# Mathematics

## College of Science and Engineering

Dean: Sheldon Axler

### Department of Mathematics

TH 937

Phone: 415-338-2251

E-mail: statmath@sfsu.edu

Web Site: http://math.sfsu.edu

Chair: David Bao

Mathematics Undergraduate Advisers: D. Ellis, E. Hsu, S. Li

Mathematics Graduate Coordinator: E. Hayashi

Statistics Undergraduate Adviser: M. Kafai

### Mathematics Faculty

Professors: Axler, Bao, Ellis, Goetz, Hayashi, Hosten, Hsu, Kafai, Kysh, Langlois, Li, Ovchinnikov, Robbins, Schuster

Associate Professors: Ardila, Beck, Cheung, Gubeladze, Krause, Vazquez

Assistant Professors: Arsuaga, Piryatinska

### Statistics Faculty

Professors: Chattopadhyay, Cholette, Eng, Kafai, Mar, Mehrotra, Mui, Soorapanth

Assistant Professor: Piryatinska

## Programs

Concentrations in:

- Mathematics for Advanced Study
- Mathematics for Liberal Arts
- Mathematics for Teaching

## Program Scope

The Bachelor of Arts is offered for students with a general interest in mathematics; Bachelor of Science programs in applied mathematics and statistics are also offered. Courses are offered in mathematics education for prospective elementary and secondary teachers. Copies of program requirements are available in the mathematics department office.

The Bachelor of Arts in Mathematics has three concentrations: Liberal Arts, Teaching, and Advanced Study.

The Liberal Arts concentration is for students who desire a broad liberal arts education with an emphasis in mathematics.

The Teaching concentration is for students whose goal is to teach mathematics in middle school or high school. These students will obtain a solid understanding of the mathematics needed for teaching and they will gain classroom experience as volunteers in local public schools. They will also have the opportunity to develop the mathematical skills, flexibility and perceptiveness in order to help future students cultivate fruitful wonderful ideas, and to help students connect their thinking to formal mathematical structures. Students who complete this concentration will have satisfied the early field experience requirement and the subject matter competency requirement for a single subject credential in mathematics.

The Advanced Study concentration is for students who plan to pursue a masters or doctoral degree in mathematics. Students who choose this concentration will obtain a solid foundation in the cornerstones of advanced mathematics: linear algebra, abstract algebra, vector analysis, real analysis and complex analysis.

The Bachelor of Science in Applied Mathematics responds to the needs of business and industry for applied mathematical scientists. The program also responds to the needs of those students who enjoy mathematics for its own sake but who also have interests in other fields to which mathematics is applied. Applied mathematicians and statisticians are employed in such areas as operations research, systems analysis, computing, data analysis, biological sciences (for example, doing research on DNA topology, mathematical cancer biology, or meeting the special needs of biostatistics), communications research, and in the management sciences.

The primary aim of applied mathematics is to elucidate scientific concepts and to describe and predict scientific phenomena through the use of mathematics. The applied mathematician is at once a mathematical specialist and a systems analyst, whose task it is to confront highly complex real-world situations with mathematical analysis. In industry the applied mathematician has an opportunity to test both background and training in solving problems of a practical nature. It is necessary to have not only a grasp of the mathematical theories involved, but also an appreciation for the specific science or technology concerned. In this way, one can arrive at usable mathematical formulations of scientific and engineering problems.

The applied mathematics program prepares students in several areas. First, students acquire a broad knowledge of the techniques and methods of applied mathematics. These techniques include differential equations, optimization, statistics, numerical analysis, computer programming, and operations research. Second, students learn to model scientific phenomena and complex real-world systems, (Mathematical Modeling, Applied Mathematics Project) and to use these models to understand and predict the behavior of these systems. Finally, they learn how to communicate these results to other scientists and managerial decision makers.

The Minor in Mathematics is available for students desiring a program of study in mathematics that is coherent but not as extensive as the B.A. program. It could provide excellent background for prospective secondary school teachers who want to be able to teach in mathematics as well as in their major area, or for students majoring in a science such as biology or economics who want to emphasize the quantitative aspects of their major.

