The Department of Mathematics provides an environment of research and comprehensive instruction in mathematics and applied mathematics at both undergraduate and graduate levels. Both a BA and a BS program in mathematics are offered, including a BS degree in applied mathematics and a BS degree in mathematics with a specialization in economics. Students in other fields of study may also complete a minor in mathematics; information follows the description of the major.

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 each bachelor's degree look to the advancement of students' general education in modern mathematics and their knowledge of its relation with the other sciences (BS) or with the other arts (BA).

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 requirements are submitted in person to the director of undergraduate studies or to one of the departmental counselors. Students should note that only one undergraduate degree may be earned from the Department of Mathematics.

At what level does an entering student begin mathematics at the University of Chicago? Every entering student must take the Mathematics Placement Test. This online test must be taken during the summer before arrival on campus. Students will be given instructions in early July on how to access more information. Scores on the Mathematics Placement Test, combined with a student’s high school record, determine the appropriate beginning mathematics course for each student: a precalculus course (MATH 10500 Fundamental Mathematics I) or one of three other courses (MATH 11200 Studies in Mathematics I, MATH 13100 Elementary Functions and Calculus I, or MATH 15100 Calculus I). Students who wish to begin at a level higher than MATH 15100 Calculus I must take the Calculus Accreditation Exam, unless they receive Advanced Placement credit as described in the following paragraphs.

During Orientation Week, the College administers the Calculus Accreditation Exam. On the basis of this exam, a student may receive credit for up to three quarters of Calculus. Students earning one quarter of credit on this exam may begin MATH 15200 Calculus II, students earning two quarters of credit may begin with MATH 15300 Calculus III, and students earning three quarters of credit may begin with MATH 19900 Introduction to Analysis and Linear Algebra, MATH 19520 Mathematical Methods for Social Sciences, MATH 19620 Linear Algebra, or MATH 20000 Mathematical Methods for Physical Sciences I. Strong students, especially those planning to continue with higher level mathematics or other disciplines requiring advanced mathematics, are urged to take this accreditation exam. The Calculus Accreditation Exam is given only during Orientation Week, and may be taken only once and only by incoming students (first-years or transfers).

On the basis of the Calculus Accreditation Exam, students may also be invited to begin MATH 16100-16200-16300 Honors Calculus I-II-III. This sequence builds on the sound computational background provided in AP courses and best prepares entering students for further study in mathematics. Students who are invited to begin Honors Calculus are encouraged to forgo credit in MATH 15100 Calculus I and/or MATH 15200 Calculus II in order to take the full Honors Calculus sequence, MATH 16100-16200-16300 Honors Calculus I-II-III. Additionally, at least one section of the MATH 16100-16200-16300 Honors Calculus I-II-III sequence each year will be offered as an inquiry-based learning (IBL) course. Interested students should have a score of 5 on the AP Calculus BC exam, placement into MATH 16100 Honors Calculus I, and fluency in spoken English.

A small number of students each year receive placement recommendations beyond Honors Calculus. Admission to MATH 20700 Honors Analysis in Rn I is by invitation only to those first-year students with superior performance on the Calculus Accreditation Exam or to those sophomores who receive a strong recommendation from their instructor in MATH 16100-16200-16300 Honors Calculus I-II-III. Students who are granted three quarters of calculus credit on the basis of the Calculus Accreditation Exam and who do not qualify for admission to MATH 20700 Honors Analysis in Rn I will place into MATH 19900 Introduction to Analysis and Linear Algebra. These students may consult with one of the departmental counselors about the option of beginning with MATH 16100 Honors Calculus I so that they would be eligible for admission to Honors Analysis the following year.

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 MATH 15100 Calculus I. Students who submit a score of 5 on the BC Advanced Placement exam in mathematics receive credit for MATH 15100 Calculus I and MATH 15200 Calculus II. Currently no course credit is offered in the Mathematics Department at Chicago for work done in an International Baccalaureate Programme or for British A-level or O-level examinations.

Four bachelor's degrees are available in the Department of Mathematics: the BA in mathematics, the BS in mathematics, the BS in applied mathematics, and the BS in mathematics with specialization in economics. Programs qualifying students for the degree of BA provide more elective freedom. Programs qualifying students for the degrees of BS require more emphasis in the physical sciences, while the BS in mathematics with specialization in economics has its own set of specialized courses. 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 first-year chemistry or first-year physics.

Except for the BS in mathematics with specialization in economics, each degree requires at least five courses outside mathematics (detailed descriptions follow for each degree). These courses must be within the Physical Sciences Collegiate Division (PSCD) or from Computational Neuroscience (CPNS). One of these courses must complete the three-quarter sequence in basic chemistry or basic physics. At least two of these courses must be from a single department and all must be chosen from among astronomy, chemistry, computer science, physics (12000s or above), geophysical sciences, statistics (22000 or above), or physical science (18100 or above). Graduate courses from these departments may also be used to fulfill these requirements. No courses from the Financial Mathematics program may be used in any of the undergraduate degree programs in mathematics. Please note in particular the requirements outside of mathematics described below in the degree program for the BS in mathematics with specialization in economics.

