Courses of Study 20192020 [ARCHIVED CATALOG]
Mathematics


In the College of Arts and Sciences .
Course Offerings
Mathematics is the language of modern science; basic training in the discipline is essential for those who want to understand, as well as for those who want to take part in, the important scientific developments of our time. Acquaintance with mathematics is also extremely useful for students in the social sciences and valuable for anyone interested in the full range of human culture and the ways of knowing the universe in which we live.
The Department of Mathematics faculty has strong and broad groups specializing in algebra, number theory, combinatorics, real and complex analysis, Lie groups, topology and geometry, logic, probability and statistics, mathematical physics, and applied mathematics. Additionally, several other departments at Cornell offer courses which involve a significant amount of advanced mathematical content. These include computer science, economics, operations research, physics, and statistics. Certain courses in these and other disciplines can be readily integrated into the math major though the various concentrations which are offered.
The department offers a rich variety of undergraduate courses. Additionally, some of the introductory graduate courses are suitable for undergraduates who have completed a rigorous foundation of 4000level coursework in mathematics. Under some conditions, a student may carry out an independent reading or research project for college credit under the supervision of a faculty member.
Members of the department are available to discuss with students the appropriate course for their levels of ability and interest, and students are urged to avail themselves of this help. Students who want to take any of the courses numbered 3000 or above are invited to confer with the instructor before enrolling.
Website: math.cornell.edu
Faculty
R. Ramakrishna, chair; X. Cao, director of undergraduate studies; M. E. Stillman, director of graduate studies; M. Aguiar, M. B.Langlois, D. Barbasch, S. Bennoun, Y. Berest, H. Chen, R. Connelly, K. Delp, R. K. Dennis, F. Fuentes, D. HalpernLeistner, T. Healey, T. Holm, J. Hubbard, M. Huntley, J. Hyde, M. Kassabov, A. Knutson, L. Levine, K. Mann, J. Manning, K. Meszaros, J. Moore, C. Muscalu, A. Nerode, M. Nussbaum, I. Peeva, J.D. Quigley, T. Riley, L. SaloffCoste, S. Sen, R. A. Shore, R. Sjamaar, S. Solecki, P. Sosoe, B. Speh, R. Strichartz, S. Strogatz, E. Swartz, N. Templier, A. Townsend, G. Uraltsev, A. Vladimirsky, M. Wegkamp, J. West, I. Zakharevich, D. Zywina. Emeritus: L. Billera, J. Bramble, K. Brown, S. Chase, M. Cohen, L. Gross, J. Guckenheimer, A. Hatcher, J. Hwang, P. Kahn, G. R. Livesay, M. Morley, A. Schatz, J. Smillie, M. Sweedler, K. Vogtmann

Course Selection Guidance:
For guidance in selecting an appropriate course, including how to factor advanced placement or transfer credit into that decision, please consult First Steps in Math. New students will have an opportunity to ask questions about math placement during fall orientation at the Arts & Sciences Open House; however, it should be noted that the Cornell placement tests are held before the open house. Students who are unsure if they need such a test should ask the director of undergraduate studies for advice in advance of the exam, which is held on the weekend before classes start.
Advanced Placement and Transfer Credit:
Students who have had some calculus should carefully read “Advanced Placement ,” and those who have not taken an advanced placement exam should take a placement test at Cornell during fall or spring orientation. Dates, times, and locations for the Cornell placement tests will be announced in the orientation guide and on the Math Department web site.
The linear algebra and multivariable calculus courses that we offer (MATH 2210 , MATH 2220 , MATH 1920 , and MATH 2940 ) cover considerably more material and in considerably greater depth than that which is covered in high school courses in these subjects. Students who have taken college courses in one or both of these subjects may wish to apply for transfer credit. Students who have completed coursework in linear algebra and/or multivariable calculus and have a strong interest in challenging theoretical mathematics may consider enrolling in MATH 2230 –MATH 2240 . Students who have completed a rigorous course in multivariable calculus may also take the Engineering Math Advanced Standing Exam. There is no placement test for linear algebra, and it should be noted that 4000level linear algebra courses are generally not regarded as meeting the prerequisites for the math major or minor.
Precalculus:
Students who need to take Calculus I (MATH 1106 or MATH 1110 ) but are lacking the necessary prerequisites may take MATH 1101 to prepare. Courses labeled “college algebra” or “precalculus” at other universities, while not eligible for transfer credit to Cornell, may be used to satisfy the prerequisites for Calculus I.
