level does an entering student begin mathematics at the University of Chicago? This question is answered individually for students on the basis of their performance on one of two placement tests in mathematics administered during Orientation in September: either a precalculus mathematics placement test or a calculus placement test. Scores on the mathematics placement test determine the appropriate beginning mathematics course for each student: a precalculus course (Mathematics 10500) or one of three other courses (Mathematics 11200, Mathematics 13100, or Mathematics 15100). Students who wish to begin at a level higher than Mathematics 15100 must take the calculus placement test, unless they receive Advanced Placement credit as described in the following paragraphs.

Students with suitable achievement on the calculus placement test are invited to begin with Honors Calculus (Mathematics 16100) or beyond. Excellent scores on the calculus placement test may give placement credit for one, two, or three quarters of calculus. Admission to Honors Analysis (Mathematics 20700) is by invitation only to those first-year students who show an exceptional performance on the calculus placement test or to those sophomores who receive a strong recommendation from their instructor in Mathematics 16100-16200-16300. Mathematics 25700-25800-25900 is designated as an honors section of Basic Algebra. Registration for this course is the option of the individual student. Consultation with the departmental counselor is strongly advised.

Students who submit a score of 5 on the AB Advanced Placement exam in mathematics or a score of 4 on the BC Advanced Placement exam in mathematics receive credit for Mathematics 15100. Students who submit a score of 5 on the BC Advanced Placement exam in mathematics receive credit for Mathematics 15100 and 15200.

Note: Mathematics students may use AP credit for chemistry or physics to meet their general education physical sciences requirement or the physical sciences component of the concentration. However, no credit designated simply as "physical science," from AP examinations or from the College's physical sciences placement examination, may be used in their general education requirement or in the concentration.

Students are required to complete both a 10000-level sequence in calculus (or to demonstrate equivalent competence on the calculus placement test) and a three-quarter sequence in analysis (Mathematics 20300-20400-20500 or 20700-20800-20900), and two quarters of a sequence in algebra (Mathematics 25400-25500 or 25700-25800). The normal procedure is to take calculus in the first year and analysis in the second.

Concentrators in mathematics or applied mathematics may take any 20000-level mathematics courses elected beyond concentration requirements for a grade of P. However, a grade of C- or better must be earned in each calculus, analysis, or algebra course, and an overall grade average of C or better must be earned in the remaining mathematics courses that a student uses to meet concentration requirements. Courses in the Physical Sciences Collegiate Division that are used to meet concentration requirements in mathematics must be taken for quality grades.

Students taking a bachelor's degree in mathematics or in applied mathematics should note that by judicious employment of courses from another field for extradepartmental requirements or for electives, a minor field can be developed that is often in itself a sufficient base for graduate or professional work in their field. A notable example is furnished by the field of statistics: the core programs are Statistics 24200 and 25100 for probability theory and Statistics 24400 and 24500 for statistical theory. For an emphasis on statistical methods, students would add Statistics 22200, 22400, or 22600; while for an emphasis on probability they would add Statistics 31200 or perhaps Statistics 38100-38200.

What is noted here for statistics can also be applied to computer science (consult following section), chemistry, geophysical sciences, physics, biophysics, theoretical biology, economics, and education.

While these remarks apply to all bachelor's degree programs in the Department of Mathematics, their force is particularly evident in programs looking to bachelor's degrees in applied mathematics, where minor fields are strongly urged.

Degree Programs in Mathematics. Candidates for the B.A. and B.S. in mathematics all take a sequence in basic algebra. Candidates for the B.S. degree must take the three-quarter sequence (Mathematics 25400-25500-25600 or Mathematics 25700-25800-25900), whereas B.A. candidates may opt for a two-quarter sequence (Mathematics 25400-25500 or Mathematics 25700-25800). The remaining mathematics courses needed in the concentration programs (three for the B.A., two for the B.S.) must be selected, with due regard for prerequisites, from the following list: Mathematics 17500, 21100, 24100, 24200, 26100, 26200, 26300, 27000, 27200, 27300, 27400, 27500, 27700, 27800, 27900, 28000, 28100, 28400, 29200, 29700 (as approved), 30000, 30100, 30200, 30300, 30900, 31000, 31200, 31300, 31400, 31700, 31800, 31900, 32500, 32600, 32700, and Statistics 25100. B.A. candidates may include Mathematics 25600 or 25900.

B.S. candidates are further required to select a minor field, which consists of an additional three-course sequence, which is outside the Department of Mathematics but within the Physical Sciences Collegiate Division, chosen in consultation with the departmental counselor.