Note: Students who are majoring in mathematics may use AP credit for chemistry and/or physics to meet their general education physical sciences requirement and/or the physical sciences component of the major. However, no credit designated simply as "physical science," from AP examinations or from the College's physical sciences placement or accreditation examination, may be used in their general education requirement or in the mathematics major.

Students who are majoring in mathematics are required to complete: a 10000-level sequence in calculus (or to demonstrate equivalent competence on the Calculus Accreditation Exam); either MATH 16300 Honors Calculus III as the third quarter of the calculus sequence or MATH 19900 Introduction to Analysis and Linear Algebra; a three-quarter sequence in analysis (MATH 20300-20400-20500 Analysis in Rn I-II-III or MATH 20700-20800-20900 Honors Analysis in Rn I-II-III); and two quarters of an algebra sequence (MATH 25400-25500 Basic Algebra I-II or MATH 25700-25800 Honors Basic Algebra I-II). The normal procedure is to take calculus in the first year, analysis in the second, and algebra in the third. Students may not use both MATH 16300 Honors Calculus III and MATH 19900 Introduction to Analysis and Linear Algebra to meet major or minor requirements. The MATH 16300 Honors Calculus III/MATH 19900 Introduction to Analysis and Linear Algebra requirement will be waived for entering students who place into MATH 20700 Honors Analysis in Rn I.

Candidates for the BA and BS in mathematics take a sequence in basic algebra. BA candidates may opt for a two-quarter sequence (MATH 25400-25500 Basic Algebra I-II or MATH 25700-25800 Honors Basic Algebra I-II), whereas candidates for the BS degree must take the three-quarter algebra sequence (MATH 25400-25500-25600 Basic Algebra I-II-III or MATH 25700-25800-25900 Honors Basic Algebra I-II-III). MATH 25700-25800-25900 Honors Basic Algebra I-II-III is designated as an honors version of Basic Algebra. Registration for this course is the option of the individual student. Consultation with one of the departmental counselors is strongly advised.

The remaining mathematics courses needed in the programs (three for the BA, two for the BS) must be selected, with due regard for prerequisites, from the following approved list of mathematics courses. STAT 25100 Introduction to Mathematical Probability also meets the requirement. BA candidates may include MATH 25600 Basic Algebra III or MATH 25900 Honors Basic Algebra III. Mathematics courses in the Paris Spring Mathematics Program may also be used to meet this requirement, and each year one of these three courses will be designated as an acceptable replacement for MATH 25600 Basic Algebra III or MATH 25900 Honors Basic Algebra III for BS candidates.

BS candidates are further required to select a minor field, which consists of three additional courses that are outside the Department of Mathematics and either are within the same department in the Physical Sciences Collegiate Division (PSCD) or are among the Computational Neuroscience (CPNS) courses in the Biological Sciences Collegiate Division (BSCD). These courses must be chosen in consultation with one of the departmental counselors.

Candidates for the BS 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 one of the departmental counselors, a secondary field, which consists of three additional courses from a single department that is outside the Department of Mathematics but within the Physical Sciences Collegiate Division, or among the Computational Neuroscience courses in the Biological Sciences Collegiate Division.

This program is a version of the BS in mathematics. The BS 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, MATH 19900 Introduction to Analysis and Linear Algebra if the calculus sequence did not terminate with MATH 16300 Honors Calculus III, a yearlong sequence in analysis (MATH 20300-20400-20500 Analysis in Rn I-II-III or MATH 20700-20800-20900 Honors Analysis in Rn I-II-III), and two quarters of abstract algebra (MATH 25400-25500 Basic Algebra I-II or MATH 25700-25800 Honors Basic Algebra I-II), and earn a grade of at least C- in each course. Students must also take STAT 25100 Introduction to Mathematical Probability. The remaining two mathematics courses must include MATH 27000 Basic Complex Variables and either MATH 27200 Basic Functional Analysis for those interested in Econometrics or MATH 27300 Basic Theory of Ordinary Differential Equations for those interested in economic theory. A C average or higher must be earned in these two courses.

In addition to the third quarter of basic chemistry or basic physics, the eight courses required outside the Department of Mathematics must include STAT 23400 Statistical Models and Methods or STAT 24400 Statistical Theory and Methods I. The remaining seven courses should be in the economics department and must include ECON 20000-20100-20200-20300 The Elements of Economic Analysis I-II-III-IV or ECON 20010-ECON 20110-ECON 20210-ECON 20310 The Elements of Economic Analysis: Honors I-II-III-IV and either ECON 20900 Econometrics: Honors or ECON 21000 Econometrics. The remaining two courses may be chosen from any undergraduate economics course numbered higher than ECON 20300 The Elements of Economic Analysis IV. A University of Chicago Booth School of Business course may be considered for elective credit if the course requires the equivalent of ECON 20100 as a prerequisite and is numbered as a Chicago Booth 40000 or higher course. Additionally, the course needs to pertain to the application of economic theory to a course subject that is not offered by the department of economics. Courses such as accounting, investments, and entrepreneurship will not be considered for economics elective credit. Consideration for elective credit must be done by petition before a student registers for the course. There will be no retroactive consideration for credit. Students must earn a grade of C or higher 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 (CMSC 15100 Introduction to Computer Science I is strongly recommended), and an additional course in statistics (STAT 24400-24500 Statistical Theory and Methods I-II 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.

