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Applied Mathematics and Statistics

2015-16 General Catalog 

Baskin School of Engineering
(831) 459-2158
http://www.soe.ucsc.edu

Faculty | Course Descriptions

Program Description

Applied mathematics and statistics are disciplines devoted to the use of mathematical methods and reasoning to solve real-world problems of a scientific or decision-making nature in a wide variety of subjects, principally (but not exclusively) in engineering, medicine, the physical and biological sciences, and the social sciences. Applied mathematical modeling often involves the use of systems of (partial) differential equations to describe and predict the behavior of complex real-world systems that unfold dynamically in time. Statistics, construed broadly, is the study of uncertainty: how to measure it (using ideas and methods in probability theory), and what to do about it (using concepts from statistical inference and decision theory). 

The Applied Mathematics and Statistics Department at UCSC offers master’s and doctoral programs in Statistics and Applied Mathematics, or Applied Mathematics and Statistics, depending on chosen emphasis, and a master's program in Scientific Computing and Applied Mathematics (SciCAM). The goal of these programs is to help students develop into independent scholars who are prepared for productive careers in research, teaching, and industry. The department also offers a designated emphasis in statistics, a minor in statistics, and a minor in applied mathematics.

Additional information on these programs can be found on the department’s web pages at www.soe.ucsc.edu.

Undergraduate Programs

Requirements of the Minor in Statistics

The statistics minor is available for students who wish to gain a quantitative understanding of how to (a) measure uncertainty and (b) make good decisions on the basis of incomplete or imperfect information, and to apply these skills to their interests in another field. This minor could also be combined with a major in mathematics as preparation for a graduate degree in statistics or biostatistics. 

Students are required to take a two-quarter basic calculus sequence:

  • Applied Mathematics and Statistics 11A-B or Economics 11A-B or Applied Mathematics and Statistics 15A-B or Mathematics 11A-B or Mathematics 19A-B or Mathematics 20A-B

  • Plus one course from each of the following seven categories:

    • Statistical Concepts: Applied Mathematics and Statistics 5 or 7/L

    • Computer Programming: Biomolecular Engineering 160/L, Computer Science 12A/L or 5C or 5J or 5P or Computer Engineering 13/L

    • Linear Algebra: Applied Mathematics and Statistics 10 or Mathematics 21 (also recommended that students take Applied Mathematics and Statistics 20 or Mathematics 24)

    • Multivariate Calculus: Mathematics 22 or both Mathematics 23A and Mathematics 23B

    • Probability: Applied Mathematics and Statistics 131 Applied Mathematics and Statistics 203 or Computer Engineering 107

    • Statistical Inference: Applied Mathematics and Statistics 132

    • Computational Methods: Applied Mathematics and Statistics 147

  • Plus two electives from the following category:

    • Statistical Elective: Applied Mathematics and Statistics 156; 198; 202; 205B; 206B; 207; 256;

    • Biomolecular Engineering 205; Computer Engineering 108; 145; Computer Science 142; Economics 104; 113; 114; 120; 161B; and 190; Electrical Engineering 151; Mathematics 114; Psychology 181; Sociology 103A; Technology and Information Management 230.

Note: Students planning graduate work in statistics are recommended to choose Mathematics 23A-B, Applied Mathematics and Statistics 205B, and Mathematics 105A-B.

Requirements of the Minor in Applied Mathematics

The applied mathematics minor is available for students who wish to develop (1) proficiency in modeling real-life problems using mathematics and (2) knowledge of standard, practical analytical and numerical methods for the solution of these models. This minor could be combined with a major in any of the physical, biological, mathematical, or engineering sciences as preparation for a graduate degree in that field or in applied mathematics.

Students are required to take the four-quarter calculus sequence:

  • Calculus Sequence: Mathematics 19A-B or Mathematics 20A-B, and Mathematics 23A-B

  • Plus one of the following sequences:

    • Applied Mathematics and Statistics 10 and 20

    • Mathematics 21 and 24

    • Physics 116A and 116B

Note: Students who complete Mathematics 21 and 24 or Physics 116A and 116B, are strongly recommended to complete the MATLAB self-paced tutorial at: http://matlab-training.soe.ucsc.edu/

  • Plus one course from each of the following categories:

    • Probability Theory: Applied Mathematics and Statistics 131 or Computer Engineering 107

    • Dynamical Systems: Applied Mathematics and Statistics 114 or Applied Mathematics and Statistics 214

    • Introduction to Numerical Methods: Applied Mathematics and Statistics 147, Physics 115, or Earth Sciences 119

    • Partial Differential Equations: Applied Mathematics and Statistics 212A, Physics 116C, or Mathematics 107

  • Plus one applied-mathematics elective from the following list:

    • Applied Mathematics elective: Applied Mathematics and Statistics 107/217, 115/215, 132, 198, 212B, 213, 216, 231, 290B

    • Electrical Engineering 103, 154; Computer Engineering 115; Mathematics 103A, 117, 121A; Physics 105, 139A, 139B, 171.

