Mathematics
Departmental Counselor: Diane L. Herrmann, E 212, 7027332, diane@math.uchicago.edu
Director of Undergraduate Studies: Paul Sally, Ry 350, 7028535, sally@math.uchicago.edu
Associate Director of Undergraduate Studies: Diane L. Herrmann, E 212,
7027332, diane@math.uchicago.edu
Secretary for Undergraduate Studies: Stephanie Walthes, E 211, 7027331, steph@math.uchicago.edu
World Wide Web: http://www.math.uchicago.edu/undergrad/
Program of Study
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 firstyear 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 161001620016300. Mathematics 257002580025900 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.
Program Requirements
Undergraduate Programs. Five bachelor's degrees are available in the Department of Mathematics: the Bachelor of Arts in mathematics, the Bachelor of Science in mathematics, the Bachelor of Science in applied mathematics, the Bachelor of Science in mathematics with specialization in computer science, and the Bachelor of Science in mathematics with specialization in economics. Course programs qualifying students for the degree of Bachelor of Arts provide more elective freedom, while programs qualifying students for the degrees of Bachelor of Science require more emphasis in the physical sciences. All degree programs, whether qualifying students for a degree in mathematics or in applied mathematics, require fulfillment of the College's general education requirements. The general education sequence in the physical sciences must be selected from either firstyear basic chemistry or firstyear basic physics. The courses that make up the concentration program include at least nine courses in mathematics (detailed descriptions follow for each degree), plus at least five courses within the Physical Sciences Collegiate Division (PSCD) but outside mathematics, one of which completes a threequarter sequence in basic chemistry or basic physics, and at least two others which should form a sequence of courses from a single department. These latter courses must be chosen from astronomy, chemistry, computer science, physics (12000s or above), geophysical sciences, or statistics (22000 or above). We particularly call attention to two degree programs that are described in more detail in the following paragraphs: (1) the Bachelor of Science in mathematics with specialization in computer science, and (2) the Bachelor of Science in mathematics with specialization in economics.
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 10000level sequence in calculus (or to demonstrate equivalent competence on the calculus placement test) and a threequarter sequence in analysis (Mathematics 203002040020500 or 207002080020900), and two quarters of a sequence in algebra (Mathematics 2540025500 or 2570025800). 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 20000level 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 3810038200.
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 threequarter sequence (Mathematics 254002550025600 or Mathematics 257002580025900), whereas B.A. candidates may opt for a twoquarter sequence (Mathematics 2540025500 or Mathematics 2570025800). 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 threecourse sequence, which is outside the Department of Mathematics but within the Physical Sciences Collegiate Division, chosen in consultation with the departmental counselor.
Summary of Requirements:
Mathematics
General CHEM 1110011200 or higher†,
Education or PHYS 1210012200 or higher†
MATH 1310013200, 1510015200, or 1610016200†
Concentration 1 CHEM 11300 or higher†,
or PHYS 12300 or higher†
1 MATH 13300, 15300, or 16300†
3 MATH 203002040020500 or
207002080020900
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
plus the following requirements:
B.A. 
B.S. 
2 MATH 2540025500 or 2570025800
1 MATH 25600, 25900, or an
approved alternative
14 
3 MATH 254002550025600 or
257002580025900
3 threequarter sequence in
a minor field outside
mathematics
17 
† 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 threecourse sequence that is outside the Department of Mathematics but within the Physical Sciences Collegiate Division.
Summary of Requirements:
Applied Mathematics
General CHEM 1110011200 or higher†,
Education or PHYS 1210012200 or higher†
MATH 1310013200, 1510015200,
or 1610016200†
Concentration 1 CHEM 11300 or higher†,
or PHYS 12300 or higher†
1 MATH 13300, 15300, or 16300†
3 MATH 203002040020500
or 207002080020900
1 MATH 21100
2 MATH 2540025500 or 2570025800
3 MATH 270002730027500
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 151001520015300 or 161001620016300 strongly recommended), in analysis (Mathematics 203002040020500 or 207002080020900), and in abstract algebra (Mathematics 254002550025600 or 257002580025900), 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 twoquarter sequence in programming is required; Computer Science 1150011600 is recommended. (Students may substitute Computer Science 1050010600, 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 1050010600 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.