General CHEM 11100-11200 or higher†,

Education or PHYS 12100-12200 or higher†

MATH 13100-13200, 15100-15200, or 16100-16200†

Concentration 1 CHEM 11300 or higher†,

or PHYS 12300 or higher†

1 MATH 13300, 15300, or 16300†

3 MATH 20300-20400-20500 or

20700-20800-20900

2 courses in mathematics chosen from
an approved list

4 courses within the PSCD but outside of

mathematics, at least two of which should

form a sequence in a single department

† Credit may be granted by examination.

Degree Program in Applied Mathematics. Candidates for the B.S. in applied mathematics all take prescribed courses in numerical analysis, algebra, complex variables, ordinary differential equations, and partial differential equations. In addition, candidates are required to select, in consultation with the departmental counselor, a minor field, which consists of a three-course sequence that is outside the Department of Mathematics but within the Physical Sciences Collegiate Division.

General CHEM 11100-11200 or higher†,

Education or PHYS 12100-12200 or higher†

MATH 13100-13200, 15100-15200,

or 16100-16200†

Concentration 1 CHEM 11300 or higher†,

or PHYS 12300 or higher†

1 MATH 13300, 15300, or 16300†

3 MATH 20300-20400-20500

or 20700-20800-20900

1 MATH 21100

2 MATH 25400-25500 or 25700-25800

3 MATH 27000-27300-27500

6 courses within the PSCD but outside of

mathematics, at least three of which

should form a sequence from a single

department

17

† Credit may be granted by examination.

Degree Program in Mathematics with Specialization in Computer Science. The concentration program leading to a B.S. in mathematics with a specialization in computer science is a version of the B.S. in mathematics. The degree is in mathematics with the designation "with specialization in computer science" included on the final transcript. Candidates are required to complete a yearlong sequence in calculus (Mathematics 15100-15200-15300 or 16100-16200-16300 strongly recommended), in analysis (Mathematics 20300-20400-20500 or 20700-20800-20900), and in abstract algebra (Mathematics 25400-25500-25600 or 25700-25800-25900), and earn a grade of at least C- in each course. The remaining two mathematics courses may be chosen from the list of approved courses in the section Degree Programs in Mathematics except for Mathematics 17500 and Statistics 25100; students are urged to take at least one of Mathematics 24200, 26200, 27700, or 28400. A C average or better must be earned in these two courses.

Besides the third quarter of basic chemistry or basic physics, the seven courses required outside the Department of Mathematics must all be in the computer science department. A two-quarter sequence in programming is required; Computer Science 11500-11600 is recommended. (Students may substitute Computer Science 10500-10600, although this is not encouraged.) Five additional courses must be selected from among computer science courses numbered 20000 or higher, except 27400. Students who take Computer Science 10500-10600 are encouraged to include Computer Science 22100 here. Students must earn a grade of C or better in each course taken in computer science to be eligible for this degree. For more information, consult the Computer Science section of this catalog.

General CHEM 11100-11200 or higher†,

Education or PHYS 12100-12200 or higher†

MATH 13100-13200, 15100-15200, or 16100-16200†

Concentration 1 CHEM 11300 or higher†,

or PHYS 12300 or higher†

1 MATH 13300, 15300, or 16300†

3 MATH 20300-20400-20500

or 20700-20800-20900

3 MATH 25400-25500-25600

or 25700-25800-25900

2 CMSC 11500-11600

2 approved courses in mathematics

5 approved courses in computer science

17

† Credit may be granted by examination.

Degree Program in Mathematics with Specialization in Economics. This concentration program is a version of the B.S. in mathematics. The B.S. degree is in mathematics with the designation "with specialization in economics" included on the final transcript. Candidates are required to complete a yearlong sequence in calculus, in analysis (Mathematics 20300-20400-20500 or 20700-20800-20900), and two quarters of abstract algebra (Mathematics 25400-25500 or 25700-25800), and earn a grade of at least C- in each course. Students must also take Statistics 25100 (Probability). The remaining two mathematics courses must include Mathematics 27000 (Complex Variables) and either Mathematics 27200 (Functional Analysis) for those interested in Econometrics or Mathematics 27300 (Ordinary Differential Equations) for those interested in economic theory. A C average or better must be earned in these two courses.

Besides the third quarter of basic chemistry or basic physics, the eight courses required outside the Department of Mathematics must include Statistics 22000 or 24400. The remaining seven courses should be in the economics department and must include Economics 20000-20100-20200-20300 and Economics 20900 or 21000 (Econometrics). The remaining two courses may be chosen from any undergraduate economics course numbered higher than Economics 20300. Students must earn a grade of C or better in each course taken in economics to be eligible for this degree.