Subject to College grading requirements and grading requirements for the major and with consent of instructor, students (except students who are majoring in mathematics or applied mathematics) may take any mathematics course beyond the second quarter of calculus for either a quality grade or for P/F grading. A Pass grade is given only for work of C- quality or higher.

All courses taken to meet requirements in the mathematics major must be taken for quality grades. A grade of C- or higher must be earned in each calculus, analysis, or algebra course; and an overall grade average of C or higher must be earned in the remaining mathematics courses that a student uses to meet requirements for the major. Students must earn a grade of C or higher in each course taken in economics for the degree in mathematics with a specialization in economics. Mathematics or applied mathematics students may take any 20000-level mathematics courses elected beyond program requirements for P/F grading.

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.

The BA or BS with honors is awarded to students who, while meeting requirements for one of the mathematics degrees, also meet the following requirements: (1) a GPA of 3.25 or higher in mathematics courses and a 3.0 or higher overall; (2) no grade below C- and no grade of W in any mathematics course; (3) completion of at least one honors sequence (either MATH 20700-20800-20900 Honors Analysis in Rn I-II-III or MATH 25700-25800-25900 Honors Basic Algebra I-II-III) with grades of B- or higher in each quarter; and (4) completion with a grade of B- or higher of at least five mathematics courses chosen from the list that follows so that at least one course comes from each group (i.e., algebra, analysis, and topology). No course may be used to satisfy both requirement (3) and requirement (4). If both honors sequences are taken, one sequence may be used for requirement (3) and one sequence may be used for up to three of the five courses in requirement (4).

The minor in mathematics requires a total of six or seven courses in mathematics, depending on whether or not MATH 16300 Honors Calculus III or MATH 19900 Introduction to Analysis and Linear Algebra is required in another degree program. If it is not used elsewhere, MATH 16300 Honors Calculus III or MATH 19900 Introduction to Analysis and Linear Algebra must be included in the minor, for a total of seven courses. The remaining six courses must include a three-course sequence in Analysis (MATH 20300-20400-20500 Analysis in Rn I-II-III or MATH 20700-20800-20900 Honors Analysis in Rn I-II-III) and a two-course sequence in Algebra (MATH 25400-25500 Basic Algebra I-II or MATH 25700-25800 Honors Basic Algebra I-II). The sixth course may be chosen from either the third quarter of Algebra (MATH 25600 Basic Algebra III or MATH 25900 Honors Basic Algebra III) or a mathematics course numbered 24000 or higher chosen in consultation with the director of undergraduate studies or one of the departmental counselors. Under special circumstances and to avoid double counting, students may also use mathematics courses numbered 24000 or higher to substitute for up to two quarters of Analysis or Algebra, if these are required in another degree program.

No course in the minor can be double counted with the student's major(s) or with other minors; nor can it be counted toward general education requirements. Students must earn a grade of at least C- in each of the courses in the mathematics minor. More than one-half of the requirements for a minor must be met by registering for courses bearing University of Chicago course numbers.

Students must meet with the director of undergraduate studies or one of the departmental counselors by Spring Quarter of their third year to declare their intention to complete a minor program in mathematics and to obtain approval for the minor on a form obtained from their College adviser. Courses for the minor are chosen in consultation with the director of undergraduate studies or one of the departmental counselors.

Qualified College students may receive both a bachelor's and a master's degree in mathematics concurrently at the end of their studies in the College. Qualification consists of satisfying all requirements of each degree in mathematics. To be eligible for the joint program, a student must excel on the Calculus Accreditation Exam (during Orientation Week of the student's first year) and must begin MATH 20700 Honors Analysis in Rn I in the Autumn Quarter of the student's first year. By following a program of prescribed undergraduate course sequences in mathematics and succeeding in all courses with grades no lower than A-, the student becomes eligible to enroll in graduate courses in mathematics in the student's third year. While only a few students complete the joint BA/MS program, many undergraduates enroll in graduate-level mathematics courses. Admission to all mathematics graduate courses requires prior written consent of the director or co-director of undergraduate studies.

Students should submit their application for the joint program to one of the departmental counselors as soon as possible, but no later than the Winter Quarter of their third year.

