The Department of Mathematics offers a strong graduate program leading to the Master of Science (M.S.) in Applied Mathematics. Specific areas of study currently available include probability and statistics, differential equations, applied analysis, algebra, and coding theory.

### Applied Mathematics, MS-Tracks

To respond to the needs of both students and employers, the Department has organized this degree program into a system of four tracks, which are intended to help students develop their programs of study:

- PhD preparation Track
- Applied and computational mathematics track
- Education Track
- Business and management track

Customized programs of study are also available.

### Student Learning Outcomes, Applied Mathematics, MS

- Students will understand core graduate mathematics material and students must demonstrate substantial comprehension of Linear Algebra and Analysis, the common core subjects required of all students in the program.
- Competence in written and oral communication is essential for most mathematical careers. Such communication skills are also important in many other aspects of life, therefore students will be able to deliver written and oral presentations demonstrating comprehension of complex mathematical content and the ability to communicate that complex mathematical content to a broad audience (general department faculty and graduate students).
- The program prepares students for a variety of mathematical careers. The current program has four identified tracks: computational and applied mathematics, education, business and management, and PhD preparation. Students should be prepared for employment requiring mathematical skill and sophistication at the Master’s level.
- Students shall develop a more sophisticated view of mathematics than is achieved in the undergraduate program. The opportunity for sophisticated learning comes both in and outside the classroom. Student exposure to mathematical research and advanced applications is an important aspect of this goal.
- Students successfully completing the PhD preparation track should be able to enter quality doctoral programs with a reasonable probability of success.
- Students should have the opportunity to hear research talks in seminars and colloquia, and participate in research projects (this might come through on campus employment, independent study, or thesis work). Students are strongly encouraged to attend our biweekly colloquium.

The admission criteria include but are NOT limited to the following requirements. Any decision regarding admission is made by the Graduate Committee on an individual basis, after taking into consideration the application material and supporting documents.

- Bachelor degree in mathematics (or a Bachelor degree in some other field, with extensive course work in mathematics), including a course in analysis comparable to the UCCS course Math 4310
- A minimum grade point average of 3.0. Under special circumstances students may be admitted with a lower grade point average (or without a course in analysis) as provisional degree students.

Please refer to the Graduate School admissions requirements.

- A 3.0 grade point average in course work applied towards the degree.
- All degree courses must be part of an approved plan of study, developed by the student and approved by the advisor (Chair of the Graduate Committee) within the first semester after being admitted to the program. This plan may be revised at any time with the approval of the advisor. The plan will require students to demonstrate some cohesiveness in the courses chosen, or to demonstrate a clear subject area of concentration.
- Courses will have graduate rank only if members of the graduate school faculty teach them and are at the 5000 level or above.
- Modern Analysis I (Math 4310) plus at least 27 semester hours of approved graduate work, including Linear Algebra (Math 5130) and Modern Analysis II (Math 5320). All students must pass the comprehensive examination in Modern Analysis. Students must complete the Analysis sequence (Math 4310 and Math 5320) in their first year of admission, and attempt the Comprehensive exam in Analysis offered in the Summer.
**Note**: The Math 4310 requirement may be waived on a case-by-case basis only for UCCS students who completed Math 4310 with a grade of B or higher within one year before admission to the program, and for students in the Accelerated Master's Program. If a waiver is approved by the Graduate Chair, then Math 4310 shall be replaced by 3 semester hours of approved graduate work.

Students completing one of the four tracks of study will automatically fulfill the requirements for the MS Applied Math degree. - All students must successfully complete one of the following: Math 5480, Math 5650, Math 5670, or Math 3670/5900 to fulfill the computational competency requirement. Please note that knowledge of a computer language such as MATLAB is necessary to complete this requirement. This requirement may be waived at the discretion of the Graduate Committee Chair upon providing evidence of previous computing experience.
- A student may complete up to 9 hours of appropriate graduate course work in departments other than the Department of Mathematics, as part of the “tracks” program. Such courses MUST be PRE-approved by the advisor.
- Students may select a thesis or non thesis option. Students in the thesis option will replace 6 hours of graduate level courses with a Master’s thesis.
- All students make an oral presentation regarding some aspect of advanced mathematics. For students pursuing the thesis option, the thesis defense will qualify as such a presentation.