It is recommended that students considering graduate work in economics use some of their electives to include at least one programming course (Computer Science 11500 is strongly recommended), and an additional course in statistics (Statistics 24400-24500 is an appropriate two-quarter sequence). Students planning to apply to graduate economics programs are strongly encouraged to meet with one of the economics undergraduate program directors before the beginning of their third year.

General CHEM 11100-11200 or higher†,

Education or PHYS 12100-12200 or higher†

MATH 13100-13200, 15100-15200, or 16100-16200†

Concentration 1 CHEM 11300 or higher†,

or PHYS 12300 or higher†

1 MATH 13300, 15300, or 16300†

3 MATH 20300-20400-20500

or 20700-20800-20900

2 MATH 25400-25500 or 25700-25800

1 MATH 27000

1 MATH 27200 or 27300

1 STAT 25100

1 STAT 22000 or 24400

4 ECON 20000-20100-20200-20300

1 ECON 20900 or 21000

2 courses in economics

numbered higher than 20300

18

† Credit may be granted by examination.

Grading. Subject to College and concentration requirements and with the consent of the instructor, all students, except concentrators in mathematics or applied mathematics, may register for regular letter grades, P/N grades, or P/F grades in any course beyond the second quarter of calculus. A Pass grade is given only for work of C- or better.

Concentrators in mathematics or applied mathematics may take any 20000-level mathematics courses elected beyond concentration requirements for a grade of P. However, a grade of C- or better must be earned in each calculus, analysis, or algebra course, and an overall grade average of C or better must be earned in the remaining mathematics courses that a student uses to meet concentration requirements. PSCD courses taken to meet concentration requirements in mathematics must be taken for quality grades.

Incompletes are given in the Department of Mathematics only to those students who have done some work of passing quality and who are unable to complete all the course work by the end of the quarter. Arrangements are made between the instructor and the student.

Honors. The B.A. or B.S. with honors is awarded to students who meet the following requirements: (1) a grade point average of 3.25 or better in concentration courses and a 3.0 or better overall; (2) completion of one honors sequence (either Mathematics 20700-20800-20900 or Mathematics 25700-25800-25900) with grades of B- or better in each quarter; and (3) completion with a grade of B- or better of at least five additional mathematics courses chosen from the list that follows so that at least one course comes from each group (algebra, analysis, and topology).

Algebra courses: Mathematics 24100, 24200, 25700, 25800, 25900, 27700, 27800, 28400, 32500, 32600, 32700

Analysis courses: Mathematics 20700, 20800, 20900, 27000, 27200, 27300, 27400, 27500, 31200, 31300, 31400, 32100, 32200, 32300

Topology courses: Mathematics 26200, 26300, 31700, 31800, 31900

As approved, Mathematics 29700 (Proseminar in Mathematics) may be chosen so that it falls in any of the three groups. Students interested in the honors degree should consult with the departmental counselor no later than the third quarter of their third year.

Jonathan L. Alperin, Professor, Department of Mathematics and the College

LASZLO BABAI, Professor, Departments of Computer Science and Mathematics, and the College

Walter L. Baily, JR., Professor, Department of Mathematics and the College

guillaume bal, L. E. Dickson Instructor, Department of Mathematics and the College

EVGUENI BALKOVSKI, L. E. Dickson Instructor, Department of Mathematics and the College

ALEXANDER BEILINSON, David and Mary Winton Green University Professor, Department of Mathematics and the College

david ben-zvi, L. E. Dickson Instructor, Department of Mathematics and the College

Roman bezrukavnikov, Assistant Professor, Department of Mathematics and the College

Spencer J. Bloch, Robert Maynard Hutchins Distinguished Service Professor, Department of Mathematics and the College

JAMES BORGER, VIGRE Dickson Instructor, Department of Mathematics and the College

Jeffrey brock, Assistant Professor, Department of Mathematics and the College

fausto cattaneo, Assistant Professor, Department of Mathematics and the College

Christopher connell, L. E. Dickson Instructor, Department of Mathematics and the College

Peter Constantin, Professor, Department of Mathematics and the College

Kevin Corlette, Professor, Department of Mathematics and the College; Chairman, Department of Mathematics

Jack D. Cowan, Professor, Department of Mathematics and the College

VLADIMIR DRINFELD, Professor, Department of Mathematics and the College

Todd Dupont, Professor, Departments of Computer Science and Mathematics, and the College

matthew emerton, Assistant Professor, Department of Mathematics and the College

Alex Eskin, Professor, Department of Mathematics and the College

Benson Farb, Professor, Department of Mathematics and the College

Robert A. Fefferman, Louis Block Professor in the Physical Sciences, Department of Mathematics and the College

dennis gaitsgory, Associate Professor, Department of Mathematics and the College

Victor Ginzburg, Professor, Department of Mathematics and the College

George I. Glauberman, Professor, Department of Mathematics and the College

JESPER GRODAL, L. E. Dickson Instructor, Department of Mathematics and the College

