Department of Applied Mathematics
Mark Hoefer, Department Chair
UCB 526
1111 Engineering Drive, ECOT 225
Boulder, CO 80309
T: 303-492-4668
The Department of Applied Mathematics in the College of Arts and Sciences offers a Bachelor of Science degree in applied mathematics through the College of Engineering and Applied Science. The BS degree is designed to prepare graduates for exciting and diverse professional careers, and for graduate study in a wide variety of disciplines.
Courses at the undergraduate level provide training in a broad range of mathematical techniques and problem-solving strategies. These courses teach the concepts and methods central to applications of linear algebra, ordinary and partial differential equations, numerical analysis, probability, statistics and data science, complex variables and nonlinear dynamics. Since applied mathematicians often are involved in interdisciplinary work, the BS degree requires an in-depth knowledge of some area of science or engineering where mathematics is used. This knowledge prepares graduates to successfully communicate and cooperate with engineers and scientists. The BS degree also requires knowledge of a programming language and skill in using the computer.
For more information, visit the department's Prospective Students webpage.
Research Opportunities
The Department of Applied Math offers a broad range of undergraduate research opportunities funded by multiple agencies including the National Science Foundation. Working with faculty, applied math students have developed solutions to a variety of problems in fluids, dynamical systems, data analysis, networks, signal processing, math biology, math education and numerics. Students can do both theoretical and experimental work in the Dispersive Hydrodynamics Lab, gain practical experience in statistics and data science through LISA, the Laboratory for Interdisciplinary Statistical Analysis or work on individual research projects with departmental and affiliated faculty.
Students can gain professional exposure through the student chapter of the Society of Industrial and Applied Mathematics (SIAM) or through the Data Buffs, the student chapter of the American Statistical Association. Applied Math also has a local chapter of AWM, the Association for Women in Mathematics.
- Requirements
- Recommended Areas of Application
- I. Actuarial
- II. Aerospace Engineering Sciences
- III. Chemical Engineering
- IV. Civil, Environmental and Architectural Engineering
- V. Computational Biology and Bioinformatics
- VI. Computer Science
- VII. Creative Technology and Design
- VIII. Electrical, Computer & Energy Engineering
- IX. Engineering Physics/Physics
- X. Finance
- XI. Geographic Information Science (GIS)
- XII. Geological Sciences
- XIII. Mechanical Engineering
- XIV. Statistics and Data Science
- XV. Advisor Approved Option
Requirements
Required Courses and Credits
The BS degree in applied mathematics requires the satisfactory completion of a minimum of 128 credit hours as follows. All prerequisite courses must be passed with a C- or better.
All AMEN-BS majors are required to take a minimum of 19 credits in APPM or STAT on the CU Boulder campus, with a minimum of 16 credits being upper-division APPM or STAT coursework.
