Mathematics Advising Guide

Mathematics encompasses the study of patterns in nature, the development of tools to understand those patterns, and the generalization of those ideas in an abstract setting. A mathematics degree teaches a student to think, to reason, to experiment, and to learn and grow. Mathematics inspires not only science, technology, and their applications, but all aspects of society.

• Students learn how to ask good questions, make connections, work with others, explain their thoughts, and find evidence to back up their reasoning.

• Professors provide a solid foundation in the subject, spark interest in mathematical topics, use technology in learning, use innovative pedagogical approaches, and provide students with resources to pursue research experiences. 

• Graduates leave Gustavus thoroughly prepared for graduate study, secondary school teaching, a life of service, or employment in government or industry. 

Mathematics Major

This section lists the requirements of the Mathematics major. A grade of C- or higher is necessary in all 11 courses used to satisfy the requirements of the major. Additionally, you can use the Mathematics Major Form, This form will help you plan out your mathematics courses and requirements. To declare a major, use this form:

1. Mathematics Core: These four courses form the core of the major and should usually be taken during the first and second years. 
   1. MCS-122 Calculus II 
   2. MCS-150 Discrete Mathematics
   3. MCS-221 Linear Algebra
   4. MCS-222 Multivariate Calculus
2. Writing: A course that introduces students to the process of constructing and writing mathematical proofs. Students must take one of the following courses:
   1. MCS-213 Intro to Algebra 
   2. MCS-220 Intro to Analysis
3. Breadth: These two courses in cognate fields to mathematics serve to give breadth to the mathematics major. 
   
   1. MCS-142 Introduction to Statistics
   2. MCS-177 Introduction to Computer Science I
4. Immersive Experience: A sequence of two 300 level courses 
   1. MCS-313 and MCS-314 Algebra
   2. MCS-331 and MCS-332 Analysis
   3. MCS-353 and MCS-357 Dynamical Systems

5. Electives: Two additional mathematics courses at the 200 or 300 level. Students should consult with their advisors to discuss which courses best fit their needs. 

Mathematics Minor

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

1. A grade point average of at least 2.333 in these four courses. 
   1. MCS-122 Calculus II or MCS-132 Honors Calculus II
   2. MCS-150 Discrete Mathematics
   3. MCS-221 Linear Algebra
   4. MCS-222 Multivariate Calculus
2. Writing: A course that introduces students to the process of constructing and writing mathematical proofs. Students must take one of the following courses:
   1. MCS-213 Intro to Algebra
   2. MCS-220 Intro to Analysis
3. At least one course from the following:
   
   1. MCS-303 Geometry
   2. MCS-313 Modern Algebra I
   3. MCS-314 Modern Algebra II
   4. MCS-321 Elementary Theory of Complex Variables
   5. MCS-331 Real Analysis
   6. MCS-344 Topics in Advanced Math
   7. MCS-353 Continuous Dynamical Systems
   8. MCS-355 Numerical Analysis
   9. MCS-357 Discrete Dynamical Systems
   10. MCS-358 Math Model Building

Sample Student Plans

All students should ideally lay out a schedule of their own showing what courses they plan to take, and when they plan to take them. The schedule may not accurately forecast the future, but it is helpful nonetheless. A printable sample plan can be found on the Mathematics Major Form 

Student Starter Plan

The sample plans below are useful starting points in developing 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 certain courses are offered on an every-other year basis; for example MCS-314 (Modern Algebra II) is offered in the spring of odd years Courses offered every other year include MCS-313, MCS-314, MCS-331, MCS-344, MCS-355, MCS-357, MCS-358, MCS-385, and MCS-394. These courses are listed with an astrix in the sample plans below. 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.

Algebra

Students interested in algebra should take *MCS-313 and *MCS-314 for their Immersive Experience and MCS-213 as their Proofs course along with two appropriate electives. 

Analysis

Students interested in analysis should take *MCS-331 and *MCS-332 for their Immersive Experience and MCS-220 as their Proofs course and two appropriate electives. 

Applied Mathematics

Students interested in applied mathematics should take *MCS-353 and *MCS-357 for their Immersive Experience, *MCS-358 as an elective, and an additional Elective. 

Thinking About Graduate School in Traditional Mathematics 

Students considering graduate school in mathematics should take *MCS-313,* MCS-314, *MCS-321, and *MCS-331 for their Immersive Experience and Electives as well as an appropriate Collaborative Experience. 

Thinking About Graduate School in Applied Mathematics

Students considering graduate school in applied mathematics should take *MCS-353 and *MCS-357 for their Immersive Experience, *MCS-358 as an Elective, and either *MCS-313 or *MCS 331 as their second Elective.

Studying Mathematics Abroad

 Students traveling abroad should speak with their advisors to discuss courses and study abroad programs. Study abroad programs are listed on the MCS Resources page. 

Honors Program

In order to graduate with Honors in Mathematics, a student must complete an application for admission to the Honors program, available through the department chair, showing that the student satisfies the admission requirements, and then the requirements of the program.

The requirements for admission to the Honors program are as follows:

1. Completion of steps 1 - 3 of the Mathematics Major with a grade point average greater than 3.14. 
2. Approval by the Mathematics Honors committee of an Honors thesis proposal. (Guidelines are available in the Mathematics Advising Guide.)

The requirements of the honors program after admission are as follows:

1. Attainment of a GPA greater than 3.14 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.
2. Approval by the Mathematics Honors Committee of an Honors thesis. The thesis should conform in general outline to the previously 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 MCS-350, whether or not they graduate with Honors. (See the Mathematics Advising Guide for the thesis guidelines.)
3. 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. 

Senior Oral Exam

 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.