Calculus and Linear Algebra:
Students should consult their advisors and keep major prerequisites in mind when planning a suitable program. The following are general recommendations. Consult First Steps in Math for more detail. The director of undergraduate studies will gladly meet with students to offer further advice.
 Students who expect to major in mathematics, economics, or a science for which a strong math background is recommended should take MATH 1110 –MATH 1120 (students with AP credit for MATH 1110 and an interest in theoretical mathematics may consider enrolling in MATH 1220 rather than MATH 1120 ) and continue with MATH 2210 –MATH 2220 . Students with AP credit for MATH 1110 MATH 1120 and a strong interest in challenging theoretical mathematics may consider MATH 2230 –MATH 2240 , especially if they have had some prior exposure to linear algebra and multivariable calculus.
 MATH 1910 –MATH 1920 –MATH 2930 –MATH 2940 is required for engineering students and recommended by some advisors in fields strongly related to the mathematical and physical sciences, such as astronomy, computer science, physics, and physical chemistry.
 MATH 1110 –MATH 1120 and MATH 2130 or MATH 2310 is a good choice for students who need to master the basic techniques of calculus but whose majors will not require a substantial amount of mathematics. MATH 1110 –MATH 2310 is an option for students who need some linear algebra but not a full year of calculus.
 MATH 1106 is an option for students whose major requires only one semester of calculus. Some topics are covered in less depth than in MATH 1110 , while more advanced topics are introduced. MATH 1106 covers differentiation, some integration, and touches on multivariable calculus. Students who may take more than one semester of calculus should take MATH 1110 rather than MATH 1106 .
 Students who are undecided about their future studies at Cornell but think they may involve a substantial amount of math can keep their options open by taking Calculus I (MATH 1110 or AP credit), Calculus II (MATH 1120 or MATH 1220 or AP credit), and Linear Algebra (MATH 2210 ). Multivariable Calculus (MATH 2220 ) would be the next step for students who are still leaning in the direction of a mathrelated major and may wish to take more advanced mathematics.
Switching between calculus sequences is often difficult, especially at the 2000level. Students should not attempt such a switch without consulting the director of undergraduate studies.
SpecialPurpose Sequences:
Students who will take no more than two semesters of mathematics can gain a broader view of the subject by taking one semester of calculus and one noncalculus mathematics course. The following options are particularly useful for students in the life and social sciences and will satisfy the mathematics requirement for most medical schools.
 MATH 1105 –MATH 1106 provides a oneyear introduction to the mathematical topics that are most useful to biologists and social scientists. (MATH 1110 may be substituted for MATH 1106 .)
 An introductory statistics course (MATH 1710 , for example), taken before or after a semester of calculus (MATH 1106 or MATH 1110 ), teaches students how to work with data and can be more useful in some disciplines than a second semester of calculus.
Students who want two semesters of calculus are advised to take the first two semesters of one of the calculus sequences, but students with excellent performance in MATH 1106 may follow that course with MATH 1120 or MATH 1220 . Minor in Mathematics:
The mathematics minor is available to students majoring in other disciplines across the university who have an interest in studying mathematics. Students planning a minor in mathematics may seek advice on course selection from the director of undergraduate studies. Applications for the minor are available in 310A Malott Hall.
Requirements:
Students must complete four 3000 or 4000level MATH courses. At least one course must be in algebra and one in analysis. Eligible algebra and analysis courses are the same as those listed for the math major requirements (1) and (2) below. At least one of the four courses must be at the 4000level or above. A course may be counted toward the minor only if it is taken for a letter grade and a grade of C– or better is received for the course. Major in Mathematics:
The mathematics major adapts to a number of purposes. It can emphasize the theoretical or the applied. It can be appropriate for professionals and nonprofessionals alike, and can be broad or narrow. It can also be combined easily with serious study in another subject in the physical, biological, or social sciences by means of a double major and/or concentration. (See “Double Majors” below for more information.) Questions concerning the major should be brought to the undergraduate coordinator. Applications for the major are available in 310A Malott Hall.