| Code | Title | Credit Hours |
|---|---|---|
| Calculus | ||
| APPM 1350 | Calculus 1 for Engineers | 4 |
| or APPM 1345 | Calculus 1 with Algebra, Part B | |
| or MATH 1300 | Calculus 1 | |
| APPM 1360 | Calculus 2 for Engineers | 4 |
| or MATH 2300 | Calculus 2 | |
| APPM 2350 | Calculus 3 for Engineers | 4 |
| or MATH 2400 | Calculus 3 | |
| Computing Experience | ||
| APPM 1650 | Python for Math and Data Science Applications | 4 |
| or CSCI 1300 | Computer Science 1: Starting Computing | |
| or CSCI 2275 | Programming and Data Structures | |
| or CHEN 1310 | Introduction to Engineering Computing | |
| or ECEN 1310 | Introduction to C Programming | |
| Science Requirement | ||
| PHYS 1110 | General Physics 1 | 4 |
| or PHYS 1115 | General Physics 1 for Majors | |
| PHYS 1120 | General Physics 2 | 4 |
| or PHYS 1125 | General Physics 2 for Majors | |
| PHYS 1140 | Experimental Physics 1 1 | 1 |
| Select one of the following options (including at least 1 credit of laboratory science): 2 | 4 | |
| General Chemistry for Engineers 1 and Laboratory in General Chemistry 1 | ||
| Accelerated Chemistry for Engineers and Engineering General Chemistry Lab | ||
| General Chemistry 1 and Laboratory in General Chemistry 1 | ||
| General Biology 1 and General Biology 2 and General Biology Laboratory 1 and General Biology Laboratory 2 | ||
| Introduction to Cellular and Molecular Biology and Principles of Genetics (and one 2-credit lab) 3 | ||
| Introduction to Quantum Mechanics and Its Applications and Experimental Physics 2 | ||
| Foundations of Modern Physics and Experimental Physics 2 | ||
| APPM Courses | ||
| APPM 2360 | Introduction to Differential Equations with Linear Algebra | 4 |
| or MATH 2130 & MATH 3430 | Introduction to Linear Algebra for Non-Mathematics Majors and Ordinary Differential Equations | |
| or MATH 2135 & MATH 3430 | Introduction to Linear Algebra for Mathematics Majors and Ordinary Differential Equations | |
| APPM 3310 | Matrix Methods and Applications | 3 |
| APPM 4350 | Methods in Applied Mathematics: Fourier Series and Boundary Value Problems | 3 |
| APPM 4360 | Methods in Applied Mathematics: Complex Variables and Applications | 3 |
| APPM 4440 | Undergraduate Applied Analysis 1 | 3 |
| or MATH 3001 | Analysis 1 | |
| or MATH 3140 | Abstract Algebra 1 | |
| APPM 4600 | Numerical Methods and Scientific Computing | 4 |
| APPM or STAT Senior Sequence | ||
| Complete a two-semester course sequence of applied mathematics or statistics courses numbered 4000 or above, chosen from the following 4 | 6 | |
| Modeling in Applied Mathematics and Modeling in Mathematical Biology | ||
| Undergraduate Applied Analysis 1 and Undergraduate Applied Analysis 2 | ||
| Numerical Methods and Scientific Computing and Numerical Differential Equations | ||
| Applied Probability and Introduction to Mathematical Statistics 4 | ||
| Applied Probability and Markov Processes, Queues, and Monte Carlo Simulations 4 | ||
| Markov Processes, Queues, and Monte Carlo Simulations and Introduction to Mathematical Statistics | ||
| Statistical Methods and Application I and Statistical Methods and Applications II | ||
| APPM or STAT Courses Numbered 3000 or Above | ||
| Select additional coursework to bring total upper-division APPM or STAT credit hours to at least 25 including required courses above. 5 | 3 | |
| Area of Application | ||
| A minimum of 24 credit hours in Engineering, Arts & Sciences, Business, or other approved courses with significant mathematical content (see "Recommended Areas of Application"). | 24 | |
| Humanities, Social Sciences and Writing Requirements 6 | 18 | |
| Free Electives | 28 | |
| Free electives should be chosen to bring the total credit hours to a minimum of 128. | ||
| Total Credit Hours | 128 | |
| 1 | Waived for students also completing B.S. in Aerospace Engineering Sciences. If waived, student completes additional Free Elective credits. |
| 2 | Students completing a B.S. in Aerospace Engineering Sciences or a B.S. in Mechanical Engineering may use MCEN 1024 to complete this requirement. If this course is used to fulfill this requirement, one additional credit of Free Electives is needed. |
| 3 | Plus one MCDB lab course for 2 credits, chosen from MCDB 1161, MCDB 1171, MCDB 1181, or MCDB 2171. |
| 4 | APPM 3570 is the only 3000-level course that can be used to satisfy this requirement. |
| 5 | No more than 3 credit hours of APPM 4840 may count towards these 25. No more than 6 credit hours of independent study are allowed for credit towards the BS degree in applied mathematics. |
| 6 | Students may choose from the list of college-approved humanities and social sciences (HSS) and writing requirements. |
Recommended Areas of Application
To fulfill their degree requirements, applied mathematics majors are required to take a coherent set of 24 credit hours in Engineering, Arts & Sciences, or Business courses with significant mathematical content. Areas of application must include:
- At least 6 credit hours in courses numbered 3000 or above.