DANIEL GROSSMAN, VIGRE Dickson Instructor, Department of Mathematics and the College

Diane L. Herrmann, Senior Lecturer, Department of Mathematics and the College

DENIS HIRSCHFELDT, L. E. Dickson Instructor, Department of Mathematics and the College

Sharon Hollander, VIGRE Dickson Instructor, Department of Mathematics and the College

PO HU, Assistant Professor, Department of Mathematics and the College

Leo P. KadaNOFF, John D. MacArthur Distinguished Service Professor, Departments of Physics and Mathematics, James Franck Institute, Enrico Fermi Institute, and the College

Carlos E. Kenig, Louis Block Professor, Department of Mathematics and the College

ROBERT KIRBY, L. E. Dickson Instructor, Departments of Computer Science and Mathematics, and the College

Alexander Kiselev, Assistant Professor, Department of Mathematics and the College

KENNETH KOENIG, L. E. Dickson Instructor, Department of Mathematics and the College

Robert Kottwitz, Professor, Department of Mathematics and the College

Norman R. Lebovitz, Professor, Department of Mathematics and the College

Arunas L. Liulevicius, Professor Emeritus, Department of Mathematics and the College

michael mandell, Assistant Professor, Department of Mathematics and the College

J. Peter May, Professor, Department of Mathematics and the College

NICOLAS MONOD, L. E. Dickson Instructor, Department of Mathematics and the College

Matam P. Murthy, Professor, Department of Mathematics and the College

NIKOLAI NADIRASHVILI, Professor, Department of Mathematics and the College

DAVID E. NADLER, VIGRE Dickson Instructor, Department of Mathematics and the College

Raghavan Narasimhan, Professor, Department of Mathematics and the College

Andrei Nies, Associate Professor, Department of Mathematics and the College

Madhav Nori, Professor, Department of Mathematics and the College

Niels O. Nygaard, Professor, Department of Mathematics and the College

robert pollack, VIGRE Dickson Instructor, Department of Mathematics and the College

Melvin G. Rothenberg, Professor Emeritus, Department of Mathematics

Leonid V. Ryzhik, Assistant Professor, Department of Mathematics and the College

Paul J. Sally, Jr., Professor, Department of Mathematics and the College

L. Ridgway Scott, Professor, Departments of Computer Science and Mathematics, and the College

Robert I. Soare, Paul Snowden Russell Distinguished Service Professor, Departments of Computer Science and Mathematics, and the College

Shankar C. Venkataramani, Assistant Professor, Department of Mathematics and the College

VADIM VOLOGODSKY, L. E. Dickson Instructor, Department of Mathematics and the College

Sidney Webster, Professor, Department of Mathematics and the College

SHMUEL WEINBERGER, Professor, Department of Mathematics and the College

KEVIN WHYTE, L. E. Dickson Instructor, Department of Mathematics and the College

Robert J. Zimmer, Max Mason Distinguished Service Professor, Department of Mathematics and the College

andrzej zuk, Assistant Professor, Department of Mathematics and the College

L refers to courses with a laboratory.

10500-10600. Fundamental Mathematics I, II. PQ: Adequate performance on the mathematics placement test. Students may not receive grades of P/N or P/F in this sequence. Students who place into this course must take it as first-year students. This two-course sequence covers basic precalculus topics. The autumn quarter course is concerned with elements of algebra, coordinate geometry, and elementary functions. The winter quarter course continues with algebraic, circular, and exponential functions. Staff. Autumn, Winter.

11200. Studies in Mathematics. PQ: MATH 10200 or 10600, or placement into MATH 13100 or higher. This course meets the general education requirement in mathematical sciences. This course covers the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. The first part addresses number theory, including a study of the rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. The second part's main topic is symmetry and geometry, including a study of polygons, Euclidean construction, polyhedra, group theory, and topology. The course emphasizes the understanding of ideas and the ability to express them through mathematical arguments. The course is at the level of difficulty of the MATH 13100-13200-13300 calculus sequence. Staff. Autumn, Spring.