Prerequisites:
Students are admitted to the major after successfully completing a semester of linear algebra — MATH 2210 , MATH 2230 , or MATH 2940 with a grade of B– or better — and a semester of multivariable calculus — MATH 2220 , MATH 2240 , or MATH 1920 with a grade of B– or better. The department recommends MATH 2210 –MATH 2220 or MATH 2230 –MATH 2240 . MATH 2130 and MATH 2310 are not recommended for students planning a math major; however, MATH 2130 with a grade of B+ or better may be accepted as a substitute for MATH 2220 , and MATH 2310 with a grade of B+ or better may be accepted as a substitute for MATH 2210 . A 3 or 4credit computer programming course is also required with a letter grade of C– or better. Eligible courses include: CS 1110 , CS 1112 , CS 1115 , CS 2110 , and CS 2112 .
Students who have taken a course in linear algebra and/or multivariable calculus should consider taking MATH 2230 –MATH 2240 . This sequence gives a more abstract, prooforiented treatment of the material. Students with an advanced background in linear algebra and/or multivariable calculus should contact the undergraduate coordinator for advice as soon as possible. Note that 4000level linear algebra courses are generally not regarded as meeting the prerequisites for the math major.
Students who receive below the minimum grade in one of these prerequisite courses should contact the undergraduate coordinator immediately.
Requirements:
Students must complete nine courses, as described in items 1–3 below, under the following constraints:
 At least 5 courses with a MATH prefix numbered 3000 or above must appear on the student’s transcript. (Double majors enrolling in crosslisted courses should pay particular attention to this constraint.)
 At least two of the MATH courses taken must be at the 4000level (or above).
 A course may be counted toward the major only if it is taken for a letter grade and a grade of C– or better is received for the course.
 No course may be used to satisfy more than one requirement for the major.
 2credit courses count as half courses.
 MATH courses numbered between 5000 and 5999 do not count toward the major.
Major advisors may make adjustments to the major requirements upon request from an advisee, provided the intent of the requirements is met. In particular, many suitable graduate courses are not listed here, but are available for undergraduates who are well prepared.
1. Two courses in algebra:
Eligible courses are: 2. Two courses in analysis:
Eligible courses are: 3. Five further highlevel mathematical courses:
The eight alternatives (a–h) below do not exhaust the possibilities. A mathematics major interested in a concentration in a subject different from those below may develop a suitable individual program in consultation with his or her major advisor.
a. Concentration in Mathematics:
i. Four additional MATH courses numbered 3000 or above:
At least one of the four courses must be among the geometry/topology courses. Eligible courses include: Note:
MATH 4500 and MATH 4560 are eligible only if not used toward the algebra requirement.
ii. One course dealing with mathematical models:
Any course from outside mathematics with serious mathematical content and dealing with scientific matters. Serious mathematical content includes, but is not limited to, extensive use of calculus or linear algebra. Eligible courses include any course from another department that would satisfy one of the concentrations, as well as: b. Concentration in Applied Mathematics:
Five additional courses from (iii) and (iv) below, of which at least three are from (iii) and one is from (iv). Of the 9 courses used to fulfill requirements (1), (2), (3 iii), and (3 iv) of the math major with an applied mathematics concentration, at least one course must be taken from three of the four Groups A, B, C, and D below. NonMATH courses in these groups may be used toward the math modeling requirement (3 iv).
iii. MATH courses numbered 3000 or above:
iv. Courses dealing with mathematical models:
Eligible courses include MATH 3610 and any course outside mathematics with serious mathematical content that deals with scientific matters. Serious mathematical content includes, but is not limited to, extensive use of calculus or linear algebra. Any course from another department that would satisfy one of the concentrations may be used. At most one of the following may be used:
Other 1000level physics courses and PHYS 2207 may not be used, but some courses in other fields may be accepted. AP credit may not be used.