- At least 15 credit hours in courses numbered 2000 or above.
Several possible options are listed below. It should be stressed that the listed courses and options are suggestions and not requirements. Students may formulate their own option to meet their educational and career goals, but final course selection should be made in consultation with an applied math advisor or faculty mentor.
Students cannot double count area of application courses with credits applying toward other requirements for their Applied Math degree, including the APPM/STAT Senior Sequence, Humanities & Social Science, computing, or science requirement. Students cannot count coursework toward their area of application that is equivalent to another course that is already applying toward degree requirements (e.g., a student who has completed CHEN 1201 for a degree requirement cannot apply CHEM 1113 toward their area of application).
For additional information and planning advice, consult the Applied Mathematics Undergraduate Curriculum Guide on the Applied Mathematics Major webpage.
I. Actuarial
Students must be admitted to the Actuarial Studies and Quantitative Finance Certificate Program to enroll in required BCOR/FNCE courses.
Students cannot double count area of application courses with credits applying toward other requirements for their Applied Math degree, including the upper division APPM, APPM/STAT Senior Sequence, Humanities & Social Science, computing, or science requirement.
| Code | Title | Credit Hours |
|---|---|---|
| BCOR 2203 | Principles of Accounting I 1 | 1.5 |
| BCOR 2204 | Principles of Financial Management 1 | 1.5 |
| ECON 3070 | Intermediate Microeconomic Theory | 4 |
| ECON 4070 | Topics in Microeconomics | 3 |
| FNCE 3010 | Corporate Finance | 3 |
| APPM 3570 | Applied Probability 2 | 3 |
| STAT 4520 | Introduction to Mathematical Statistics | 3 |
| STAT 4540 | Introduction to Time Series | 3 |
| At least one of the following courses must be taken | 3 | |
| FNCE 3030 | Investment and Portfolio Management | - |
| FNCE 4040 | Derivative Securities | - |
| ECON 4818 | Introduction to Econometrics | - |
| Total Credit Hours | 25 | |
| 1 | The prerequisites for BCOR 2203 and BCOR 2204 are waived for students enrolled in the Actuarial Studies and Quantitative Finance Certificate Program. |
| 2 | The first actuarial examination, Exam P/1, can be taken after completing this course. |
II. Aerospace Engineering Sciences
Students who pursue this area of application are usually double majors. Students who wish to enroll in ASEN courses without being a double major should see their applied mathematics advisor for next steps.
III. Chemical Engineering
Students completing this area of application must take at least 24 credits of coursework in this area. See recommended courses below:
| Code | Title | Credit Hours |
|---|---|---|
| Recommended courses include: | ||
| Introduction to Engineering Computing | ||
| Chemical Engineering Material and Energy Balances | ||
| Chemical Engineering Fluid Mechanics | ||
| Chemical Engineering Heat and Mass Transfer | ||
| Chemical Engineering Separations | ||
| Chemical Engineering Thermodynamics 1 | ||
| Energy Fundamentals | ||
| Kinetics and Reactor Design | ||
| Physical Chemistry for Engineers | ||
| Organic Chemistry 1 | ||
IV. Civil, Environmental and Architectural Engineering
Students completing this area of application must take at least 24 credits of AREN/CVEN/EVEN coursework (see recommended courses below). Students completing this area of emphasis may wish to complete a minor in Architectural or Civil Engineering.
