Applied Mathematical and Computational Sciences

Program in
Applied Mathematical
and Computational Sciences

Graduate
College

Monday, November 23 2009 @ 10:50 AM CST

The Interdisciplinary Program January 20, 2009

Applied Mathematical and Computational Sciences (AMCS) at The University of Iowa is a broad-based interdisciplinary Ph.D. program for students desiring to study mathematics and a companion science so that they can apply their mathematical skills to significant scientific problems. The main goal of the program is to develop applied mathematicians with sufficient professional experience and versatility to meet the research, teaching, and industrial needs of our technology-based society.

While building a base in the Mathematical Sciences, students acquire skills in another area of their own interest, chosen from the behavioral, biological, business, engineering, medical, physical, or social science areas. Most students concentrate on applied mathematics such as differential equations, numerical analysis or optimization, but a few students have used statistics as their mathematical science base.

The University of Iowa has become a center for the computational sciences. Because of expertise in fields such as numerical analysis, mathematical programming, parallel and vector processing, hydraulics and fluid mechanics, heat transfer, dynamic simulation of mechanical systems, optimization in management sciences and industrial engineering, discrete event simulation, robotics, atmospheric and environmental studies, climate/chemistry modeling, geographical decision making, theoretical and plasma physics, and pharmacological and biological modeling, the computational sciences are now an important part of the program. There is a demand for mathematical scientists who are trained to use a computational sciences approach in relevant problems. Our fifty-eight faculty in fourteen different departments are working on exciting research projects and are eager to train students. The diversity of the areas of application is manifest in the descriptions provided by the faculty associated with the program in the AMCS Faculty Personal Pages.

New AMCS Graduate Curriculum

AMCS students study theoretical mathematics, applied mathematics, and an outside area in which mathematics is applied. Then they do dissertation research on a problem involving the outside area. The goals of the new AMCS graduate curriculum are:

To develop the competence of all AMCS students in four core areas.
To give AMCS students the opportunity to get early starts on their research.
To have an AMCS curriculum that facilitates transfers to and from Mathematics.

Implementation timeline:

AMCS students starting in Fall of 2005 or later must follow the new curriculum.
Students starting before Fall of 2005 can follow the old AMCS curriculum or may chose to follow the new AMCS graduate curriculum.

Section 1 Summary of Requirements

Section 1.1 Required Courses in Core Areas

22M:115-116 Analysis
22M:132-133 Topology
22M:142-144 Differential Equations with Numerical Methods
22M:170-171 Numerical Analysis

Every student in the AMCS Ph.D. program must pass all four core course sequences (or be exempted -- see details in Section 2) in the first two years of graduate study. Detailed course descriptions and sample syllabuses are available at the Department of Mathematics home page http://www.math.uiowa.edu/ by following the link to Courses or to Graduate Programs and then Core Courses under New Graduate Curriculum. These core courses are accessible to students who have completed single and multivariable calculus, linear algebra, and an introduction to analysis.

Section 1.2. Ph.D. Qualifying Examination

Every student in the AMCS Ph.D. program must pass a Ph.D. qualifying examination consisting of area examinations in three of the four core areas listed above. These area examinations are based on the 100-level core sequences given in Section 1.1. The AMCS Qualifying Examination must be passed within two and a half years after beginning graduate study.

Section 1.3. Ph.D. Comprehensive Examination

All AMCS students must pass a Ph.D. Comprehensive Examination over their outside research area within three and a half years after beginning graduate study. An outside area is defined as an area outside of mathematics, in which mathematics is applied. The Ph.D. Comprehensive Examination is based on a sequence of 200-level courses in the outside area or an equivalent combination of courses and directed reading.

Section 1.4. Advanced Mathematics Course Requirement

In order to establish a solid foundation in mathematics, all AMCS students must pass at least 18 credit hours of graduate mathematics courses numbered from 22M:200 to 22M:371 with the exception of the seminars 22M:224 and 22M:225. The courses in the student’s written plan of study should be chosen to obtain mathematical breadth and must be approved by the AMCS Director.