The department graduate committee must approve exceptions to these requirements.

- Generally, up to 9 semester hours of graduate level mathematics courses completed prior to admission to the program may be applied towards the credit requirements of the MS Applied Mathematics degree.
- At least one mathematics course is usually offered at the 400/500 level during each Summer semester. "Full time" students normally complete the MS Applied Mathematics program in two calendar years. Prospective students holding full time jobs are also encouraged to enroll. Such students typically complete one course per semester, and normally complete the degree in four to five years.
- For a complete description of all courses taught by the mathematics department, as well as for a complete description of the requirements for the degree MS Applied Mathematics, students should obtain a copy of the UCCS Course Catalog, available at the UCCS Bookstore.

The Ph.D Preparation track is designed for those individuals who are interested in obtaining a Masters Degree in Mathematics that will enable them to pursue a Ph.D degree in pure mathematics at a later time. It has been our experience that students interested in this track tend to emphasize pure mathematics rather than applied mathematics. To that end, the Ph.D Preparation track includes a solid exposure to pure mathematics with special emphasis in algebra, analysis and probability.

The Ph.D Preparation track consists of a core list of required courses together with an elective strategy. These are described below.

The core course list for the Ph.D Preparatory Track (All 5 courses)

- MATH 5130 – Linear Algebra
- MATH 5150 – Modern Algebra II
- MATH 5320 – Modern Analysis II
- MATH 5330 – Real Analysis
- MATH 5450 – Complex Variables

The elective course list for the Ph.D Preparatory Track (Choose 5 courses)

- MATH 5170– Graduate Modern Algebra I
- MATH 5270 – Algebraic Coding Theory
- MATH 5430 – Ordinary Differential Equations
- MATH 5620 – Complex Variables II
- MATH 5850 – Stochastic Modeling

or any other 5000-level mathematics course offered by University of Colorado at Colorado Springs, approved by the Mathematics Graduate Advisor.

The Applied Mathematics and Scientific Computation Track for the Masters Degree in Applied Mathematics is designed for those individuals pursuing careers in science and engineering with major mathematical or computational content. A mathematician working in an applied environment should have some knowledge of another discipline as well as modern computational skills. A typical problem requires construction of mathematical models from the concepts and data of the relevant branch of science, and reporting solutions and conclusions in the language of the discipline. To insure that graduates have some background in the major topics, a year sequence in Methods of Applied Mathematics and Scientific Computation is recommended. Courses from the core make up 18 of the required 30 hours for a Master's Degree.

Core course list for the Applied Science Track

- MATH 5130 - Linear Algebra
- MATH 5470 - Methods of Applied Mathematics
- MATH 5420 - Optimization
- MATH 5900 - Fourier Analysis
- MATH 5320 - Modern Analysis II
- MATH 5670 - Scientific Computation
- MATH 5850 - Stochastic Modeling
- MATH 5430 - Ordinary Differential Equations
- MATH 5650 - Numerical Analysis

Beyond the core, any graduate course offered by the Department of Mathematics or by other Science and Engineering departments is suitable for the degree, but students must develop a program of study with a Mathematics Graduate Advisor. The topics and corresponding courses from mathematics most likely to be encountered in applications are listed below.

