The Mathematics and Computer Science Department offers majors in mathematics, mathematics education, and computer science. This document is designed as a resource for students in navigating through the requirements of the math or math education major. (There is a companion guide for computer science.) It should also be valuable to freshman advisors outside the department as well as advisors within the department. The first section gives a brief overview of the discipline; the remaining sections describe the major, the honors program, and the minor.

As an additional resource, you can also use the Mathematics Major Form, which is intended for students majoring (or considering majoring) in Mathematics, and their advisors. This form will help you plan out your mathematics courses.

# Mathematics

Mathematics is a field that is rich in both theoretical analysis and practical application. It is also quite broad in scope, encompassing subfields such as statistics, applied mathematics, and the classical subjects of analysis, algebra, and geometry. This diversity within mathematics makes most definitions of mathematics either too narrow or too general. However, one can say that mathematicians deal with objects (e.g. numbers, triangles, function), and their patterns and relationships (e.g. prime numbers, isosceles triangles, calculus of functions). The search for patterns and relationships involves the process of abstraction, that is forming a generalization from a set of examples that reflects shared properties of these examples. Mathematicians use the skills of creative and analytical thinking to hypothesize the existence of patterns and use logical argument to show the validity of these postulates. Mathematics is also a science, a science of patterns and relationships. Mathematicians experiment in various laboratories -- the mind, the computer, and the natural world.

Mathematicians are always in demand in industry, business, government, and academia. The breadth of subject matter and the logical, analytical training required provide math majors with flexibility in their choice of career.

Mathematicians are involved in activities such as:

• mathematical modeling of semiconductors for a research laboratory
• studying cryptology schemes for secure communication networks
• teaching in the public schools
• teaching at a college or university
• researching the role that chaotic systems play in the regulation of the heart
• devising new fractal algorithms for the display of realistic natural objects
• economic forecasting and model building for the government and industry
• devising better ways to solve the differential equations arising in the flow of turbulent fluids
• working as an actuary for a large insurance firm
• carrying out foundational mathematical research at a research university or research laboratory
• working as a statistician for a governmental agency
This is only the tip of the iceberg as far as career opportunities in the mathematical sciences. For more information talk to someone in the Math/Computer Science department or see the departmental secretary for brochures describing careers in math.

Mathematics majors who are interested in research opportunities or in teaching at the college level should choose a set of courses, in consultation with a department member, that will prepare them for graduate study in mathematics or a related field (such as economics, mathematical physics, statistics, etc.)

# The Major

This section lists the requirements of the math major and describes the senior oral which is an optional component of the major. Qualified majors may additionally participate in the honors program, which is described in the next section.

A grade of C- or higher is necessary in all courses used to satisfy the requirements of the major, which are as follows:

1. MCS-121, MCS-122 or MCS-132, MCS-220, MCS-221, and MCS-222, with a grade point average of at least 2.333 in these five courses. These five courses form the core of the major and should usually be taken during the freshman and sophomore years.
2. MCS-142 and MCS-177. These two courses in cognate fields to mathematics serve to give breadth to the math major.
3. At least four courses chosen from MCS-242, MCS-253, MCS-256, MCS-265, MCS-303, MCS-313, MCS-314, MCS-321, MCS-331, MCS-332, MCS-341, MCS-342, MCS-344, MCS-357, MCS-355, and MCS-358, subject to the following constraints:
1. At least one sequence chosen from (i) MCS-331 & MCS-321 (Real and Complex Analysis), (ii) MCS-313 & MCS-314 (Abstract Algebra), (iii) MCS-331 & MCS-332 (Analysis and Topology), (iv) MCS-341 & MCS-342 (Probability and Statistics), or (v) MCS-253 & MCS-357 (Differential Equations and Discrete Dynamical Systems.)
2. Completion of at least one course from the classical core of mathematics: MCS-321 (Complex), MCS-313 (Algebra), and MCS-331 (Real Analysis). This course can count toward 3a.
3. Completion of at least one course from the applied areas of mathematics: MCS-242, MCS-253, MCS-256, MCS-342, MCS-357, MCS-355, and MCS-358. This course can count toward 3a.
4. Either complete one of the capstone courses MCS-314, MCS-332, MCS-342, MCS-344, MCS-357, MCS-358 (beyond any used for requirements 3) or alternatively pass a successful senior oral examination covering the student's knowledge of mathematics. Successful completion of an honors thesis also will fulfill this requirement.

## Senior oral

As described above, every math major must either take an additional upper level math course from a specified list or alternatively submit to oral examination during the Spring semester of their final year.