Students may also propose other electives which use applied mathematical methods, subject to approval by the department.

Graduate Programs (M.S., Ph.D.)

The department of Applied Mathematics and Statistics at UCSC offers both masters and doctoral degrees within the Statistics and Applied Mathematics (SAM) graduate program, with respective emphasis in statistics (for a degree title in “Statistics and Applied Mathematics”) or in applied mathematics (for a degree title in “Applied Mathematics and Statistics”). In addition, the department also offers an independent masters program in “Scientific Computing and Applied Mathematics (SciCAM).”

The goal of the two tracks of the SAM graduate program is to help students develop into independent scholars specialized in statistics or applied mathematics respectively, who are prepared for productive careers in research, teaching, and industry. The master of science (M.S.) degree may be used as a terminal degree or as the first step toward the associated doctor of philosophy degree (Ph.D.).

The goal of the SciCAM M.S. program is primarily to prepare students interested in scientific computing for productive careers in industry. However, it also serves as an excellent further preparation for students who prefer to pursue an academic career and wish to transfer into a Ph.D. program later, either in scientific computing/applied mathematics, or in the field related to their undergraduate degree.

Graduate Program in Statistics and Applied Mathematics

Requirements for a Graduate Degree in Statistics and Applied Mathematics

This track is for students emphasizing statistics. All students must complete the core courses described below.

Required Core Courses:

200    Research and Teaching in Applied Mathematics and Statistics

203    Introduction to Probability Theory

205B   Intermediate Classical Inference

206B   Intermediate Bayesian Inference

207    Intermediate Bayesian Statistical Modeling

211    Foundations of Applied Mathematics

256    Linear Statistical Models

280B   Seminar in Statistics and Applied Mathematical Modeling

In addition to these 35 credits, master of science (M.S.) students must complete two additional 5-credit courses from the approved list, for a total requirement of 45 credits; doctor of philosophy (Ph.D.) students must complete four additional 5-credit courses from the approved list, for a total requirement of 55 credits.

Requirements for a Graduate Degree in Applied Mathematics and Statistics

This track is for students emphasizing Applied Mathematics. All students must complete the core courses described below.

Required core courses:

200    Research and Teaching in Applied Mathematics and Statistics

203    Introduction to Probability Theory

211    Foundations of Applied Mathematics

212A   Applied Mathematical Methods I

212B   Applied Mathematical Methods II

213B   Numerical Solutions of Differential Equations

214    Applied Dynamical Systems

280B   Seminar in Statistical and Applied Mathematical Modeling

In addition to these 35 credits, master of science (M.S.) students must complete two additional 5-credit courses from the approved list, for a total requirement of 45 credits; doctor of philosophy (Ph.D.) students must complete four additional 5-credit courses from the approved list, for a total requirement of 55 credits.

For both emphasis tracks, M.S. students will be allowed to substitute up to two courses with their required research project in which they conduct a research program in one or two of the quarters of their second year. The project will consist of solving a problem or problems from the selected area of application and will be presented to the sponsoring faculty member as a written document.

Ph.D. students will be required to serve as teaching assistants for at least two quarters during their graduate study.  Certain exceptions may be permitted for those with extensive prior teaching experience, for those who are not allowed to be employed due to visa regulations, or for other reasons approved by the graduate director.

Qualifying Examinations

At the end of the first year, all students will take a pre-qualifying examination covering the six (non-seminar) core courses. This examination will have two parts: an in-class written examination, followed by a take-home project involving data analysis. Students who do not pass this examination will be allowed to retake it before the start of the following fall quarter; if they fail the second examination they will be dismissed from the program.

Ph.D. students must complete the oral proposal defense, through which they advance to candidacy, by the end of the spring quarter of their third year. The proposal defense is a public seminar as part of an oral qualifying examination given by the qualifying committee.

Thesis and/or Dissertation Requirements

A capstone project is required for the M.S. degree and a dissertation for the Ph.D. degree.

For the M.S. degree, students will conduct a capstone research project in their second year (up to three quarters).  Students must submit a proposal to the potential faculty sponsor by the start of the fourth academic quarter. If the proposal is accepted, the faculty member will become the sponsor and will supervise the research and writing of the project. The project will involve the solution of a problem or problems from the selected area of application. When the project is completed and written, it will be submitted to and must be accepted by a committee of two individuals, consisting of the faculty adviser and one additional reader. Additional readers will be chosen appropriately from within the Applied Mathematics and Statistics Department or outside of it. Either the adviser or the additional reader must be from within the Applied Mathematics and Statistics Department.