13100-13200-13300. Elementary Functions and Calculus I, II, III. PQ: Invitation only based on appropriate performance on the mathematics placement test or MATH 10200 or 10600. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. MATH 13100-13200 meets the general education requirement in mathematical sciences. This sequence provides the opportunity for students who are somewhat deficient in their precalculus preparation to complete the necessary background and cover basic calculus in three quarters. This is achieved through three regular one-hour class meetings and two mandatory one-and-one-half hour tutorial sessions each week. A class is divided into tutorial groups of about eight students each, and these meet with an undergraduate junior tutor for problem solving related to the course. The autumn quarter component of this sequence covers real numbers (algebraic and order properties), coordinate geometry of the plane (circles and lines), and real functions, and introduces the derivative. Topics examined in the winter quarter include differentiation, applications of the definite integral and the fundamental theorem, and antidifferentiation. In the spring quarter, subjects include exponential and logarithmic functions, trigonometric functions, more applications of the definite integral, and Taylor expansions. Students are expected to understand the definitions of key concepts (limit, derivative, and integral) and to be able to apply definitions and theorems to solve problems. In particular, all calculus courses require students to do proofs. Students completing MATH 13100-13200-13300 have a command of calculus equivalent to that obtained in MATH 15100-15200-15300. MATH 13300 is only offered in the spring quarter. Staff. Autumn, Winter, Spring.

15100-15200-15300. Calculus I, II, III. PQ: Superior performance on the mathematics placement test, or MATH 10200 or 10600. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. MATH 15100-15200 meets the general education requirement in mathematical sciences. This is the regular calculus sequence in the department. Students entering this sequence are to have mastered appropriate precalculus material and, in many cases, have had some previous experience with calculus in high school or elsewhere. MATH 15100 undertakes a careful treatment of limits, the differentiation of algebraic and transcendental functions, and applications. Work in MATH 15200 is concerned with integration and additional techniques of integration. MATH 15300 deals with techniques and theoretical considerations, infinite series, and Taylor expansions. MATH 15100 is offered only in the autumn quarter. Staff. Autumn, Winter, Spring.

16100-16200-16300. Honors Calculus I, II, III. PQ: Invitation only based on an outstanding performance on the calculus placement test. Students may not receive grades of P/N or P/F in the first two quarters of this sequence. MATH 16100-16200 meets the general education requirement in mathematical sciences. MATH 16100-16200-16300 is an honors version of MATH 15100-15200-15300. A student with a strong background in the problem-solving aspects of one-variable calculus may, by suitable achievement on the calculus placement test, be permitted to register for MATH 16100-16200-16300. This sequence emphasizes the theoretical aspects of one-variable analysis and, in particular, the consequences of completeness in the real number system. Staff. Autumn, Winter, Spring.

17500. Elementary Number Theory. PQ: Two quarters of calculus. This course covers basic properties of the integers following from the division algorithm, primes and their distribution, congruences, existence of primitive roots, arithmetic functions, quadratic reciprocity, and other topics. Some transcendental numbers are covered. The subject is developed in a leisurely fashion, with many explicit examples. Staff. Autumn.

19500-19600. Mathematical Methods for Biological or Social Sciences I, II. PQ: MATH 15300 or equivalent. This sequence includes some linear algebra and three-dimensional geometry, a review of one-variable calculus, ordinary differential equations, partial derivatives, multiple integrals, partial differential equations, sequences, and series. Staff. Summer; Autumn, Winter; Winter, Spring.

20000-20100-20200. Mathematical Methods for Physical Sciences I, II, III. PQ: MATH 15300 or equivalent. Entering students who have placement for MATH 15100-15200 may begin MATH 20000; such students have the requirement for MATH 15300 waived, but do not receive placement for MATH 15300. This sequence is designed for students intending to major in the physical sciences (other than mathematics). MATH 20000 covers linear algebra and multivariable calculus. Topics include linear systems of equations, vector spaces, matrices, eigenvalue problems, partial derivatives, minimum and maximum problems, coordinate transformations, and multiple integrals. MATH 20100 deals with vector differential calculus, line integrals, theorems of Green, Gauss, and Stokes, complex numbers, introduction to ordinary differential equations, Fourier series, and partial differential equations. MATH 20200 is concerned with functions of a complex variable, Laplace and Fourier transforms, and an introduction to tensor calculus. Staff. Autumn, Winter, Spring; Winter, Spring.

21100. Basic Numerical Analysis. PQ: MATH 20000 or 20300. This course covers direct and iterative methods of solution of linear algebraic equations and eigenvalue problems. Topics include numerical differentiation and quadrature for functions of a single variable; approximation by polynomials and piece-wise polynomial functions; approximate solution of ordinary differential equations; and solution of nonlinear equations. Staff. Spring.

22000. Introduction to Mathematical Methods in Physics. PQ: MATH 15200 or 16200, and PHYS 13200. This course is required for prospective Physics concentrators with concurrent enrollment in PHYS 13300. Topics include infinite and power series, complex numbers, linear equations and matrices, partial differentiation, multiple integrals, vector analysis, and Fourier series. Applications of these methods include Maxwell’s equations, wave packets, and coupled oscillators. Staff. Spring.