Majors in mathematics or applied mathematics seeking to prepare for secondary school teaching and possible futures in mathematics education may be eligible for admission to the University of Chicago Urban Teacher Education Program (UChicago UTEP) Master of Arts in Teaching (MAT). Students completing the program receive a master of arts in teaching (MAT) degree and an Illinois teaching certificate and endorsement to teach high school mathematics (grades 6 to 12). During the fourth year of undergraduate study, MAT candidates take a Foundations of Education sequence, SOSC 25501-25502-25503 Schools and Communities; Human Development and Learning; Teaching and Learning in Urban Schools. Candidates enter into focused content area course work and small group instruction during the summer following graduation from the College, before working with entire classes during the internship year and following summer. Graduates are assisted with job placement in the Chicago Public Schools and have continued support for an additional three years through personalized coaching and workshops provided by UChicago UTEP staff. Interested students should consult with one of the departmental counselors no later than the Autumn Quarter of their third year.

### Mathematics Courses

**MATH 10500. Fundamental Mathematics I. 100 Units.**

Students who place into this course must take it in their first year in the College. Must be taken for a quality grade. MATH 10500 will count only as one elective. This course does not meet the general education requirement in mathematical sciences. This course covers basic precalculus topics. It is concerned with elements of algebra, coordinate geometry, and elementary functions, including trigonometric, and exponential functions.

Terms Offered: Autumn

Prerequisite(s): Adequate performance on the mathematics placement test

**MATH 11200-11300. Studies in Mathematics I-II.**

MATH 11200 AND 11300 cover the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. MATH 11200 addresses number theory, including a study of the rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. MATH 11300’s main topic is symmetry and geometry, including a study of polygons, Euclidean construction, polyhedra, group theory, and topology. These courses emphasize the understanding of ideas and the ability to express them through rigorous mathematical arguments. While students may take MATH 11300 without having taken MATH 11200, it is recommended that MATH 11200 be taken first. Either course in this sequence meets the general education requirement in mathematical sciences. These courses are at the level of difficulty of the MATH 13100-13200-13300 calculus sequence.

**MATH 11200. Studies in Mathematics I. 100 Units.**

MATH 11200 AND 11300 cover the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. MATH 11200 addresses number theory, including a study of the rules of arithmetic, integral domains, primes and divisibility, congruences, and modular arithmetic. These courses emphasize the understanding of ideas and the ability to express them through rigorous mathematical arguments. While students may take MATH 11300 without having taken MATH 11200, it is recommended that MATH 11200 be taken first. Either course in this sequence meets the general education requirement in mathematical sciences. These courses are at the level of difficulty of the MATH 13100-13200-13300 calculus sequence.

Terms Offered: Autumn, Spring

Prerequisite(s): MATH 10500, or placement into MATH 13100 or higher

**MATH 11300. Studies in Mathematics II. 100 Units.**

MATH 11200 AND 11300 cover the basic conceptual foundations of mathematics by examining the ideas of number and symmetry. MATH 11300’s main topic is symmetry and geometry, including a study of polygons, Euclidean construction, polyhedra, group theory, and topology. These courses emphasize the understanding of ideas and the ability to express them through rigorous mathematical arguments. While students may take MATH 11300 without having taken MATH 11200, it is recommended that MATH 11200 be taken first. Either course in this sequence meets the general education requirement in mathematical sciences. These courses are at the level of difficulty of the MATH 13100-13200-13300 calculus sequence.

Terms Offered: Winter

Prerequisite(s): MATH 10500, or placement into MATH 13100 or higher; MATH 11200 recommended

**MATH 13100-13200-13300. Elementary Functions and Calculus I-II-III.**

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. MATH 13100 component of this sequence gives a careful treatment of limits and the continuity and differentiability of algebraic functions. Topics examined in MATH 13200 include applications of differentiation; exponential, logarithmic, and trigonometric functions; the definite integral and the fundamental theorem, and applications of the integral. In MATH 13300, subjects include more applications of the definite integral, infinite sequences and series, and Taylor expansions. Students are expected to understand the definitions of key concepts (i.e., limit, derivative, 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 15100-15200-15300. Students may not take the first two quarters of this sequence for P/F grading. MATH 13100-13200 meets the general education requirement in mathematical sciences.

**MATH 13100. Elementary Functions and Calculus I. 100 Units.**

Terms Offered: Autumn, Winter

Prerequisite(s): Invitation only, based on adequate performance on the mathematics placement test; or MATH 10500.

**MATH 13200. Elementary Functions and Calculus II. 100 Units.**

Topics examined in MATH 13200 include applications of differentiation; exponential, logarithmic, and trigonometric functions; the definite integral and the fundamental theorem, and applications of the integral. Students are expected to understand the definitions of key concepts 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 15100-15200-15300. Students may not take the first two quarters of this sequence for P/F grading. MATH 13100-13200 meets the general education requirement in mathematical sciences.

Terms Offered: Winter, Spring

Prerequisite(s): MATH 13100

**MATH 13300. Elementary Functions and Calculus III. 100 Units.**

In MATH 13300, subjects include more applications of the definite integral, infinite sequences and series, and Taylor expansions. Students are expected to understand the definitions of key concepts 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 15100-15200-15300.