A dissertation is required for the Ph.D. degree. Ph.D. students must select a faculty research adviser by the end of the second year. A written dissertation proposal will be submitted to the adviser, and filed with the graduate secretary. A qualifying examination committee will be formed, consisting of the adviser and three additional members, approved by the Chair of the Graduate Program and the Dean of the Graduate Division. The student will submit the written dissertation proposal to all members of the committee and the graduate secretary no less than one month in advance of the qualifying examination. The dissertation proposal will be formally presented in a public oral qualifying examination with the committee, followed by a private examination.

Students will advance to candidacy after they have completed all course requirements (including removal of all incompletes), passed the qualifying examination, and paid the filing fee. Under normal progress, a student will advance to candidacy by the end of the spring quarter of her/his third year. A student who has not advanced to candidacy by the start of the fourth year will be subject to academic probation. Upon advancement to candidacy, a dissertation reading committee will be formed, consisting of the dissertation supervisor and at least two additional readers appointed by the Graduate Program chair upon recommendation of the dissertation supervisor. At least one of these additional readers must be in the Applied Mathematics and Statistics Department. The committee is subject to the approval of the Graduate Division.

The dissertation will consist of a minimum of three chapters composed of material suitable for submission and publication in major professional journals in statistics or applied mathematics. The completed dissertation will be submitted to the reading committee at least one month before the dissertation defense, which consists of a public presentation of the research followed by a private examination by the reading committee.  Successful completion of the dissertation defense is the final requirement for the Ph.D. degree.

Relationship of Masters and Doctoral Programs

The M.S. and Ph.D. programs are freestanding and independent, so that students can be admitted to either. Students completing the M.S. program may proceed into the Ph.D. program, and students in the Ph.D. program can receive a M.S. degree upon completion of M.S. requirements, including the capstone research project. Each Ph.D. student will be required to have knowledge of statistics and applied mathematics equivalent to that required for the M.S. degree. In addition, Ph.D. candidates will be required to complete coursework beyond the M.S. level.

Transfer Credit

Up to three School of Engineering courses fulfilling the degree requirements of either the M.S. or Ph.D. degrees may be taken before beginning the graduate program through the concurrent enrollment program. Ph.D. students who have previously earned a master’s degree in a related field at another institution may substitute courses from their previous university with approval of the adviser and the graduate committee. Courses from other institutions may not be applied to the M.S. degree course requirements.

Petitions should be submitted along with the transcript from the other institution or UCSC Extension. For courses taken at other institutions, copies of the syllabi, exams, and other course work should accompany the petition. Such petitions are not considered until the completion of at least one quarter at UCSC. At most, a total of three courses may be transferred from concurrent enrollment and other institutions.

Requirements for a Designated Emphasis to an External Degree Program

Students from another degree program who meet the following requirements can have the designated emphasis of “statistics” annotated to their degree title.  For example, a Ph.D. student in electrical engineering who meets the requirements would get a certification that read “Ph.D. Electrical Engineering (Statistics).” The course requirements are:

Required Core Courses:

203    Introduction to Probability Theory

206    Classical and Bayesian Inference (or 206B Intermediate Bayesian Inference)

207    Intermediate Bayesian Statistical Modeling

256    Linear Statistical Models

and one other statistics course from a list of approved courses in AMS (currently 202, 205B, 221, 223, 225, 241, 245, 261, 263, 274, and 291).

Graduate program in Scientific Computing and Applied Mathematics (SciCAM)

Program learning outcomes (PLOs)

M.S. PLOs:

1. The ability to take a real-life science or engineering problem, and create a mathematical model of it, under supervision or with the help of discussions with colleagues.

2. Proficiency in analytical methods for the solution of linear algebra problems, ordinary and partial differential equations.

3. Proficiency in the construction of numerical algorithms for the solution of linear algebra problems, as well as ordinary and partial differential equations.

4. Proficiency in at least two scientific computing languages such as: Fortran, C, Python, R, Matlab, etc. Familiarity with Unix-type operating systems, the use of compilers, professional scientific computing libraries, efficient IO algorithms, data visualization tools, etc.

5. Proficiency in the two main parallel computing paradigms (shared vs. distributed memory) and in the use of OpenMP and MPI. Familiarity with parallel architectures and with supercomputing environments such as batch submission scripts, data transfer protocols, scripting, etc.

6. The ability to identify and implement, among all of the methods and languages known, the most appropriate and efficient approach for the original problem posed.