22100. Mathematical Methods in Physics. PQ: MATH 22000 and PHYS 13300, or PHYS 14200 and 14300. This course is required for all Physics concentrators. Topics include ordinary and partial differential equations, calculus of variations, coordinate transformations, series solutions and orthogonal functions, integral transforms, and elements of complex analysis. Staff. Autumn.

24100. Topics in Geometry. PQ: MATH 25500. This course focuses on the interplay between abstract algebra (group theory, linear algebra, and the like) and geometry. Several of the following topics are covered: affine geometry, projective geometry, bilinear forms, orthogonal geometry, and symplectic geometry. Not offered 2001-02; will be offered 2002-03.

24200. Algebraic Number Theory. PQ: MATH 25500. Factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree are covered. Staff. Spring.

25000. Elementary Linear Algebra. PQ: MATH 15200 or equivalent. This course takes a concrete approach to the subject and includes some applications in the physical and social sciences. Topics covered in the course include the theory of vector spaces and linear transformations, matrices and determinants, and characteristic roots and similarity. Staff. Autumn, Spring.

25400-25500-25600. Basic Algebra I, II, III. PQ: MATH 13300 or 15300. This sequence covers groups, subgroups, and permutation groups; rings and ideals; some work on fields; vector spaces, linear transformations and matrices, and modules; and canonical forms of matrices, quadratic forms, and multilinear algebra. MATH 25600 is offered only in spring quarter. Staff. Autumn, Winter, Spring; Winter, Spring (MATH 25400-25500).

25700-25800-25900. Honors Basic Algebra I, II, III. PQ: MATH 15300 or 16300. This is an accelerated version of MATH 25400-25500-25600. Topics include the theory of finite groups, commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms. The course also covers basic field theory, the structure of padic fields, and Galois theory. Staff. Autumn, Winter, Spring.

26100. Set Theory and Metric Spaces. PQ: MATH 25400, or 20300 and 25000. This course covers sets, relations, and functions; partially ordered sets; cardinal numbers; Zorn's lemma, well-ordering, and the axiom of choice; metric spaces; and completeness, compactness, and separability. Staff. Autumn.

26200. Point-Set Topology. PQ: MATH 20300 and 25400. This course examines topology on the real line, topological spaces, connected spaces and compact spaces, identification spaces and cell complexes, and projective and other spaces. With MATH 27400, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Winter.

26300. Introduction to Algebraic Topology. PQ: MATH 26200. Some of the topics covered are the fundamental group of a space; Van Kampen's theorem; covering spaces and groups of covering transformation; existence of universal covering spaces built up out of cells; and theorems of Gauss, Brouwer, and Borsuk-Ulam. Staff. Spring.

27000. Basic Complex Variables. PQ: MATH 20500. Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping. Staff. Autumn, Spring.

27300. Basic Theory of Ordinary Differential Equations. PQ: MATH 20200 or 22100 or 27000. This course covers first-order equations and inequalities, Lipschitz condition and uniqueness, properties of linear equations, linear independence, Wronskians, variation-of-constants formula, equations with constant coefficients and Laplace transforms, analytic coefficients, solutions in series, regular singular points, existence theorems, theory of two-point value problem, and Green's functions. Staff. Winter.

27400. Introduction to Differentiable Manifolds and Integration on Manifolds. PQ: MATH 27200. Topics include exterior algebra, differentiable manifolds and their basic properties, differential forms, integration on manifolds, Stoke's theorem, DeRham's theorem, and Sard's theorem. With MATH 26200, this course forms a foundation for all advanced courses in analysis, geometry, and topology. Staff. Spring.

27500. Basic Theory of Partial Differential Equations. PQ: MATH 27300. This course covers classification of second-order equations in two variables, wave motion and Fourier series, heat flow and Fourier integral, Laplace's equation and complex variables, second-order equations in more than two variables, Laplace operators, spherical harmonics, and associated special functions of mathematical physics. Staff. Spring.

27700. Mathematical Logic I (=CMSC 27700, MATH 27700). PQ: MATH 25400. This course provides an introduction to mathematical logic. Topics include propositional and predicate logic and the syntactic notion of proof versus the semantic notion of truth, including soundness and completeness. We also discuss the Goedel completeness theorem, the compactness theorem, and applications of compactness to algebraic problems. Staff. Autumn.

27800. Mathematical Logic II (=CMSC 27800, MATH 27800). PQ: MATH 27700 or equivalent. Some of the topics examined are number theory, Peano arithmetic, Turing compatibility, unsolvable problems, Gödel's incompleteness theorem, undecidable theories (e.g., the theory of groups), quantifier elimination, and decidable theories (e.g., the theory of algebraically closed fields). Staff. Winter.