Terms Offered: Spring

Prerequisite(s): MATH 13200

**MATH 15100-15200-15300. Calculus I-II-III.**

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 the mean value theorem, integration, techniques of integration, and applications of the integral. MATH 15300 deals with additional techniques and theoretical considerations of integration, infinite sequences and series, and Taylor expansions. All Autumn Quarter offerings of MATH 15100, 15200, and 15300 begin with a rigorous treatment of limits and limit proofs. Students may not take the first two quarters of this sequence for P/F grading. MATH 15100-15200 meets the general education requirement in mathematical sciences.

**MATH 15100. Calculus I. 100 Units.**

This is the first course in 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. All Autumn Quarter offerings of MATH 15100 begin with a rigorous treatment of limits and limit proofs. Students may not take the first two quarters of this sequence for P/F grading. MATH 15100-15200 meets the general education requirement in mathematical sciences.

Terms Offered: Autumn

Prerequisite(s): Superior performance on the mathematics placement test, or MATH 10500

**MATH 15200. Calculus II. 100 Units.**

This is the second course in 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. Work in MATH 15200 is concerned with the mean value theorem, integration, techniques of integration, and applications of the integral. All Autumn Quarter offerings of MATH 15200 begin with a rigorous treatment of limits and limit proofs. Students may not take the first two quarters of this sequence for P/F grading. MATH 15100-15200 meets the general education requirement in mathematical sciences.

Terms Offered: Autumn, Winter

Prerequisite(s): MATH 15100; or placement based on the Calculus Accreditation Exam or appropriate AP score

**MATH 15300. Calculus III. 100 Units.**

This is the third course in the regular calculus sequence in the department. MATH 15300 deals with techniques and theoretical considerations of integration, infinite sequences and series, and Taylor expansions. All Autumn Quarter offerings of MATH 15300 begin with a rigorous treatment of limits and limit proofs.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 15200; or placement based on the Calculus Accreditation Exam or appropriate AP score

**MATH 16100-16200-16300. Honors Calculus I-II-III.**

Students may not take the first two quarters of this sequence for P/F grading. 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 Accreditation Exam, 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. MATH 16300 also includes an introduction to linear algebra. At least one section of this sequence is offered as an inquiry-based learning (IBL) course. Students interested in IBL should have an AP score of 5 on the BC Calculus exam and fluency in spoken English.

**MATH 16100. Honors Calculus I. 100 Units.**

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 Accreditation Exam, 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. MATH 16300 also includes an introduction to linear algebra. At least one section of this sequence is offered as an inquiry-based learning (IBL) course. Students interested in IBL should have an AP score of 5 on the BC Calculus exam and fluency in spoken English. Students may not take the first two quarters of this sequence for P/F grading. MATH 16100-16200 meets the general education requirement in mathematical sciences.

Terms Offered: Autumn

Prerequisite(s): Invitation only based on superior performance on the Calculus Accrediation Examination

**MATH 16200. Honors Calculus II. 100 Units.**

This sequence emphasizes the theoretical aspects of one-variable analysis and, in particular, the consequences of completeness in the real number system.

Terms Offered: Winter

Prerequisite(s): MATH 16100

**MATH 16300. Honors Calculus III. 100 Units.**

This sequence emphasizes the theoretical aspects of one-variable analysis and, in particular, the consequences of completeness in the real number system. MATH 16300 also includes an introduction to linear algebra.

Terms Offered: Spring

Prerequisite(s): MATH 16200

**MATH 17500. Basic Number Theory. 100 Units.**

This course covers basic properties of the integers following from the division algorithm, primes and their distribution, and congruences. Additional topics include existence of primitive roots, arithmetic functions, quadratic reciprocity, and transcendental numbers. The subject is developed in a leisurely fashion, with many explicit examples.

Terms Offered: Autumn

Prerequisite(s): MATH 16300 or 19900

**MATH 17600. Basic Geometry. 100 Units.**

This course covers advanced topics in geometry, including Euclidean geometry, spherical geometry, and hyperbolic geometry. We emphasize rigorous development from axiomatic systems, including the approach of Hilbert. Additional topics include lattice point geometry, projective geometry, and symmetry.

Terms Offered: Winter

Prerequisite(s): MATH 16300 or 19900

**MATH 19520. Mathematical Methods for Social Sciences. 100 Units.**

This course takes a concrete approach to the basic topics of multivariable calculus. Topics include a brief review of one-variable calculus, parametric equations, alternate coordinate systems, vectors and vector functions, partial derivatives, multiple integrals, and Lagrange multipliers.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 13300 or MATH 15300 or MATH 16300

**MATH 19620. Linear Algebra. 100 Units.**

This course takes a concrete approach to the basic topics of linear algebra. Topics include vector geometry, systems of linear equations, vector spaces, matrices and determinants, and eigenvalue problems.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 13300 or MATH 15200 or MATH 16200.

Note(s): Recommended sequence for ECON majors: MATH 19620, STAT 23400, ECON 21000 in consecutive quarters.

