1. The Mathematics Major
   1. Lower Division Requirements
   2. Upper Division Requirements
   3. Additional Requirements for BSMath Degree in IT
   4. Requirements for the BA in CLA
   5. Specializations
      1. Actuarial
      2. Mathematics Education
      3. Computer Applications
      4. Graduate School Track
      5. Mathematical Biology Track
2. The Mathematics Minor

Mathematics is a subject with many facets and many applications. The School of Mathematics offers a major that allows many choices. We have well-defined tracks in: mathematics applicable to industry and the biological sciences, mathematics applicable to computer science, mathematics useful in actuarial practice, and mathematics necessary for certification as a secondary mathematics teacher. We of course have programs that lead to graduate study in both pure and applied mathematics, and we offer a general major for those students who do not choose to specialize. We offer coursework that could lead to a career in business or finance. The Actuarial, Education and Computer Science specializations are recorded on the transcript and noted on the diploma. At the moment of writing, the other specializations are not so formally recognized. However, they all exist in the School of Mathematics and, as will be made clear in the descriptions of the different programs, they tend to require more coursework (and more credits) than the minimum needed to complete the general major. Students who do not choose a specialized major are able to graduate with the minimum 120 credits if they plan their programs carefully.

We offer mathematics majors in both the Institute of Technology and the College of Liberal Arts. What basic differences are there between the majors in CLA and in IT? IT mathematics majors face broader technical requirements, and must take Physics and Computer Science. Everyone graduating from CLA is advised to take at least one course in Computer Science and also must be proficient in a foreign language. CLA students must also complete a Senior Project in the major. CLA majors must take at least six upper division courses while IT majors must take at least eight of them. In addition, IT majors must take a "Technical Elective" as described below.

All of these different requirements and approaches to the subject are described in detail in this handout.

This document addresses only the mathematics departmental requirements for the major. In particular, it does not discuss the University's liberal education requirements or the foreign language requirement in CLA. The Undergraduate Catalogue or College Advising Office is the proper source of information on these and other Collegiate and all-University requirements.

All courses that are required for the major or as prerequisites for major courses must be passed with a grade of at least C-.

Lower Division Requirements

1271-1272-2243-2263 (for CLA or IT students) --or-- 1371-1372-2373-2374 (for IT students only) --or-- 1571H-1572H-2573H-2574H (for IT honors students)--or--3592H-3593H (Advanced Placement and Math Honors).

Currently, Math 1371 can only be taken in the Fall.

All students in the major who have not taken the Honors Calculus must start their approach to proofs by taking one of the following courses.

2283 Sequences, Series and Foundations --or--

3283 The writing-intensive version of Math 2283.

This requirement is waived for students who have completed the IT honors calculus sequence or the Advanced Placement sequence.

Note that, while Math 3283 is a 3-xxx level course, and so satisfies the requirement of a writing-intensive course in the major, it may not be counted as part of the Upper Division Math Requirements.

Additional Requirements: IT majors must take at least one semester of Computer Science, as discussed below, and must also take Physics. (CLA majors are strongly advised to take at least one semester of Computer Science.)

• Computer Science: At least one semester. The most common choices include:
  • CSci 1103: Introduction to Computer Programming in Java: For students with little or no programming experience who expect to be able to design web pages and do other web-related tasks.
  • CSci 1107: Introduction to FORTRAN Programming for Scientists and Engineers. For students with limited programming experience who expect to take Numerical Analysis courses or to be involved with scientific computing.
  • CSci 1113: Introduction to C/C++ Programming for Scientists and Engineers. A somewhat higher level course for the same audience as CSci 1107. Students with little or no programming experience often have difficulty in this course. Such students who want to learn C/C++ might consider taking CSci 1107 first.
  • CSci 1901 or 1901-1902: Structure of Computer Programs I, II. For students who want a deeper grasp of programming. After the two semesters the student should have mastered Java at a deeper level than 1103, and be prepared to master C/C++ or FORTRAN.
• Physics: Three semesters, except that students in the Actuarial or Mathematical Biology tracks need only take two semesters. The courses most commonly taken are: Phys 1301-1302 plus one of the following: 2303, 2311, or 2503. The IT Honors Physics, Phys 1401H-1402H-2403H also satisfies the requirement. Transfer students with precisely one year of transferable physics should consult the physics department as to what course or courses to take.

Upper Division Math Requirements

Each Math major in CLA or IT must take two courses from the algebra list and two courses from the analysis list.

Algebra List: At least one of the courses on this list must be chosen from column X.