27900. Logic and Logic Programming (=CMSC 21500, MATH 27900). PQ: MATH 25400, or CMSC 27700, or consent of instructor. Programming knowledge not required. Predicate logic is a precise logical system developed to formally express mathematical reasoning. Prolog is a computer language intended to implement a portion of predicate logic. This course covers both predicate logic and Prolog, which are presented to complement each other and to illustrate the principles of logic programming and automated theorem proving. Topics include syntax and semantics of propositional and predicate logic, tableaux proofs, resolution, Skolemization, Herbrand's theorem, unification, and refining resolution. It includes weekly classes and programming assignments in Prolog (e.g., searching, backtracking, and cut). This course overlaps only slightly with MATH 27700; students are encouraged to take both courses. This course offered in alternate years. R. Soare. Winter.

28000. Introduction to Formal Languages (=CMSC 28000, MATH 28000). PQ: MATH 25000 or 25500, and experience with mathematical proofs. Topics include automata theory, regular languages, CFL's, and Turing machines. This course is a basic introduction to computability theory and formal languages. Staff. Autumn.

28100. Introduction to Complexity Theory (=CMSC 28100, MATH 28100). PQ: MATH 25000 or 25500, and experience with mathematical proofs. Computability topics are discussed, including the s-m-n theorem and the recursion theorem. We also discuss resource-bounded computation. This course introduces complexity theory. Relationships between space and time, determinism and nondeterminism, NP-completeness, and the P versus NP question are investigated. This course is offered in alternate years. Not offered 2001-02; will be offered 2002-03.

28400. Honors Combinatorics and Probability (=CMSC 27400, MATH 28400). PQ: MATH 25000 or 25400, or CMSC 17400, or consent of instructor. Experience with mathematical proofs. Methods of enumeration, construction, and proof of existence of discrete structures are discussed in conjunction with the basic concepts of probability theory over a finite sample space. Enumeration techniques are applied to the calculation of probabilities and, conversely, probabilistic arguments are used in the analysis of combinatorial structures. Topics include basic counting, linear recurrences, generating functions, extremal set systems, Latin squares, finite projective planes, graph theory, Ramsey theory, coloring graphs and set systems, random variables, independence, expected value, standard deviation, Chebyshev's and Chernoff's inequalities, the structure of random graphs and tournaments, probabilistic proofs of existence, and linear algebra methods to prove inequalities in combinatorics and geometry. L. Babai. Spring.

29200. Chaos, Complexity, and Computers (=CMSC 27900, MATH 29200, PHYS 25100). PQ: One year of calculus and two quarters of physics at any level. Knowledge of computer programming not required. In this course we use the computer to investigate the question of how patterns and complexity arise in nature. The systems studied are drawn from physics, biology, and other areas of science. This course also is intended to be an introduction to the use of computers in the physical sciences. Staff. Winter. L.

29700. Proseminar in Mathematics. PQ: General education mathematics sequence. Consent of instructor and departmental counselor. Open to mathematics concentrators only. Students are required to submit the College Reading and Research Course Form. Must be taken for a letter grade. Staff. Autumn, Winter, Spring.

30000-30100. Set Theory I, II. PQ: Consent of instructor. MATH 30000 is a course on axiomatic set theory with applications to the undecidability of mathematical statements. Topics include axioms of Zermelo-Fraenkel (ZF) set theory; Von Neumann rank and reflection principles; the Levy hierarchy and absoluteness; inner models; Gödel's Constructible sets (L), the consistency of the Axiom of Choice (AC), and the Generalized Continuum Hypothesis (GCH); and Souslin's Hypothesis in L. MATH 30100 deals with models of set theory coding of syntax; Cohen's method of forcing and the unprovability of AC and GCH; Martin's axiom and the unprovability of Souslin's Hypothesis; Solovay's model in which every set of reals is Lebesgue Measurable; inaccessible and measurable cardinals; and analytic determinateness, Silver indiscernibles for L (O-Sharp), larger cardinals (elementary embeddings and compactness), and the axiom of determinateness. Staff. Winter, Spring.

30200. Computability Theory I (=CMSC 38000, MATH 30200). PQ: MATH 25500 or consent of instructor. MATH 30200 begins with models for defining computable functions such as the recursive functions and those computable by a Turing machine. Topics include the Kleene normal form theorem for representing computable functions and computably enumerable (c.e.) sets; the enumeration and s-m-n theorem, unsolvable problems, classification of c.e. sets, the Kleene arithmetic hierarchy, coding of information from one set to another, various degrees for measuring noncomputability, many-one, truth-table, and Turing degrees. The course also includes the Kleene recursion theorem and its applications, other fixed point theorems such as the Arslanov completeness criterion, elementary properties of Turing degrees, generic sets, and the construction of various non-c.e. degrees by oracle Kleene-Post constructions. Staff. Winter.