A student who chooses to take the oral examination selects, in consultation with a faculty member, a topic to research. They then present a 20-minute talk on that topic to an examining committee of three faculty members. At the conclusion of the talk, the faculty question the student about the talk, and also about fundamental topics from the student's full four years' of courses. The goal is not to require recollection of details, but rather to make sure that the student is leaving with the essentials intact.

The examination committee confers privately immediately after the examination and delivers the results to the student at the conclusion of their deliberations. The outcome is either that the student is deemed to have satisfied the requirement or alternatively that the student is requested to retry the examination at a later date. In the latter case, specific suggestions for areas of improvement are provided by the faculty committee.

More information about the oral examination procedures and schedule are provided routinely to those fourth-year majors who will likely choose to take the examination.

## Mathematics Education major

The requirements for the mathematics education major are met by the completion of the mathematics major as listed above, with the additional requirement that MCS-303(Geometry) and MCS-313 (Modern Algebra), must be completed. Math education majors may substitute the sequence MCS-313-MCS-303 for the sequence requirement in part 3 of the math major.

## Concentrations

Within the math major there are two concentrations or tracks available to students with specific interests.
Applied Mathematics
This concentration is for those students interested in the scientific applications of mathematics, and who are planning on entering fields that require training in mathematical modeling and the analysis of physical problems. MCS-253(Diff Eqns) and MCS-357(Discrete Dynamical Systems) form the core of this track. Other highly recommended courses would be MCS-321(Complex), MCS-358(Math Model Building), and MCS-355(Numerical Analysis).
Statistics
This concentration is intended for students who wish to pursue a career in actuarial science, or who will do graduate studying statistics, biostatistics, or a related field. MCS-242(Appd Stats), MCS-341(Prob Math Stat I), and MCS-342(Prob Math Stat II) form the core of this track. For those interested in actuarial science, MCS-355 (Numerical Analysis) is recommended. Also, a minor in economics or management is recommended. For those interested in graduate study in biostatistics, epidemiology, or public health, a minor in biology is recommended. For those interested in pursuing a PhD in statistics, MCS-331(Real Analysis) and MCS-332(Topology) are strongly recommended.

## Sample student plans

All students should ideally lay out a schedules of their own showing what courses they plan to take when. This schedule may not accurately forecast the future, but it is helpful none the less. The sample plans below are a useful starting point in developing such an individual plan. You can select the sample plan that comes closest to fitting your own situation and then tailor it as necessary. Note that these sample plans show only courses within the Math and Computer Science Department, but in some cases exceed the requirements of the major. Also note that certain courses are offered on an every-other year basis; for example MCS-314 (Algebra II) is offered in the spring of odd years and MCS-332 (Topology) is offered in the spring of even years. Courses offered every other year include MCS 242, 313, 314, 331, 332, 341, 342, 344, 358, 385, and 394. Please keep these course alterations in mind when planning out your major. Check the college catalog for when the courses you are interested in will be scheduled.
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
3rd year  222  321
4th year  313
357
314
Traditional (MCS 331-332 (Real Analysis) sequence)
Fall  Spring
1st year  121
177
122
2nd year  220  221
222
3rd year  142
313
256
4th year  331  332
Traditional (MCS 341-342 (Prob Math Stat) sequence)
Fall  Spring
1st year  121  122
142
2nd year  220
177
221
3rd year  222
331
256
4th year  341  342
Statistics concentration
Fall  Spring
1st year  121  122
142
2nd year  220
177
221
3rd year  222
331
242
4th year  341  342
Applied concentration
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
3rd year  222
331
253
4th year  357  355
321
Math Education
(Even Year Graduation; Student Teach in Spring)
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
242
3rd year  222
313
253
4th year  303
358 J-term
student
teaching
Math Education
(Odd Year Graduation; Student Teach in Spring)
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
256
3rd year  222
358 J-term
242
313
4th year  303  student
teaching
Math Education
(Even Year Graduation; Student Teach in Fall)
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
242
3rd year  222
303
253
4th year  student
teaching
358 J-term
313
Math Education
(Odd Year Graduation; Student Teach in Fall)
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
3rd year  222
303
358 J-term
242
313
4th year  student
teaching
253
Fall  Spring
1st year  121
177
122
2nd year  220
142
221
222
3rd year  331  332
321
4th year  313  314
Fall  Spring
1st year  121
142
122
2nd year  220
177
221
222
3rd year  331  242
332
4th year  341  342
Fall  Spring
1st year  121
177
122
2nd year  220
142
221
222
3rd year  331  321
253
4th year  357
313
355
Fall  Spring
1st year  120  121
177
2nd year  122  220
221
3rd year  222
142
321
4th year  331  256
313
Fall  Spring
1st year  121  122
177
2nd year  220
142
221
256
4th year  321
331
313
Fall  Spring
1st year  121  122
142
2nd year  220
177
221
256
3rd year  331
222
4th year  341  342
Fall  Spring
1st year  121
177
122
2nd year  220
142
221
222
4th year  313
331
321
256

# Honors Program

In order to graduate with honors in mathematics, a student must complete an application for admission to the honors program, showing that the student satisfies the admission requirements, and then must satisfy the requirements of the program.