COLUMN X	COLUMN Y
Math 4281: Introduction to Modern Algebra	Math 4242: Applied Linear Algebra
Math 5248: Cryptography and Number Theory	Math 5705: Enumerative Combinatorics
Math 5251: Error-Correcting Codes, Finite Fields, Algebraic Curves	Math 5707:Graph Theory and Non-enumerative Combinatorics
Math 5285: Honors: Fundamental Structures of Algebra I	Math 5711: Linear Programming and Combinatorial Optimization
Math 5286: Honors: Fundamental Structures of Algebra II	Math 5485: Introduction to Numerical Methods I
Math 5385: Introduction to Computational Algebraic Geometry

Analysis List:

Math 4606 Advanced Calculus
Math 4653 Elementary Probability
Math 5486 Introduction to Numerical Methods II
Math 5525 Introduction to Ordinary Differential Equations
Math 5535 Dynamical Systems and Chaos
Math 5583 Complex Analysis
Math 5587 Elementary Partial Differential Equations
Math 5588 Elementary Partial Differential Equations II
Math 5615 Honors: Introduction to Analysis I
Math 5616 Honors: Introduction to Analysis II
Math 5651 Basic Theory of Probability and Statistics (or the equivalent Stat 5101)
Math 5652 Stochastic Processes
Math 5654 Prediction and Filtering

Additional Requirements for the BSMath Degree in IT

• A prerequisite for the courses shall be Calculus I (Math 1271 or 1371) or higher.
• The courses shall be numbered 3XXX or higher.
• The courses shall have a "theme" or "connection". Courses in the same department are assumed to be linked. If the courses are in different departments, they must have visible "connections" in the subject matter, or to the student's special interests/career plans. For example, a student who has abandoned a major in some engineering department will be allowed to offer EE 3025-3161 (same department), but not the interdepartmental mix of EE 3061-Stat 3021 (no visible connections). An acceptable package for anyone would be Math 5651-Stat 5041 (obvious connections in the subject matter). On the other hand, a known actuarial student might offer the diverse package of Econ 3101-Stat 3021 (career-related). Advisor approval of the package is required for the technical elective. If the package is questionable, and receives advisor approval, the student should ask the advisor to make a note of that fact in his or her file. Where a dispute arises, and the DUGS is called to arbitrate, he or she will honor such written approval.

Credit and coursework requirements

Courses from other departments that count as part of the eight course upper division requirement are:
Stat 5101 Theory of Statistics I (mentioned above as equivalent to Math 5651
Stat 5102 Theory of Statistics II (which is different from Math 5652)

No other courses from other departments may be used to satisfy the eight course upper division requirement, though they may be used as part of the Technical Elective.

The following courses may not be used to satisfy any upper division requirement in mathematics and may not be used to satisfy any part of the Technical Elective:

Math 3113 or 4113 Topics in Elementary Mathematics I
Math 3116 or 4116 Topics in Elementary Math II: Short Course
Math 3118 or 4118 Topics in Elementary Mathematics II
Math 4005 Calculus Refresher

Note again that while Math 3283 is a 3-xxx level course, and so satisfies the requirement of a writing-intensive course in the major, it may not be counted as part of the Upper Division Math Requirement or as technical elective.

Senior project. All CLA students must complete a senior project. See a mathematics advisor to discuss procedures for working on that project.

Credit and coursework requirements. The student must complete at least 120 credits, including at least 41 in the major. In addition to the Lower Division Requirements, the student must complete six upper division mathematics courses of at least 3 credits each, including two from the algebra list and two from the analysis list. The courses available for satisfying the six-course upper division requirement are the same as those available for the BSMath degree, as described on the previous page.

The School of Mathematics offers specializations in actuarial mathematics, computer applications and in preparation for teaching mathematics in the secondary school. These specializations earn a designation that appears on the diploma. There are also specializations in industrial mathematics and mathematical biology, and a track that is designed as an optimal preparation for graduate study in mathematics. Students in any of these specializations must meet the requirements of the major in their college as described above (with some exceptions noted in the descriptions of the specializations).

These are the additional requirements to obtain a specialization in Actuarial Science (both CLA and IT):

• All Actuarial Math students must take one of CSci 1103 (Java) or CSci 1113 (Programming in C/C++)
• Economics : 1104-1105-3101-4751 (Elementary Macroeconomics and Microeconomics, Intermediate Microeconomics, Financial Economics). Note: 1101-1102 will be accepted in place of 1104-1105. Also 3101-4751 can be used as the technical elective for IT students. Also, Fina 4241 (Corporate Financing Decisions) may be substituted for Econ 4751. IT majors should understand, however, that since Fina 4241 does not have a calculus prerequisite, it cannot be used as part of the Technical Elective.
• Business : Acct 2050, Fina 3001, and 2 out of 3 of Ins 4100, 4101, 4200.
• Probability and Statistics : Math 5651 or Stat 5101, and either Math 5652 or Stat 5102. Note: Math 5651 and Stat 5101 are considered identical; either is a sufficient prerequisite for either sequel; either can be used as part of the analysis requirement. Of the sequels, we recommend Math 5652 over Stat 5102. A secondary advantage to this choice is that Math 5652 will complete the analysis requirement but Stat 5102 will not.
• Actuarial Mathematics : The student must complete Math 4065, Theory of Interest and Math 5067-5068, Actuarial Mathematics I and II.
• Algebra : Math 4242, Applied Linear Algebra. It is strongly recommended that the student also take 5711 (Optimization). Either one of these courses counts as part of the Algebra Requirement.