**MATH 19900. Introduction to Analysis and Linear Algebra. 100 Units.**

This course is intended for students who are making the transition from MATH 15300 to 20300, or for students who need more preparation in learning to read and write proofs. This course covers the fundamentals of theoretical mathematics and prepares students for upper-level mathematics courses beginning with MATH 20300. Topics include the construction of the real numbers, completeness and the least upper bound property, the topology of the real line, the structure of finite-dimensional vector spaces over the real and complex numbers. Students who are majoring or minoring in mathematics may not use both MATH 16300 and 19900 to meet program requirements.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): Superior performance on the Calculus Accreditation Exam or MATH 15300 or 13300

**MATH 20000-20100. Mathematical Methods for Physical Sciences I-II.**

This sequence is intended for students who are majoring in a department in the Physical Sciences Collegiate Division other than mathematics. MATH 20000 covers multivariable calculus, including the algebra and geometry of Euclidean space, differentiation and integration of functions of several variables, vector valued functions and the classical theorems of vector analysis (i.e., theorems of Green, Gauss, and Stokes). MATH 20100 introduces ordinary differential equations (e.g., first and second order linear differential equations, series solutions, and the Laplace transform) and complex analysis (i.e., basic properties of the complex plane and analytic functions through Cauchy’s theorem).

**MATH 20000. Mathematical Methods for Physical Sciences I. 100 Units.**

This sequence is intended for students who are majoring in a department in the Physical Sciences Collegiate Division other than mathematics. MATH 20000 covers multivariable calculus, including the algebra and geometry of Euclidean space, differentiation and integration of functions of several variables, vector valued functions and the classical theorems of vector analysis (i.e., theorems of Green, Gauss, and Stokes).

Terms Offered: Autumn, Winter

Prerequisite(s): MATH 13300 or 15300 or 19620; entering students by invitation only, based on superior performance on the Calculus Accreditation Exam

**MATH 20100. Mathematical Methods for Physical Sciences II. 100 Units.**

This sequence is intended for students who are majoring in a department in the Physical Sciences Collegiate Division other than mathematics. MATH 20100 introduces ordinary differential equations (e.g., first and second order linear differential equations, series solutions, and the Laplace transform) and complex analysis (i.e., basic properties of the complex plane and analytic functions through Cauchy’s theorem).

Terms Offered: Winter, Spring

Prerequisite(s): MATH 20000, OR both 19520 AND 19620 or equivalent.

**MATH 20300-20400-20500. Analysis in Rn I-II-III.**

This three-course sequence is intended for students who plan to major in mathematics or who require a rigorous treatment of analysis in several dimensions. Both theoretical and problem solving aspects of multivariable calculus are treated carefully. Topics in MATH 20300 include metric spaces, the topology of Rn, compact sets, the geometry of Euclidean space, and limits and continuous mappings. MATH 20400 deals with partial differentiation, vector-valued functions, extrema, and the inverse and implicit function theorems. MATH 20500 is concerned with multiple integrals, line and surface integrals, and the theorems of Green, Gauss, and Stokes. This sequence is the basis for all advanced courses in analysis and topology.

**MATH 20300. Analysis in Rn I. 100 Units.**

The three-course sequence MATH 20300-20400-20500 is intended for students who plan to major in mathematics or who require a rigorous treatment of analysis in several dimensions. Both theoretical and problem solving aspects of multivariable calculus are treated carefully. Topics in MATH 20300 include metric spaces, the topology of Rn, compact sets, the geometry of Euclidean space, and limits and continuous mappings. This sequence is the basis for all advanced courses in analysis and topology.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 16300 or 19900

**MATH 20400. Analysis in Rn II. 100 Units.**

MATH 20400 deals with partial differentiation, vector-valued functions, extrema, and the inverse and implicit function theorems. The sequence MATH 20300-20400-20500 is the basis for all advanced courses in analysis and topology.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 20300

**MATH 20500. Analysis in Rn III. 100 Units.**

MATH 20500 is concerned with multiple integrals, line and surface integrals, and the theorems of Green, Gauss, and Stokes. The sequence MATH 20300-20400-20500 is the basis for all advanced courses in analysis and topology.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): MATH 20400

**MATH 20700-20800-20900. Honors Analysis in Rn I-II-III.**

This highly theoretical sequence in analysis is intended for the most able students. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.

**MATH 20700. Honors Analysis in Rn I. 100 Units.**

This is the first course in a highly theoretical sequence in analysis, and is intended for the most able students. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.

Terms Offered: Autumn

Prerequisite(s): Invitation only, based on performance on the Calculus Accreditation Exam

**MATH 20800. Honors Analysis in Rn II. 100 Units.**

This is the second course in a highly theoretical sequence in analysis. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.

Terms Offered: Winter

Prerequisite(s): MATH 20700

**MATH 20900. Honors Analysis in Rn III. 100 Units.**

This is the third course in a highly theoretical sequence in analysis. Topics include the real number system, metric spaces, basic functional analysis, and the Lebesgue integral.