Summary of Requirements:
Mathematics with Specialization in Computer Science
General CHEM 1110011200 or higher†,
Education or PHYS 1210012200 or higher†
MATH 1310013200, 1510015200, or 1610016200†
Concentration 1 CHEM 11300 or higher†,
or PHYS 12300 or higher†
1 MATH 13300, 15300, or 16300†
3 MATH 203002040020500
or 207002080020900
3 MATH 254002550025600
or 257002580025900
2 CMSC 1150011600
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 203002040020500 or 207002080020900), and two quarters of abstract algebra (Mathematics 2540025500 or 2570025800), 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 20000201002020020300 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 2440024500 is an appropriate twoquarter 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.
Summary of Requirements:
Mathematics with Specialization in Economics
General CHEM 1110011200 or higher†,
Education or PHYS 1210012200 or higher†
MATH 1310013200, 1510015200, or 1610016200†
Concentration 1 CHEM 11300 or higher†,
or PHYS 12300 or higher†
1 MATH 13300, 15300, or 16300†
3 MATH 203002040020500
or 207002080020900
2 MATH 2540025500 or 2570025800
1 MATH 27000
1 MATH 27200 or 27300
1 STAT 25100
1 STAT 22000 or 24400
4 ECON 20000201002020020300
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 20000level 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 207002080020900 or Mathematics 257002580025900) 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.
Joint Degree Program
B.A. (B.S.)/M.S. in Mathematics. Qualified College students may receive both a bachelor's and a master's degree in mathematics concurrently at the end of their years in the College. Qualification consists of satisfying all the requirements of each degree in mathematics. With the help of placement tests and honors sequences, able students can be engaged in 30000level Mathematics as early as their third year and, through an appropriate use of free electives, can complete the master's requirements by the end of their fourth year. Interested students should apply to the departmental counselor as soon as possible and in any event no later than the winter quarter of the third year.
Faculty
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 benzvi, 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
Courses
For a description of the numbering guidelines for the following courses, consult the section on reading the catalog on page 15.
L refers to courses with a laboratory.
Students must fulfill their requirements for biological, mathematical, and physical sciences by taking one quarter of an approved mathematical sciences course. In addition, a sixth quarter of an approved course in biological, mathematical, or physical sciences must be taken to complete the general education requirements. NOTE: In order to get general education credit for calculus, two quarters must be taken. This will count as two quarters toward fulfilling the general education requirement in science.
1050010600. 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 firstyear students. This twocourse 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 131001320013300 calculus sequence. Staff. Autumn, Spring.
131001320013300. 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 1310013200 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 onehour class meetings and two mandatory oneandonehalf 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 131001320013300 have a command of calculus equivalent to that obtained in MATH 151001520015300. MATH 13300 is only offered in the spring quarter. Staff. Autumn, Winter, Spring.
151001520015300. 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 1510015200 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.
161001620016300. 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 1610016200 meets the general education requirement in mathematical sciences. MATH 161001620016300 is an honors version of MATH 151001520015300. A student with a strong background in the problemsolving aspects of onevariable calculus may, by suitable achievement on the calculus placement test, be permitted to register for MATH 161001620016300. This sequence emphasizes the theoretical aspects of onevariable 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.
1950019600. Mathematical Methods for Biological or Social Sciences I, II. PQ: MATH 15300 or equivalent. This sequence includes some linear algebra and threedimensional geometry, a review of onevariable calculus, ordinary differential equations, partial derivatives, multiple integrals, partial differential equations, sequences, and series. Staff. Summer; Autumn, Winter; Winter, Spring.
200002010020200. Mathematical Methods for Physical Sciences I, II, III. PQ: MATH 15300 or equivalent. Entering students who have placement for MATH 1510015200 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.