Section 2. Details on the Ph.D. Qualifying Examination

Every student in the AMCS Ph.D. program is required to demonstrate competence in each of the four core areas within the first two years of graduate study, either by passing the four 100-level AMCS core course sequences, or by passing the relevant portion of the Ph.D. Qualifying Examination. To ensure that students plan appropriately, all AMCS students must have their written individual plans of study approved by the AMCS Director. Graduate courses transferred from other universities may be used to satisfy the core course and advanced mathematics course requirements, subject to approval of the AMCS Director. The Qualifying Examination is given in all areas at the beginning of each fall and spring semester.

Every student in the AMCS Ph.D. program must pass a Ph.D. Qualifying Examination.

The Qualifying Examination must be passed within two and a half years after beginning graduate study.
The Qualifying Examination consists of area examinations in three of the four AMCS core areas (taken in the same examination period).
The area examinations are based on the 100-level core course sequences listed above.
Each area examination is a three-hour written examination.
A student may take the Qualifying Examination at most twice.
For each area examination, a student will receive a grade of Ph.D. qualifying level pass, Master’s level pass or not pass.
In order to pass the Qualifying Examination, a student must receive Ph.D. qualifying level passes in at least two areas and at least a Master’s level pass in the third area.
If a student needs to take the Qualifying Examination a second time, then the student may carry forward area examination score(s) of Ph.D. qualifying level pass. A student who is carrying forward two of these passes must obtain a Ph.D. qualifying level pass in the third examination area.
AMCS does not offer a Master’s Degree; AMCS students often obtain applied mathematics Master’s Degrees in the Department of Mathematics.

Student may satisfy core course requirements via a Qualifying Examination in three ways.

A student who receives a Ph.D. qualifying level pass in an area on the Qualifying Examination is exempted from the core course requirement in that area.
A student may take all four area examinations at once, having registered for three as a Ph.D. Qualifying Examination. A score of Ph.D. qualifying level pass in the fourth area is necessary to satisfy the core course requirement in that area.
A student who has already passed a Ph.D. Qualifying Examination may take an individual area examination in a subsequent Ph.D. Qualifying Examination. A score of Ph.D. qualifying level pass is necessary to satisfy a core course requirement.

Section 3. Details on the Ph.D. Comprehensive Examination

Each student must submit a written Ph.D. Comprehensive Examination proposal to the AMCS Director for approval. This proposal must list at least two 200-level courses to be taken in the outside area, or describe an equivalent combination of courses and directed readings to be done as preparation. The examining committee specified in the proposal must contain at least three people with some members from the outside area. The Ph.D. Comprehensive Examination may be written, oral, or a combination of written and oral examinations. Existing outside Ph.D. comprehensive examinations or standard practices for Ph.D. comprehensive examinations in the outside area may be used. All AMCS students must pass a Ph.D. Comprehensive Examination over their outside area within three and a half years after beginning graduate study.

The new AMCS curriculum allows the following graduate student paths. These are parallel to possible paths in the Department of Mathematics (go to http://www.math.uiowa.edu/ and follow the link to Graduate Programs and then to New Graduate Curriculum and Sample Programs).