Note again that IT students in the actuarial specialization are not required to take the third semester of Physics.

IT students (and CLA students who have room in their programs) might use some of the following as technical electives or to fill out their program:

Math 4428, Mathematical Modeling

Math 5485, Numerical Methods I

Math 5705 or 5707, Combinatorics A or B

Math 5075/5076, Mathematics of Options, Futures and Derivatives

Stat 5302,Applied Regression Analysis

Fina 4241, Corporate Financing Decisions

Fina 4121, Financial Markets and Interest Rates;

Econ 3101, Intermediate Microeconomics;

Econ3102, Intermediate Macroeconomics

CSci 4061, Introduction to Operating System

CSci 5103, Operating Systems

CSci 5301/2, Numerical Analysis.

These are the additional requirement to obtain a specialization in Secondary Mathematics Education.

• CLA majors must take seven upper division mathematics courses satisfying major requirements.
• Each student must include the following four courses in the major program:

Math 5651: Basic Theory of Probability and Statistics or the equivalent Stat 5101: Theory of Statistics I, or Math 4653, Elementary Probability

Math 4281: Introduction to Modern Algebra or Math 5285: Honors: Fundamental Structures of Algebra I or Math 4242:Applied Linear Algebra.

Math 5335: Geometry I

Math 4707 or 5707 or 5705 (Combinatorics)

Note that, of the courses on this list, the Algebra Requirement declares that you must take Math 4281 or Math 5285. Any of Math 4242, 4707, 5705 or 5707 will then complete the Algebra Requirement. Only one course from the analysis list is included however. NOTE: Math 4707 will be accepted as an algebra course only for students completing this specialization.

There are other courses that are recommended but not required. These include Math 5652: Stochastic Processes or Stat 5102: Theory of Statistics II, which are strongly recommended. Another strong recommendation is Math 5336: Geometry II. A second course from the analysis list could be Math 4606: Advanced Calculus or Math 5615: Honors: Introduction to Analysis I.

IT majors should be able to satisfy the technical elective requirement with courses in Mathematics Education. These may include two of Math E 5011, Math E 5021 and MathE 5031, but the student's mathematics advisor should be consulted to approve the technical elective.

All students who attempt this specialization should consult the staff in Mathematics Education about other courses that must be completed in order to obtain secondary licensure in Minnesota. To start this, call (612) 625-6372 and ask for a referral for information on the program in Mathematics Education. This consultation should be done by the end of the junior year or at the very beginning of the senior year.

These are the additional requirements for an emphasis in Computer Applications.

• Lower level computer science courses:
  • One of the following:
    • CSci 1901-1902 Structure of Computer Programming I-II (recommended)
    • CSci 1103 (Programming in Java) and CSci 1113 (Programming in C/C++)

    Either choice satisfies the requirement of an elementary computer science course for the mathematics major.

  • CSci 2011 Discrete Structures of Computer Science
• Math 5485-5486 Intro to Numerical Methods I-II. Note that Math 5486 counts as one of the courses on the Analysis list and that Math 5485 counts as one of the courses on the Algebra
• Math 5165 (Mathematical Logic I)
• Upper level computer science courses:
  • Any three of the following:
  • CSci 4041 Algorithms and Data Structures
  • CSci 5107 Computer Graphics I
  • CSci 5108 Computer Graphics II
  • CSci 5403 Computational Complexity
  • CSci 5421 Advanced Algorithms and Data Structures
  • CSci 5442 Computational Geometry and Applications
  • CSci 5511 Artificial Intelligence I
  • CSci 5512W Artificial Intelligence II or CSci 5519 Artificial Intelligence II (non-W)
  • CSci 5521 Pattern Recognition
  .

  These courses can be counted as part of the Technical Elective package for the IT mathematics major.

• Additional computing-related mathematics course:

  Any one of the following:

  • Math 4242 Applied Linear Algebra
  • Math 5166 Mathematical Logic II
  • Math 5248 Cryptography and Number Theory
  • Math 5251 Error-Correcting Codes, Finite Fields, Algebraic Curves
  • Math 5285 Honors: Fundamental Structures of Algebra I
  • Math 5286 Honors: Fundamental Structures of Algebra II
  • Math 5385 Introduction to Computational Algebraic Geometry
  • Math 5487 Computational Methods for Differential and Integral Equations in Engineering and Science
  • Math 5705 Enumerative Combinatorics
  • Math 5707 Graph Theory and Non-enumerative Combinatorics
  • Math 5711 Linear Programming and Combinatorial Optimization

If the course for this category is from the algebra list, then it must be in addition to the two courses from that list that are required for all mathematics majors.