Students wishing to enroll in AREN or CVEN courses that are restricted to majors only must do so through the departmental course request form.
| Code | Title | Credit Hours |
|---|---|---|
| Recommended Basic Courses | ||
| AREN 2110 | Thermodynamics | 3 |
| CVEN 2121 | Analytical Mechanics 1 | 3 |
| CVEN 3161 | Mechanics of Materials 1 | 3 |
| CVEN 3313 | Theoretical Fluid Mechanics | 3 |
| or AREN 2120 | Fluid Mechanics and Heat Transfer | |
| Additional Courses | ||
| Select two courses from any one of the following groups plus additional AREN, CVEN, or EVEN courses to bring the total credit hours to 24: | 12 | |
| Environmental and Water Resources Engineering | ||
| Fundamentals of Environmental Engineering | ||
| Engineering Hydrology | ||
| Structural Engineering | ||
| Structural Analysis | ||
| Geotechnical Engineering 1 | ||
| Steel Design | ||
| Reinforced Concrete Design | ||
| Architectural Engineering | ||
| Building Materials and Systems | ||
| Energy Efficient Buildings | ||
| Illumination I | ||
| Total Credit Hours | 24 | |
V. Computational Biology and Bioinformatics
The following concentration of selected courses from computer science, biology and chemistry provide the foundation for work in mathematical biology, computational biology and/or bioinformatics.
Complete at least 24 credits from the following:
| Code | Title | Credit Hours |
|---|---|---|
| CSCI 2270 | Computer Science 2: Data Structures | 4 |
| or APPM 3650 | Algorithms and Data Structures in Python | |
| CHEM 3311 & CHEM 3321 | Organic Chemistry 1 and Laboratory in Organic Chemistry 1 | 5 |
| MCDB 1150 & MCDB 1152 | Introduction to Cellular and Molecular Biology and Problem Solving Co-Seminar for Introduction to Molecular and Cellular Biology | 4 |
| MCDB 2150 & MCDB 2152 | Principles of Genetics and Problem Solving Co-Seminars for Genetics | 4 |
| MCDB 3135 & MCDB 3140 | Molecular Biology and Cell Biology Laboratory | 5 |
| CSCI 3104 | Algorithms | 4 |
| CSCI 4314 | Dynamic Models in Biology | 3 |
VI. Computer Science
Students completing this area of application should have a minor in computer science plus take additional CSCI coursework to bring the total to at least 24 credits.
VII. Creative Technology and Design
Students interested in this area of application should complete requirements for the Creative Technology and Design Minor plus additional ATLS or CSCI coursework to bring the total to at least 24 credits.
Students must declare the Creative Technology and Design Minor to enroll in ATLS classes.
VIII. Electrical, Computer & Energy Engineering
Students interested in this area of application should consult with the Electrical, Computer and Energy Engineering website, as several minors are available. A minimum of 24 credits is required.
IX. Engineering Physics/Physics
Students completing the physics area of application should complete the Minor in Physics, plus additional PHYS coursework to bring the total to at least 24 credits.
Note: PHYS 1230, PHYS 1240, PHYS 2010 and PHYS 2020 cannot count toward this area of application.
X. Finance
Students must be admitted to the Actuarial Studies and Quantitative Finance Certificate Program to enroll in required BCOR/FNCE courses.
Students cannot double count area of application courses with credits applying toward other requirements for their Applied Math degree, including the upper division APPM, APPM/STAT Senior Sequence, Humanities & Social Science, computing, or science requirement.
| Code | Title | Credit Hours |
|---|---|---|
| Required Courses | ||
| BCOR 2203 | Principles of Accounting I | 1.5 |
| BCOR 2204 | Principles of Financial Management | 1.5 |
| FNCE 3010 | Corporate Finance | 3 |
| ECON 3070 | Intermediate Microeconomic Theory | 4 |
| ECON 4818 | Introduction to Econometrics | 3 |
| Complete a minimum of two of the following courses | 6 | |
| Corporate Financial Reporting 1 | ||
| Investment and Portfolio Management | ||
| Derivative Securities | ||
| Topics in Finance | ||
| Financial Markets and Institutions | ||
| Additional courses that may be taken to bring the total to 24 credits | 5 | |
| Corporate Financial Reporting 2 | ||
| Financial Institutions Management | ||
| Capital Investment Analysis | ||
| Special Topics in Finance | ||
| Total Credit Hours | 24 | |
XI. Geographic Information Science (GIS)
Students interested in this area of application must complete the requirements for the Certificate in GIS and Computational Science plus additional GIS Electives to bring the total to at least 24 credits.