7. The ability to analyze critically the results from the model obtained, and to present them clearly and coherently to peers (orally and/or in writing).

Assessment:

The progress of each student towards the PLOs will be monitored by the program faculty through:

  • regular formative and summative assessments in the classroom context (participation, homework, quizzes, midterms);

  • final examinations or final projects in each of the individual core course;

  • the comprehensive capstone requirement (whether thesis or comprehensive exam).

Requirements for a Graduate Degree in Scientific Computing and Applied Mathematics

Required core courses and foundational requirements:

209    Foundations of Scientific Computing

212A   Applied Mathematical Methods I

213A   Numerical Linear Algebra

213B    Numerical Solutions of Differential Equations

214      Applied Dynamical Systems

250      Introduction to parallel computing

Students in the SciCAM program must also demonstrate mastery in the foundations of Scientific Computing and Applied Mathematics, either by producing evidence through undergraduate transcripts, or by taking some or all of the following foundational courses upon entry to the M.S. program: AMS 147 (Computational Methods and Applications), AMS 210 (Foundations of Scientific Computing) and AMS 211 (Foundations of Applied Mathematics).

Electives and capstone requirements:

Approved elective courses: Any regular graduate AMS course not already in the core except AMS 200 and supervised research courses; a list of current allowable electives in other departments is maintained by the graduate director, and updated on a yearly basis. Students may also petition the program to add one or more courses to the elective list. Note that some upper-division electives are allowed, bearing in mind that no more than a total of 15 credits of upper-division courses may be used to satisfy the degree requirements.

Students in the SciCAM program may pursue a Plan I (thesis capstone) or a Plan II (comprehensive examination capstone) curriculum.

Candidates for a Plan I capstone must, in addition to the 25 credits required from core courses, (1) complete one additional 5-credit course from the approved elective list, (2) complete 10 credits of supervised research (in the form of AMS 297 or AMS 298 with one of the program faculty), and (3) write a thesis.

The thesis requirements are as follows. Students must submit a thesis proposal to the potential faculty sponsor after completion of all core courses. If the proposal is accepted, the faculty member will become the sponsor and will supervise the research and writing of the thesis project. The project will involve the solution of a problem or problems from the selected area of application. The thesis must consist of at least 30 pages and no more than 60 pages of printed written work and accompanying pertinent figures, consisting of a coherent introduction and presentation of the current state of the field, a clear presentation of the questions raised, of the methodology used to solve them, and a discussion of the results obtained. The thesis will be read by at least two faculty from the AMS Department, one of which must be the student’s adviser. The student will then be required to give a short (20-minute) public oral presentation of their thesis, which will be evaluated by the reading committee. The reading committee will assess the quality of both written work and oral presentation in making their recommendation for awarding the M.S. degree to the student.

Candidates for a Plan II capstone must, in addition to the 25 credits required from core courses, (1) complete three additional 5-credit courses from the approved elective list, and (2) successfully pass the SciCAM comprehensive examination. The latter takes place in June at the end of the academic year. Students may only take this exam following completion of the last core course. The exam will take the form of a take-home project covering all core and foundational courses.

Accelerated 1-Year Program Plan

The expected time to completion of the SciCAM degree program is two years. However, AMS also offers a one-year accelerated track for interested students who can demonstrate sufficient proficiency in the foundational subjects. Information on the minimum requirements to join the accelerated track can be found on the program web site. Requests to join the accelerated track must be made to the graduate director by e-mail no later than August 31 of each year.

Relationship of SciCAM Masters program and AMS Doctoral Program

Students in the SciCAM M.S. program interested in an academic career will be strongly encouraged to apply to the SAM Ph.D. program. Applications are reviewed in the standard academic cycle, so that students interested in applying to the SAM program are encouraged to discuss this option with the graduate director in the fall of each year.

Transfer Credit

Up to three UCSC courses fulfilling the degree requirements of the SciCAM degree may be taken before beginning the graduate program.

Up to one course from other institutions may be applied to the M.S. degree course requirements. Petitions should be submitted along with the transcript from the other institution or UCSC Extension. For courses taken at other institutions, copies of the syllabi, exams, and other course work should accompany the petition. Such petitions are not considered until the completion of at least one quarter at UCSC.

Review of Progress

Each year, the faculty reviews the progress of every student in all programs and tracks. Students not making adequate progress toward completion of degree requirements are subject to dismissal from the program (see the Graduate Handbook for the policy on satisfactory academic progress). For specific guidelines on the annual student reviews, please refer to http://www.soe.ucsc.edu/programs/ssm/graduate/index.html.

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Revised: 09/01/15