Group A. Differential equations
Group B. Discrete mathematics and combinatorics
Group C. Numerical and computational methods
Group D. Probability and statistics
c. Concentration in Computer Science:
Five additional courses from (v) and (vi) below, of which at least one is from (v) and three are from (vi).
v. MATH courses numbered 3000 or above:
vi. Computer science courses with significant mathematical content:
Eligible courses are:
Note:
There are also many CS graduate courses with significant mathematical content that may be used. Interested students should discuss these options with their math faculty advisor (after being admitted to the math major).
d. Concentration in Economics:
Five additional courses from (vii), (viii), and (ix) below, as follows: one course from (vii), three courses from (viii), and a fifth course from any of (vii), (viii), or (ix).
vii. MATH courses numbered 3000 or above:
viii. Economics courses with significant mathematical content:
Eligible courses are:
Note:
Undergraduate enrollment in ECON graduate courses requires permission of instructor.
ix. Courses in operations research with significant mathematical content and dealing with material of interest in economics:
Eligible courses are:
e. Concentration in Mathematical Biology:
Five additional courses from (x) and (xi) below, with three courses from (x) and two courses from (xi).
x. Biology courses that have mathematical content and provide background necessary for work at the interface between biology and mathematics:
Eligible courses are:
xi. MATH courses numbered 3000 or above:
Particularly appropriate are:
f. Concentration in Mathematical Physics:
Five additional courses from (xii) and (xiii) below, of which at least one is from (xii) and three are from (xiii).
xii. MATH courses numbered 3000 or above:
xiii. Physics courses that make significant use of advanced mathematics:
Eligible courses are:
g. Concentration in Operations Research:
Five additional courses from (xiv) and (xv) below, of which at least one is from (xiv) and three are from (xv).
xiv. MATH courses numbered 3000 or above:
xv. Courses in operations research in which the primary focus involves mathematical techniques:
Eligible courses are:
h. Concentration in Statistics:
Five additional courses from (xvi), (xvii), and (xviii) below. No substitutions are allowed for MATH 4710 or MATH 4720 . Students who have already taken a course with overlapping content should consult the undergraduate coordinator. (For students who have not had experience with realworld data, MATH 1710 is recommended before or concurrent with MATH 4710 . It will not, however, count toward any of the math major requirements.)
xvii. One additional MATH course numbered 3000 or above:
xviii. Two courses in other departments with significant content in statistics, complementing (xvii):
Eligible courses are:
Double Majors:
A double major with computer science, economics, or physics can be facilitated by the corresponding concentrations described above. The Departments of Computer Science and Economics permit double majors to use courses in the corresponding concentrations to satisfy the requirements of both majors.
Double majors with physics may count eligible physics courses toward both the physics major and the math major’s math physics concentration; however, math courses that are being used for an outside concentration for the physics major may not also be counted for the math major.
When enrolling in crosslisted courses, double majors must take care that at least 5 courses with a MATH prefix numbered 3000 or above will appear on their transcript. Students should consult other major departments about any further conditions they may have.
Senior Thesis:
A senior thesis can form a valuable part of a student’s experience in the mathematics major. It is intended to allow students to conduct an indepth investigation not possible in regular course work. The work should be independent and creative. It can involve the solution of a serious mathematics problem, or it can be an expository work, or variants of these. Conducting independent research, paying careful attention to exposition in the finished written product, and the delivery of an optional oral presentation can have a lasting positive impact on a student’s educational and professional future. Graduate Courses:
Some exceptional undergraduates, upon completing a rigorous foundation of 4000level math courses, may wish to further develop their understanding of the material in subsequent graduate courses that the math department offers. The core courses from the mathematics graduate program — MATH 6110 , MATH 6120 , MATH 6310 , MATH 6320 , MATH 6510 , and MATH 6520 — represent a good first exposure to graduatelevel mathematics. MATH 6150 , MATH 6160 , MATH 6210 , MATH 6220 , MATH 6710 , and MATH 6720 cover some additional material in a manner suitable to advanced undergraduates. Undergraduates taking graduate courses should have completed advanced undergraduate courses on the same topic with a grade of A– or better. Interested students should discuss the possibility of taking graduate courses with their faculty advisor in the Math Department prior to enrolling in the course.
Honors:
The Department of Mathematics awards honors (cum laude) and high honors (magna cum laude and summa cum laude) to graduating mathematics majors who have performed outstandingly in the major program.
The awards are determined by the Mathematics Major Committee in the latter part of the semester before graduation. The committee will primarily be looking for excellent performance in mathematics courses, particularly in challenging courses at the 4000level or beyond. Independent study at a high performance level can also contribute to honors. Students interested in any level of honors should consult their major advisors or a member of the Mathematics Major Committee concerning suitable courses. Outstanding performance in the core graduate classes (MATH 6110 –MATH 6120 , MATH 6310 –MATH 6320 , MATH 6510 MATH 6520 ) or an excellent senior thesis can contribute to high honors.