Terms Offered: Spring

Prerequisite(s): MATH 20800

**MATH 21100. Basic Numerical Analysis. 100 Units.**

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.

Terms Offered: Spring

Prerequisite(s): MATH 20000 or 20300

**MATH 21200. Advanced Numerical Analysis. 100 Units.**

This course covers topics similar to those of Math 21100 but at a more rigorous level. The emphasis is on proving all of the results. Previous knowledge of numerical analysis is not required. Programming is also not required. The course makes extensive use of the material developed in the analysis sequence (ending in Math 20500 or Math 20900) and provides an introduction to other areas of analysis such as functional analysis and operator theory.

Terms Offered: Autumn

Prerequisite(s): MATH 20500 or 20900

**MATH 22000. Introduction to Mathematical Methods in Physics. 100 Units.**

This course, with concurrent enrollment in PHYS 13300, is required of students who plan to major in physics. Topics include infinite series 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.

Terms Offered: Spring

Prerequisite(s): MATH 15200 or 16200, and PHYS 13200

**MATH 24100. Topics in Geometry. 100 Units.**

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.

Terms Offered: Spring

Prerequisite(s): MATH 25500 or 25800

Note(s): This course is offered in alternate years.

**MATH 24200. Algebraic Number Theory. 100 Units.**

Topics include factorization in Dedekind domains, integers in a number field, prime factorization, basic properties of ramification, and local degree.

Terms Offered: Spring

Prerequisite(s): MATH 25500 or 25800

**MATH 24300. Introduction to Algebraic Curves. 100 Units.**

This course covers the projective line and plane curves, both affine and projective. We also study conics and cubics, as well as the group law on the cubic. Abstract curves associated to function fields of one variable are discussed, along with the genus of a curve and the Riemann-Roch theorem. Curves of low genus are emphasized. Although the formal prerequisite is MATH 25500 or 25800, MATH 25600 or 25900 is strongly recommended.

Terms Offered: Spring

Prerequisite(s): MATH 25500 or 25800, or consent of instructor

Note(s): This course is offered in alternate years.

**MATH 25400-25500-25600. Basic Algebra I-II-III.**

This sequence covers groups, subgroups, and permutation groups; rings and ideals; fields; vector spaces, linear transformations and matrices, and modules; and canonical forms of matrices, quadratic forms, and multilinear algebra.

**MATH 25400. Basic Algebra I. 100 Units.**

This course covers groups, subgroups, permutation groups, rings and ideals.

Terms Offered: Autumn, Winter

Prerequisite(s): MATH 16300 or 19900

**MATH 25500. Basic Algebra II. 100 Units.**

This course covers fields, vector spaces, linear transformations and matrices, modules and canonical forms of matrices, quadratic forms, and multilinear algebra.

Terms Offered: Winter, Spring

Prerequisite(s): MATH 25400

**MATH 25600. Basic Algebra III. 100 Units.**

This course covers Sylow Theorems and the fundamentals of Galois theory.

Terms Offered: Spring

Prerequisite(s): MATH 25500

**MATH 25700-25800-25900. Honors Basic Algebra I-II-III.**

This sequence is an accelerated version of MATH 25400-25500-25600 that is open only to students who have achieved a B- or better in prior mathematics courses. Topics include the theory of finite groups, commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms. We also cover basic field theory, the structure of p-adic fields, and Galois theory.

**MATH 25700. Honors Basic Algebra I. 100 Units.**

Topics in MATH 25700 include the theory of finite groups, including the proofs of the Sylow Theorems.

Terms Offered: Autumn

Prerequisite(s): MATH 16300 or 19900; no entering student may begin this sequence in their first term

**MATH 25800. Honors Basic Algebra II. 100 Units.**

Topics in MATH 25800 include commutative and noncommutative ring theory, modules, linear and multilinear algebra, and quadratic forms.

Terms Offered: Winter

Prerequisite(s): MATH 25700

**MATH 25900. Honors Basic Algebra III. 100 Units.**

Topics in this course include basic field theory, the structure of p-adic fields, and Galois theory.

Terms Offered: Spring

Prerequisite(s): MATH 25800

**MATH 26200. Point-Set Topology. 100 Units.**

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, it forms a foundation for all advanced courses in analysis, geometry, and topology.

Terms Offered: Winter

Prerequisite(s): MATH 20300 or 20700, and 25400 or 25700

**MATH 26300. Introduction to Algebraic Topology. 100 Units.**

Topics include 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.

Terms Offered: Spring

Prerequisite(s): MATH 26200

**MATH 26700. Introduction to Representation Theory of Finite Groups. 100 Units.**

Topics include group algebras and modules, semisimple algebras and the theorem of Maschke; characters, character tables, orthogonality relations and calculation; and induced representations and characters. Applications to permutation groups and solvability of groups are also included.