The College of Business, and the College of Science and Engineering offer the Bachelor of Science in Statistics for students who are planning careers as statisticians in a wide variety of industrial and business activities, and government services.

The Bachelor of Science in Statistics is for students who are planning careers as statisticians in industry, business, government, or biomedical research. Statistics is basic to quantitative research in the biological, physical, and social sciences. Because its methods are based on mathematics, it requires a firm understanding of mathematical methods as well as an appreciation of scientific method, computation, and practical problems. To give the student both breadth and depth and to introduce the student to a variety of fields where statistics may be applied, three emphases are offered: science, business, and economics.

The Master of Arts in Mathematics is offered with the purpose of extending students' experience in mathematics. A student's goal may be to prepare for a career in government, industry, or community college teaching, to enhance competency as an elementary or secondary school teacher, or to prepare for further graduate study.

The described degrees, when supplemented by the appropriate courses, can also prepare students for graduate study in other fields such as accounting, mathematical biology, computer science, economics, engineering, physics, and statistics.

## Career Outlook

The degree programs in Mathematics and statistics prepare students for additional graduate work; teaching careers; and work in business, industry, and government that apply mathematical and statistical concepts. In addition, specific careers in actuarial science, investment firms, computer industry, biomedical research and the government sector (such as NASA and the NSA) are especially attractive.

## Bachelor of Arts in Mathematics

Undergraduate degree programs in mathematics presuppose a background equivalent to two years of high school algebra, one year of plane geometry, and at least one semester of trigonometry. Students in doubt as to their satisfaction of these prerequisites should consult the mathematics department. MATH 60, 70, and 109 may be used to remedy deficiencies. Because of the sequential nature of mathematics courses, all students must consult with a departmental adviser at regular intervals during their degree programs. MATH 500 through MATH 599 may not be counted as electives toward the mathematics major or minor. CR/NC grades are not acceptable in courses to be counted for a mathematics major or minor program.

The Bachelor of Arts in Mathematics can be earned by completing any one of three concentrations:

- Concentration in Mathematics for Advanced Study (48 units): this concentration prepares students for graduate study of mathematics.
- Concentration in Mathematics for Liberal Arts (42 units): this concentration is intended for students planning non-technical careers.
- Concentration in Mathematics for Teaching (45 units): this concentration matches the state-approved subject matter program for the single-subject credential in mathematics.

Mathematics majors who successfully complete MATH 300 GW spring 2010 or thereafter or MATH 301 GW in spring 2009 or thereafter will have satisfied the University’s Graduation Writing Assessment Requirement (GWAR).

Courses are 3 units unless otherwise indicated. On-line course descriptions are available.

#### Core Courses

Course | Title | Units |
---|---|---|

MATH 226-228 | Calculus I-III (4 units each) | 12 |

MATH 301 GW | Exploration and Proof - GWAR | 3 |

MATH 325 | Linear Algebra | 3 |

MATH 335 | Modern Algebra I | 3 |

MATH 370 | Real Analysis I | 3 |

Total for Core Courses: 24 units

#### Concentration in Mathematics for Advanced Study

Course | Title | Units |
---|---|---|

CSC 210 or CSC 309 |
Introduction to Computer Programming Computer Programming for Scientists and Engineers |
3 |

MATH 340 | Probability and Statistics I | 3 |

MATH 310 or MATH 376 |
Elementary Number Theory Ordinary Differential Equations |
3 |

MATH 380 | Introduction to Functions of a Complex Variable | 3 |

MATH 435/735 | Modern Algebra II | 3 |

MATH 470 or MATH 471 |
Real Analysis II - Several Variables Fourier Series and Wavelets |
3 |

Two elective courses numbered 400 or above except MATH 475: 6

Total for Concentration in Mathematics for Advanced Study: 24 units

#### Concentration in Mathematics for Liberal Arts

Course | Title | Units |
---|---|---|

CSC 210 or CSC 309 |
Introduction to Computer Programming Computer Programming for Scientists and Engineers |
3 |

MATH 300 GW | History of Mathematics - GWAR | 3 |

Four elective courses numbered 300 or above except MATH 475: 12 units

Total for Concentration in Mathematics for Liberal Arts: 18 units

**Note:** A minimum of 40 upper division units must be completed for the degree (including upper division units required for the major, general education, electives, etc.). A student can complete this major yet not attain the necessary number of upper division units required for graduation. In this case additional upper division courses will be needed to reach the required total.