203002040020500. Analysis in Rn I, II, III. PQ: MATH 13300 or 15300 or 16300. This threecourse sequence is for students who intend to concentrate in mathematics or who require a rigorous treatment of analysis in several dimensions. Here, both the theoretical and problemsolving aspects of multivariable calculus are treated carefully. Topics covered in MATH 20300 include the topology of Rn, compact sets, the geometry of Euclidean space, limits and continuous mappings, and partial differentiation. MATH 20400 deals with vectorvalued functions, extrema, the inverse and implicit function theorems, and multiple integrals. MATH 20500 is concerned with line and surface integrals, and the theorems of Green, Gauss, and Stokes. One section of this course is intended for students who have taken MATH 13300 or who had a substandard performance in MATH 15300. This sequence is the basis for all advanced courses in analysis and topology. Staff. Autumn, Winter, Spring; Winter, Spring, Autumn.
207002080020900. Honors Analysis in Rn I, II, III. PQ: Invitation only. This highly theoretical sequence in analysis is reserved for the most able students. The sequence covers the real number system, metric spaces, basic functional analysis, the Lebesgue integral, and other topics. Staff. Autumn, 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 piecewise 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 200102; will be offered 200203.
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.
254002550025600. 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 2540025500).
257002580025900. Honors Basic Algebra I, II, III. PQ: MATH 15300 or 16300. This is an accelerated version of MATH 254002550025600. 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, wellordering, and the axiom of choice; metric spaces; and completeness, compactness, and separability. Staff. Autumn.
26200. PointSet 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 BorsukUlam. 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.
27200. Basic Functional Analysis. PQ: MATH 20900, or 26100 and 27000. Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, Lpspaces, Fourier transforms, Plancherel's theorem for Rn, and spectral properties of bounded linear operators are some of the topics discussed. Staff. Winter.
27300. Basic Theory of Ordinary Differential Equations. PQ: MATH 20200 or 22100 or 27000. This course covers firstorder equations and inequalities, Lipschitz condition and uniqueness, properties of linear equations, linear independence, Wronskians, variationofconstants formula, equations with constant coefficients and Laplace transforms, analytic coefficients, solutions in series, regular singular points, existence theorems, theory of twopoint 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 secondorder equations in two variables, wave motion and Fourier series, heat flow and Fourier integral, Laplace's equation and complex variables, secondorder 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 smn theorem and the recursion theorem. We also discuss resourcebounded computation. This course introduces complexity theory. Relationships between space and time, determinism and nondeterminism, NPcompleteness, and the P versus NP question are investigated. This course is offered in alternate years. Not offered 200102; will be offered 200203.
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.
3000030100. 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 ZermeloFraenkel (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 (OSharp), 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 smn 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, manyone, truthtable, 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 nonc.e. degrees by oracle KleenePost 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 ChurchTuring 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, NPcomplete problems, polynomial time hierarchy, and Pspace complete problems. Staff. Autumn. Not offered 200102; will be offered 200203.
3090031000. 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 CantorBendixson and Morley derivatives; the Morley theorem and the BaldwinLachlan 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 200102; will be offered 200203.
312003130031400. Analysis I, II, III. PQ: MATH 26200, 27000, 27200, and 27400; and consent of director or associate director of undergraduate studies. Topics include Lebesgue measure, abstract measure theory, and Riesz representation theorem; basic functional analysis (Lpspaces, elementary Hilbert space theory, HahnBanach, open mapping theorem, and uniform boundedness); RadonNikodym theorem, duality for Lpspaces, Fubini's theorem, differentiation, Fourier transforms, locally convex spaces, weak topologies, and convexity; compact operators; spectral theorem and integral operators; Banach algebras and general spectral theory; Sobolev spaces and embedding theorems; Haar measure; and PeterWeyl theorem, holomorphic functions, Cauchy's theorem, harmonic functions, maximum modulus principle, meromorphic functions, conformal mapping, and analytic continuation. Staff. Autumn, Winter, Spring.
317003180031900. 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 MayerVietoris theorem, Kunneth theorem, cup products, duality, and geometric applications. Staff. Autumn, Winter, Spring.
320003210032200. Mathematical and Statistical Methods for the NeuroSciences 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 threequarter 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.
325003260032700. Algebra I, II, III. PQ: MATH 254002550025600, 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.