Terms Offered: Autumn

Prerequisite(s): MATH 25900 or 25600

**MATH 26800. Introduction to Commutative Algebra. 100 Units.**

Topics include basic definitions and properties of commutative rings and modules, Noetherian and Artinian modules, exact sequences, Hilbert basis theorem, tensor products, localizations of rings and modules, associated primes and primary decomposition, Artin-Rees Lemma, Krull intersection theorem, completions, dimension theory of Noetherian rings, integral extensions, normal domains, Dedekind domains, going up and going down theorems, dimension of finitely generated algebras over a field, Affine varieties, Hilbert Nullstellensatz, dimension of affine varieties, product of affine varieties, and the dimension of intersection of subvarieties.

Terms Offered: Winter

Prerequisite(s): MATH 25900 or 25600

**MATH 27000. Basic Complex Variables. 100 Units.**

Topics include complex numbers, elementary functions of a complex variable, complex integration, power series, residues, and conformal mapping.

Terms Offered: Autumn, Spring

Prerequisite(s): MATH 20500 or 20900

**MATH 27200. Basic Functional Analysis. 100 Units.**

Topics include Banach spaces, bounded linear operators, Hilbert spaces, construction of the Lebesgue integral, Lp-spaces, Fourier transforms, Plancherel's theorem for Rn, and spectral properties of bounded linear operators.

Terms Offered: Winter

Prerequisite(s): MATH 20900 or 27000

**MATH 27300. Basic Theory of Ordinary Differential Equations. 100 Units.**

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.

Terms Offered: Winter

Prerequisite(s): MATH 20500 or MATH 20900 or PHYS 22100

**MATH 27400. Introduction to Differentiable Manifolds and Integration on Manifolds. 100 Units.**

Topics include exterior algebra; differentiable manifolds and their basic properties; differential forms; integration on manifolds; and the theorems of Stokes, DeRham, and Sard. With MATH 26200, this course forms a foundation for all advanced courses in analysis, geometry, and topology.

Terms Offered: Spring

Prerequisite(s): MATH 26200

**MATH 27500. Basic Theory of Partial Differential Equations. 100 Units.**

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.

Terms Offered: Spring

Prerequisite(s): MATH 27300

**MATH 27700-27800. Mathematical Logic I-II.**

Mathematical Logic I-II

**MATH 27700. Mathematical Logic I. 100 Units.**

This course introduces mathematical logic. Topics include propositional and predicate logic and the syntactic notion of proof versus the semantic notion of truth (e.g., soundness, completeness). We also discuss the Gödel completeness theorem, the compactness theorem, and applications of compactness to algebraic problems.

Terms Offered: Autumn

Prerequisite(s): MATH 25400 or 25700; open to students who are majoring in computer science who have taken CMSC 15400 along with MATH 16300 or MATH 19900

Equivalent Course(s): CMSC 27700

**MATH 27800. Mathematical Logic II. 100 Units.**

Topics include 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).

Terms Offered: Winter

Prerequisite(s): MATH 27700 or equivalent

Equivalent Course(s): CMSC 27800

**MATH 28000. Introduction to Formal Languages. 100 Units.**

This course is a basic introduction to computability theory and formal languages. Topics include automata theory, regular languages, context-free languages, and Turing machines.

Instructor(s): K. Mulmuley Terms Offered: Winter

Prerequisite(s): CMSC 15400, or MATH 19900 or MATH 25500

Equivalent Course(s): CMSC 28000

**MATH 28100. Introduction to Complexity Theory. 100 Units.**

Computability topics are discussed (e.g., the s-m-n theorem and the recursion theorem, resource-bounded computation). This course introduces complexity theory. Relationships between space and time, determinism and non-determinism, NP-completeness, and the P versus NP question are investigated.

Instructor(s): K. Mulmuley Terms Offered: Spring

Prerequisite(s): CMSC 27100, or MATH 19900 or 25500; and experience with mathematical proofs

Equivalent Course(s): CMSC 28100

**MATH 28410. Honors Combinatorics. 100 Units.**

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. Other topics include basic counting, linear recurrences, generating functions, Latin squares, finite projective planes, graph theory, Ramsey theory, coloring graphs and set systems, random variables, independence, expected value, standard deviation, and Chebyshev's and Chernoff's inequalities.

Instructor(s): L. Babai Terms Offered: Winter

Prerequisite(s): MATH 19900 or 25400, or CMSC 27100, or consent of instructor

Note(s): This course is offered in alternate years.

**MATH 29520. Introduction to Error-Correcting Codes. 100 Units.**

Cyclic codes, BCH codes, Golay codes, Shannon's Theorem, and codes approaching Shannon's bounds will be covered. Applications to electrical engineering, combinatorics, and group theory will be discussed.

Terms Offered: Winter in alternate years

Prerequisite(s): MATH 25500 or 25800

**MATH 29700. Proseminar in Mathematics. 100 Units.**

Consent of instructor and departmental counselor. Students are required to submit the College Reading and Research Course Form. Must be taken for a quality grade.

Terms Offered: Autumn, Winter, Spring

Prerequisite(s): Completion of general education mathematics sequence