XII. Geological Sciences
Students interested in this area of application must complete the requirements for the Minor in Geology plus additional GEOL Electives to bring the total to at least 24 credits.
XIII. Mechanical Engineering
Students completing this area of application must take at least 24 credits of MCEN coursework (see recommended courses below).
Students wishing to enroll in MCEN courses that are restricted to majors must do so through the departmental course request form.
| Code | Title | Credit Hours |
|---|---|---|
| Recommended Courses | ||
| MCEN 2023 | Statics and Structures | 3 |
| MCEN 2043 | Dynamics | 3 |
| MCEN 2063 | Mechanics of Solids | 3 |
| MCEN 3012 | Thermodynamics | 3 |
| MCEN 3021 | Fluid Mechanics | 3 |
| MCEN 3022 | Heat Transfer | 3 |
| MCEN 3025 | Component Design | 3 |
| MCEN 4043 | System Dynamics | 3 |
XIV. Statistics and Data Science
Students interested in this area of application must complete the requirements for the Statistics and Data Science Minor plus additional Minor electives to bring the total to at least 24 credits.
Note: the 12 upper-division statistics credits required for the minor may not be counted toward the 25 credits of upper-division math courses for the bachelor's degree.
XV. Advisor Approved Option
Students may formulate their own area of application to meet their educational and career goals.
The area of application must have a focused theme that is significantly different from the areas of application listed above. Final course selection must be approved by an applied math advisor or faculty mentor.
Recommended Four-Year Plan of Study
Students must complete 128 hours for graduation.
| Year One | ||
|---|---|---|
| Fall Semester | Credit Hours | |
| APPM 1350 | Calculus 1 for Engineers | 4 |
| CHEN 1201 | General Chemistry for Engineers 1 | 4 |
| CHEM 1114 | Laboratory in General Chemistry 1 | 1 |
| APPM 1650 | Python for Math and Data Science Applications | 4 |
| COEN 1500 | CEAS First Year Seminar | 1 |
| Humanities or Social Sciences Elective 1 | 2 | |
| Credit Hours | 16 | |
| Spring Semester | ||
| APPM 1360 | Calculus 2 for Engineers | 4 |
| PHYS 1110 | General Physics 1 | 4 |
| Free Electives | 3 | |
| Area of Application Course | 3 | |
| Humanities or Social Sciences Elective 1 | 3 | |
| Credit Hours | 17 | |
| Year Two | ||
| Fall Semester | ||
| APPM 2350 | Calculus 3 for Engineers | 4 |
| PHYS 1120 | General Physics 2 | 4 |
| PHYS 1140 | Experimental Physics 1 | 1 |
| APPM 3170 | Discrete Applied Mathematics (Recommended, but not required) | 3 |
| Humanities or Social Sciences Elective 1 | 3 | |
| Credit Hours | 15 | |
| Spring Semester | ||
| APPM 2360 | Introduction to Differential Equations with Linear Algebra | 4 |
| APPM 2460 | Differential Equations Computer Lab (Recommended, but not required) | 1 |
| APPM 3310 | Matrix Methods and Applications | 3 |
| Area of Application course | 3 | |
| Free Electives | 3 | |
| Humanities or Social Sciences Elective 1 | 3 | |
| Credit Hours | 17 | |
| Year Three | ||
| Fall Semester | ||
| APPM 4350 | Methods in Applied Mathematics: Fourier Series and Boundary Value Problems | 3 |
| APPM 4440 | Undergraduate Applied Analysis 1 | 3 |
| Area of Application Course | 3 | |
| Area of Application Course | 3 | |
| College-approved writing course 2 | 3 | |
| Credit Hours | 15 | |
| Spring Semester | ||
| APPM 4360 | Methods in Applied Mathematics: Complex Variables and Applications | 3 |
| Area of Application Course | 3 | |
| Free Electives | 7 | |
| Humanities or Social Sciences Elective 1 | 3 | |
| Credit Hours | 16 | |
| Year Four | ||
| Fall Semester | ||
| APPM 4600 | Numerical Methods and Scientific Computing | 4 |
| Senior Sequence Course #1 | 3 | |
| Area of Application Course | 3 | |
| Free Electives | 6 | |
| Credit Hours | 16 | |
| Spring Semester | ||
| Senior Sequence Course #2 | 3 | |