## Admission to the honors program

The requirements for admission to the honors program are as follows:
1. Completion of MCS-121(Calc I), MCS-122(Calc II) or MCS-132(Honors Calc II), MCS-220(Intro to Analysis), MCS-221(Linear Algebra), MCS-222(Advanced Calculus), MCS-142(Intro Stats), and MCS-177(Intro CS I) with a quality point average greater than pi.
2. Approval by the Mathematics Honors Committee of an honors thesis proposal. (See the Mathematics Honors Thesis Guidelines, reprinted below.)

## Requirements for graduation with honors

The requirements of the honors program, after admission to the program, are as follows:
• Attainment of a quality point average greater than pi in courses used to satisfy the requirements of the major. If a student has taken more courses than the major requires, that student may designate for consideration any collection of courses satisfying the requirements of the major.
• Approval by the Mathematics Honors Committee of an honors thesis. The thesis should conform in general outline to the approved proposal (or an approved substitute proposal), should include approximately 160 hours of work, and should result in an approved written document. Students completing this requirement will receive credit for the course MC96 (Honors Thesis), whether or not they graduate with honors. (See the Mathematics Honors Thesis Guidelines, below.)
• Oral presentation of the thesis in a public forum, such as the departmental seminar. This presentation will not be evaluated as a criterion for thesis approval, but is required.

## Honors thesis guidelines

Mathematics honors thesis proposals should be written in consultation with the faculty member who will be supervising the work. The proposal and thesis must each be approved by the Mathematics Honors Committee. These guidelines are intended to help students, faculty supervisors, and the committee judge what merits approval.

The thesis should include creative work, and should not reproduce well-known results; however, it need not be entirely novel. It is unreasonable for an undergraduate with limited time and library resources to do a thorough search of the literature, such as would be necessary to ensure complete novelty. Moreover, it would be rare for any topic to be simultaneously novel, easy enough to think of, and easy enough to do.

The thesis should include use of primary-source reference material. As stated above, an exhaustive search of the research literature is impractical. None the less, the resources of inter-library loan, the faculty supervisor's private holdings, etc. must be tapped if the thesis work is to go beyond standard classroom/textbook work.

The written thesis should sufficiently explain the project undertaken and results achieved that someone generally knowledgeable about mathematics, but not about the specific topic, can understand it. The quality of writing and care in citing sources should be adequate for external distribution without embarrassment.

The thesis must contain a substantial mathematical component, though it can include other disciplines as well. If a single thesis simultaneously satisfies the requirements of this program and some other discipline's honors program, it can be used for both (subject to the other program's restrictions). However, course credit will not be awarded for work which is otherwise receiving course credit.

The Mathematics Honors Committee will maintain a file of past proposals and theses, which may be valuable in further clarifying what constitutes a suitable thesis. In order to provide some guidance of the sort before the program gets under way, here are some possible topics that appear on the surface to be suitable:

• A student could study the history surrounding Fermat's last theorem, and discuss and explain past failed attempts and the recent successful attempt to prove this theorem.
• A student could research the topic of knot theory and discuss the implications of this theory to the study of DNA and other biological materials.
• A student could study the use of wavelets in signal analysis, and the general usefulness of orthonormal families of functions in signal analysis.

# The Minor

## The Mathematics minor

As with the major in mathematics, a minimum grade of C- must be attained in all courses used to satisfy the minor. The necessary courses are
• MCS-121(Calc I), MCS-122(Calc II) or MCS-132(Honors Calc II), MCS-220(Intro to Analysis), MCS-221(Linear Algebra), and MCS-222(Advanced Calculus), with an average grade of at least C+ in these five courses.
• At least two course chosen from MCS-242(Appd Stats), MCS-303(Geometry), MCS-265(Theory Computation), MCS-256(Discrete), MCS-321(Complex), MCS-358(Math Model Building), MCS-313(Algebra), MCS-314(Algebra II), MCS-357(Discrete Dynamical Systems), MCS-355(Numerical Analysis), MCS-331(Real Analysis), MCS-332(Topology), MCS-344(Topics Adv Math), MCS-341(Prob Math Stat I), and MCS-342(Prob Math Stat II).