Entering students with excellent preparation have the opportunity to pass the Ph.D. Qualifying Examination in August of their beginning year, and then move directly to research related activities and preparation for their Ph.D. Comprehensive Examination over their outside area.
Entering students with very strong preparation have the opportunity to pass some of the area examinations in August of their beginning year, and then concentrate on the remaining areas. These students could pass the Ph.D. Qualifying Examination in August of the beginning of their second year and then move quickly to research related activities and preparation for their Ph.D. Comprehensive Examination over their outside area.
Entering students with good preparation have the opportunity to start three core sequences in the first year and possibly pass the Ph.D. Qualifying Examination in August of the beginning of their second year. Then they could move quickly to research related activities as soon as their second year, finishing their Core Courses, Ph.D. Qualifying Examination, and Ph.D. Comprehensive Examination by the end of their second year.
Entering students with less preparation have the opportunity to start two core sequences in the first year and another two in the second year, so that they can pass the Ph.D. Qualifying Examination in August of the beginning of their third year. By starting preparation for their Ph.D. Comprehensive Examination in their second and third years, they could finish their Core Courses, Ph.D. Qualifying Examination, and Ph.D. Comprehensive Examination by the end of their third year.
The time restrictions could be modified for students with weak backgrounds or special circumstances, but the requirements regarding Core Courses, Ph.D. Qualifying Examination, and Ph.D. Comprehensive Examination must be met by all AMCS students.

Graduate Study

In 1992 the program name was changed from Applied Mathematical Sciences to Applied Mathematical and Computational Sciences, which better describes the current nature of the program. Some aspect of the computational sciences has been a part of the dissertation research of nearly all recent graduates. Although it is a separate, independent academic unit in the Graduate College, the program cooperates with the Department of Mathematics. Many of the courses taken by students are in the Department of Mathematics, and most students in the program have teaching assistantships in the Department of Mathematics.

The AMCS program differs from other Ph.D. programs because it is flexible and individualized and because it requires study in both a science and a mathematical science. It is not designed to replace existing departmental Ph.D. programs at The University of Iowa. For example, individuals interested primarily in mathematical aspects of applicable mathematics should apply to the graduate program in the Department of Mathematics, which has many faculty members interested in ordinary and partial differential equations, numerical analysis, optimization, mathematical physics, and biomathematics. Students interested primarily in a science may fit into another departmental Ph.D. program since many of these programs involve some aspects of applied mathematics, statistics, or computer science.

The program is suitable for those who are capable of graduate study in both a mathematical science and another science and who want to do dissertation research on a problem in the scientific area which involves the use of graduate-level mathematics.

Currently, there are about 30 students enrolled in the program. This small size means that students have more direct contact with faculty members. Each student's faculty committee helps plan a program consistent with the student's background, interests, and goals, which should develop expertise in methods of application of mathematics, build a good foundation in related topics of theoretical mathematics, and provide sufficient knowledge in a particular science so the student can use mathematical techniques in that science.

Each student takes comprehensive examinations in three areas: in a theoretical foundation area, in the applied mathematics that is useful in the student's chosen field, and in the particular area of the student's specialization. Each student's dissertation research should include the activities of a mathematical scientist. For example, this could involve formulation of a model, quantitative analysis of the model, and interpretation of the results.

Research topics of students have included geometric programming and entropy optimization problems, the computational finite analytic method for three-dimensional fluid mechanics problems, the effects of monetary policy on economic optimization problems, global optimization problems in manufacturing management, efficient algorithms for computer-aided design problems, effective numerical algorithms for mechanical systems simulation, a modified finite analytic method to solve concavity flow problems, computational exterior flow problems in fluid mechanics, digital signal processing, neural networks, computer-aided simulation of automobile performance, optimization in robotic trajectory design, and chaotic dynamics in physics.

Employment Opportunities

There are many uncertainties in the national economy and in the funding of colleges and universities. However, our doctoral graduates have been successful in finding suitable professional positions. Our graduates (see recent graduates), have found positions in colleges, universities, businesses and industries, or they have taken postdoctoral positions.

Future prospects for employment in mathematics and applied mathematics are difficult to predict, but the clear trend is toward quantitative and computational approaches to solving problems in numerous areas. Mathematical scientists with a sound training in mathematical and computational skills and with an interest and background in an area of application usually find a demand for their services. Academic opportunities in Departments of Mathematics and Applied Mathematics still exist for those trained in applied and computational mathematics. Opportunities in business and industry are available, but require an intensive job search. Most applied mathematical scientists find an intellectually challenging job in a healthy environment with stimulating colleagues and reasonably good wages.

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