| Area of Application Course | 3 | |
| Area of Application Course | 3 | |
| Free Electives | 7 | |
| Credit Hours | 16 | |
| Total Credit Hours | 128 | |
| 1 | Students may choose courses from the list of college-approved humanities and social sciences (HSS) electives. |
| 2 | Students may choose a course from the list of college-approved writing courses. |
Learning Outcomes
Content Knowledge
Students completing the undergraduate degree in Applied Mathematics will be broadly knowledgeable in a number of mathematical areas including:
- Differential and integral calculus in one and several variables.
- Vector spaces and matrix algebra.
- Ordinary and partial differential equations.
- At least one programming language.
- At least one application software package in either mathematics or statistics.
- Methods of complex variables as used in applications.
- Numerical solutions of linear and nonlinear problems.
- An in-depth knowledge of an area of application (statistics, an engineering discipline, a natural science field, or one of the quantitative areas of business and economics).
Student Outcomes
Upon graduation, students will:
- Acquire foundational knowledge in calculus, ordinary and partial differential equations, vector spaces and matrix methods, analysis, numerical analysis, complex variables, and probability and statistics.
- Develop proficiency in at least one programming language.
- Acquire an in-depth knowledge of an area of application (statistics, an engineering or natural science field, or one of the quantitative areas of finance and economics).
- Acquire problem-solving and modeling skills that allow them to formulate a real-world problem in a mathematical setting and implement a numerical solution.
- The ability to clearly and concisely, in oral and in written forms, communicate analytic arguments.
Bachelor's–Accelerated Master's Degree Program(s)
The bachelor's–accelerated master's (BAM) degree program options offer currently enrolled CU Boulder undergraduate students the opportunity to receive a bachelor's and master's degree in a shorter period of time. Students receive the bachelor's degree first but begin taking graduate coursework as undergraduates (typically in their senior year).
Because some courses are allowed to double count for both the bachelor's and the master's degrees, students receive a master's degree in less time and at a lower cost than if they were to enroll in a stand-alone master's degree program after completion of their baccalaureate degree. In addition, staying at CU Boulder to pursue a bachelor's–accelerated master's program enables students to continue working with their established faculty mentors.
BS and MS in Applied Mathematics
Admissions Requirements
In order to gain admission to the BAM program named above, a student must meet the following criteria:
- Have a cumulative GPA of 3.40 or higher.
- Have a minimum GPA of 3.40 in APPM, STAT, and MATH courses.
- Have at least junior class standing.
- Completion of all MAPS requirements and no deficiencies remaining (students admitted to CU Boulder prior to Summer 2023 only).
- Satisfactory completion of at least two APPM courses numbered 3000 or higher.
- Two letters of recommendation from CU Boulder Department of Applied Mathematics faculty.
Program Requirements
Students may take up to and including 12 graduate credit hours while in the undergraduate program which can later be used toward the master’s degree. However, only six credits may be double counted toward the bachelor’s degree and the master’s degree. Students must apply to graduate with the bachelor’s degree, and apply to continue with the master’s degree, early in the semester in which the undergraduate requirements will be completed.
Please see the Applied Mathematics/Applied Mathematics BAM degree program webpage for more information.