Note: Students who complete this emphasis will also satisfy the requirements for a minor in Computer Science.

(No claim is made or implied that taking another track and doing well precludes going to graduate school. In some sense this is a "Pure Mathematics" track.)

1. Lower Division:

   First year calculus;

   Most preferable: Math 3592H, 3593H: Honors Mathematics

   Next: Math 1571H, 1572H: Honors Calculus

   Next: Math 1371, 1372: IT Calculus

   Next: Math 1281, 1282: Calculus for the Biological Sciences

   Next: Math 1271, 1272: Standard calculus

   Second year calculus: Continue the sequence that the student is in: Math 2573H, Math 2574H or Math 2373, Math 2374 or Math 2243, 2263.

   Math 3283: Sequences, Series and Foundations; Writing Intensive (for those students who have not taken the Honors sequence)

   Some of the students who choose this option will enter with some sort of advanced placement. Individual consultation will be necessary to place them properly.

2. Analysis and Algebra requirements:

   One of the sequences Math 5615-5616 Honors: Introduction to Analysis I, II and Math 5285-5286 Honors: Fundamental Structures of Algebra I, II must be taken. Many students in this track may wish to take both of these sequences.

   If Math 5615-5616 is taken and Math 5285-5286 is not taken, the algebra requirement must be chosen from Math 4281, Introduction to Modern Algebra and one of Math 5385, Introduction to Computational Algebraic Geometry or one of Math 5705: Enumerative Combinatorics or Math 5707: Graph Theory and Non-Enumerative Combinatorics.

   If Math 5285-5286 is taken and Math 5615-5616 is not taken, the analysis requirement must be chosen from Math 5525, Introduction to Ordinary Differential Equations, Math 5535, Dynamical Systems and Chaos, Math 5588, Elementary Partial Differential Equations II, Math 5652, Introduction to Stochastic Processes, and Math 5654, Prediction and Filtering.

3. Other mathematics courses to make a total of ten, including the technical elective (for CLA majors replace "ten" by "eight") 5-xxx or higher level mathematics courses must be chosen from the courses listed above (in section 2) or from among the following courses:

   Math 5165-5166, Mathematical Logic I, II

   Math 5345, Introduction to Topology

   Math 5378, Differential Geometry

   Math 5467, Introduction to the Mathematics of Wavelets

   Math 5485-5486, Introduction to Numerical Methods I, II

   Math 5487-5488, Computational Methods for Differential and Integral Equations in Engineering and Science

   Math 5587, Elementary Partial Differential Equations I

   Math 5583, Complex Analysis

   Math 5651, Basic Theory of Probability and Statistics or Stat 5101

4. Other requirements:

   Students are expected to gain some expertise in and understanding of computer programming. Among the ways do this, taking CSci 1113, Introduction to C/C++ Programming for Scientists and Engineers and/or CSci 1901/1902, Structure of Computer Programming I, II are frequently used.

   Physics: 3 semesters as described in the mathematics major description in the undergraduate catalogue (p.290 in the 2002-2004 undergraduate catalogue).

   Either Math 4997W Senior Project (required for CLA majors) and/or at least one

   8-xxx level course in mathematics.

These are the additional requirements for an emphasis in Mathematical Biology.

• Lower Division Requirements:
  Either Biology 1001 and 1002, or Biol 1009. Note again that IT students with this emphasis are not required to take the third semester of Physics.
• Upper Division requirements
  All students seeking a degree in math biology must complete the following requirements:
  • Senior Seminar (1 credit)
  • Math 4242 (Applied Linear Algebra).
  • Math 4428 (Mathematical Modeling)
  • Math 5651 (Basic Theory of Probability and Statistics) or Stat 5101
  • Research internship (4 credits; recommended but not required). Selected sites may include university departments, companies, or national laboratories
  • Math 5525 (Intro to Ordinary Differential Equations)
  • Math 5535 (Dynamical Systems and Chaos)

• Chemistry 1021-1022 (Chemical Principles I and II)
• GCD 3022 (Genetics) or Biol 4003 (Genetics)

and at least two courses from the following list:

• Pbio/PlPa 5301 (Plant Genomics)
• EEB 5221 (Molecular and Genetic Evolution
• CSci 5481 (Computational Techniques for Genomics)
• BioC 5361 (Microbial Genomics and Bioinformatics) or Phcl 5111 (Pharmacogenomics)
• Math 5445 Mathematical Analysis of Biological Networks