30300. Computability Theory II (=CMSC 38100, MATH 30300). PQ: MATH 30200. MATH 30300 develops the deeper properties of computability and the classification of relative computability on sets and (Turing) degrees. It begins with the finite injury priority method of Friedberg and Muchnik, continues with the infinite injury priority method of Sacks, and minimal pair of computably enumerable (c.e) degrees method by Lachlan. It introduces the tree method of Lachlan for classifying more difficult priority constructions, and it works out many properties of the c.e. degrees and the algebraic structure of the c.e. sets. It presents results on the relationship between a c.e. set and the degree of information it encodes such as the high maximal set theorem of Martin. R. Soare. Spring.

30500. Computability and Complexity Theory (=CMSC 38500, MATH 30500). PQ: Consent of instructor. Part one consists of models for defining computable functions, such as primitive recursive functions, (general) recursive functions, and Turing machines; and their equivalence, the Church-Turing Thesis, unsolvable problems, diagonalization, and properties of computably enumerable (c.e.) sets. Part two deals with Kolmogorov complexity (resource bounded complexity) that studies the quantity of information in individual objects and uses the book by Li and Vitanyi. The third part covers functions computable with special bounds on time and space of the Turing machine, such as polynomial time computability, the classes P and NP, nondeterministic Turing machines, NP-complete problems, polynomial time hierarchy, and P-space complete problems. Staff. Autumn. Not offered 2001-02; will be offered 2002-03.

30900-31000. Model Theory I, II. PQ: MATH 25500. MATH 30900 covers completeness and compactness; elimination of quantifiers; omission of types; elementary chains and homogeneous models; two cardinal theorems by Vaught, Chang, and Keisler; categories and functors; inverse systems of compact Hausdorf spaces; and applications of model theory to algebra. In MATH 31000, the following subjects are studied: saturated models; categoricity in power; the Cantor-Bendixson and Morley derivatives; the Morley theorem and the Baldwin-Lachlan theorem on categoricity; rank in model theory; uniqueness of prime models and existence of saturated models; indiscernibles; ultraproducts; and differential fields of characteristic zero. Not offered 2001-02; will be offered 2002-03.

31700-31800-31900. Topology and Geometry I, II, III. PQ: Consent of director or associate director of undergraduate studies. MATH 31700 covers smooth manifolds, tangent bundles, vector fields, Frobenius theorem, Sard's theorem, Whitney embedding theorem, and transversality. MATH 31800 considers fundamental group and covering spaces; Lie groups and Lie algebras; and principal bundles, connections, introduction to Riemannian geometry, geodesics, and curvature. Topics in MATH 31900 are cell complexes, homology, and cohomology; and Mayer-Vietoris theorem, Kunneth theorem, cup products, duality, and geometric applications. Staff. Autumn, Winter, Spring.

32000-32100-32200. Mathematical and Statistical Methods for the Neuro-Sciences I, II, III (MATH 32100=STAT 24700/31000). PQ: Students must have completed the equivalent of one year of college calculus and a course in linear algebra such as MATH 25000 and preferably a course in differential equations such as MATH 27300, and at least one course in neurobiology such as BIOS 14106 or 24236, or NURB 31800. This three-quarter sequence is for students interested in computational and theoretical neuroscience. It introduces various mathematical and statistical ideas and techniques used in the analysis of brain mechanisms. The first quarter introduces mathematical ideas and techniques in a neuroscience context. Topics include some coverage of matrices and complex variables; eigenvalue problems, spectral methods, and Greens functions for differential equations; and some discussion of both deterministic and probabilistic modeling in the neurosciences. The second quarter treats statistical methods that are important in understanding nervous system function. It includes basic concepts of mathematical probability; and information theory, discrete Markov processes, and time series. The third quarter covers more advanced topics that include perturbation and bifurcation methods for the study of dynamical systems, symmetry, methods, and some group theory. A variety of applications to neuroscience are described. Staff. Autumn, Winter, Spring.

32500-32600-32700. Algebra I, II, III. PQ: MATH 25400-25500-25600, and consent of director or associate director of undergraduate studies. MATH 32500 deals with groups and commutative rings. MATH 32600 investigates elements of the theory of fields and of Galois theory, as well as noncommutative rings. MATH 32700 introduces other basic topics in algebra. Staff. Autumn, Winter, Spring.