Applied and Applicable Courses

- MATH 5450 - Complex Variables
- MATH 5520 - Perturbation Theory in Astrodynamics
- MATH 5350 - Applied Functional Analysis
- MATH 5480 - Mathematical Modeling
- MATH 5820 - Introduction to Mathematical Statistics
- MATH 5270 - Algebraic Coding Theory
- MATH 5250 - Introduction to Chaotic Dynamical Systems

The Education track is designed for those individuals who are interested in pursuing a career in teaching mathematics at the K- 14 level, but who desire the rigor and quality of a Master of Science (M.S.) degree in Applied Mathematics. Secondary schools and community colleges are increasingly interested in mathematics teachers who can demonstrate a competent use of technology within the classroom and who can integrate this technology with meaningful applications of mathematics. To that end, the Education track includes a core set of three courses that emphasize the use of applications and technology in the mathematics curriculum. A student successfully completing the Education track will have a solid background in the fundamental areas of mathematics (algebra, analysis), an excellent exposure to the applications of mathematics, particularly those that can be taught in the K-14 classroom, and robust training in the use of technology in the mathematics classroom.

The Education track consists of a core list of required courses together with an elective strategy. These are described below. Please visit the Math Course Descriptions page for details about the courses.

The core course list for the Education Track

- MATH 5130 – Linear Algebra
- MATH 5210 -- Modern Geometry
- MATH 5320 – Modern Analysis II
- MATH 5820 – Introduction to Mathematical Statistics I

Recommended

- One from CURR 5301 -- Mathematical Connections and Concepts or T ED 5920 Secondary Math Methods

The elective strategies for the Education Track

After successfully passing the core list of courses above, the student must complete 12 additional graduate credit hours in mathematics. It is important to develop some depth in the areas of algebra, analysis, applied mathematics, geometry or probability. To that end, each student should specialize in two of the following areas by taking at least one course in each of the areas.

Algebra

- MATH 5150 – Field theory,
- MATH 5170 – Groups, rings and modules,
- MATH 5270 – Coding theory

Analysis

- MATH 5330 – Real Analysis I
- MATH 5350 – Applied Functional Analysis
- MATH 5420 -- Optimization
- MATH 5430 – Ordinary Differential Equations
- MATH 5450 – Complex Variables,
- MATH 5620 – Complex Variables II

Applied Mathematics and Computing

- MATH 5470 – Methods of Applied Mathematics
- MATH 5480 – Mathematical Modeling
- MATH 5520 – Perturbation Theory in Astrodynamics
- MATH 5650 – Numerical Analysis
- MATH 5840 – Computer vision

Geometry

- MATH 5210 – Higher Geometry,
- MATH 5230 – Fractal Geometry,
- MATH 5250 – Introduction to Chaotic Dynamical Systems,
- MATH 5260 – Elementary Differential Geometry

Probability and Statistics

- MATH 5850 – Stochastic modeling
- MATH 5830 – Linear Statistical Models
- MATH 5910 – Theory of Probability

The Business Track for the Master Degree in Applied Mathematics is designed for those individuals pursuing careers as actuaries, in financial services, or in other aspects of business and economics. Some of the core mathematical concepts in these areas include linear algebra, optimization, probability, statistics, and stochastic processes. Most applications require modern computational techniques. Please visit the Math Course Descriptions page for details about the courses.

Core Math courses for the Business Track

- MATH 5130 - Linear Algebra
- MATH 5320 - Analysis
- MATH 5810 - Mathematical Statistics I
- MATH 5820 - Mathematical Statistics II

Courses from the core make up 12 of the required 30 hours for a Masters Degree. Other topics and corresponding courses from Mathematics most likely to be encountered in business are listed below.

Specialty courses from Mathematics (Choose at least 2)

- MATH 5530 - Ordinary Differential Equations
- MATH 5650 - Numerical Analysis
- MATH 4480 - Mathematical Modeling
- MATH 5420 - Optimization

Both financial services and Actuarial options require courses from Economics and Finance. These options contain adequate preparation for the first two actuarial exams.

Additional required courses for Actuarial or Finance options

- MATH 5850 - Stochastic Modeling

Two of four economics and finance courses:

- ECON 4010 - Advanced Micro Economic Theory
- ECON 4020 - Advanced Macro Economic Theory
- FIN 4010 - Advanced Corporate Finance
- FIN 4020 - Cases and Concepts in Finance

Other Appropriate Courses From Business:

- Marketing
- Operations Management
- Information Systems
- Risk Management