#### Concentration in Mathematics for Teaching

Course | Title | Units |
---|---|---|

MATH 300 GW | History of Mathematics - GWAR | 3 |

CSC 210 or CSC 309 |
Introduction to Computer Programming Computer Programming for Scientists and Engineers |
3 |

MATH 310 | Elementary Number Theory | 3 |

MATH 324 | Probability and Statistics with Computing | 3 |

MATH 350 | Geometry | 3 |

MATH 375 | Field Study for Secondary Teachers | 3 |

MATH 475 | Capstone Course for Secondary Teachers of Mathematics | 3 |

Total for Concentration in Mathematics for Teaching: 21 units

## Bachelor of Science in Applied Mathematics

Courses are 3 units unless otherwise indicated. On-line course descriptions are available.

#### Required Courses (39 units)

Course | Title | Units |
---|---|---|

CSC 210 or CSC 309 |
Introduction to Computer Programming Computer Programming for Scientists and Engineers |
3 |

MATH 226-228 | Calculus I-III (4 units each) | 12 |

MATH 324 or MATH 441 |
Probability and Statistics with Computing Probability and Statistics II |
3 |

MATH 325 | Linear Algebra | 3 |

MATH 335 or MATH 370 |
Modern Algebra Real Analysis I |
3 |

MATH 340 | Probability and Statistics I | 3 |

MATH 376 | Ordinary Differential Equations I | 3 |

MATH 400 | Numerical Analysis | 3 |

MATH 460 | Mathematical Modeling | 3 |

MATH 696-697 | Applied Mathematics Project (1-2) | 3 |

#### Electives: 15 units

Course | Title | Units |
---|---|---|

Select from the following: | 6 | |

MATH 301 GW | Exploration and Proof - GWAR | |

MATH 380 | Introduction to Functions of a Complex Variable | |

MATH 430 | Operations Research: Deterministic Methods | |

MATH 442 | Probability Models | |

PHYS 385 | Introduction to Theoretical Physics I |

A coherent collection of three courses emphasizing applications of mathematics, chosen with the consent of the applied mathematics adviser: 9 units

Total units for Bachelor of Science in Applied Mathematics: 54 units

## Bachelor of Science in Statistics

The Bachelor of Science in statistics is an interdisciplinary program offered for students who intend to pursue an advanced degree, or who are planning careers as statisticians in industry, business, government, or scientific research.

To give the students both breath and depth and to introduce them to a variety of fields where statistics may be applied, we offer three emphases for the degree: science, business, and economics.

Courses are 3 units unless otherwise indicated. On-line course descriptions are available.

#### Core Requirements

Course | Title | Units |
---|---|---|

CSC 210 or CSC 309 |
Introduction to Computer Programming Computer Programming for Scientists and Engineers |
3 |

MATH 226-228 | Calculus I-III (4 units each) | 12 |

MATH 301 GW | Exploration and Proof - GWAR | 3 |

MATH 325 | Linear Algebra | 3 |

MATH 338 | Introduction to SAS | 3 |

MATH 340 | Probability and Statistics I | 3 |

MATH 441 | Probability and Statistics II | 3 |

Total for Core Requirements: 30 units

Select one emphasis:

#### Business Emphasis

Course | Title | Units |
---|---|---|

DS 312 | Data Analysis with Computer Applications | 3 |

DS 412 | Operations Management | 3 |

ECON 101 | Introduction to Microeconomic Analysis | 3 |

FIN 350 | Business Finance | 3 |

ISYS 363 | Information Systems for Management | 3 |

Upper division quantitative course chosen in consultation with the statistics major adviser 3

Elective units selected with approval of adviser 6

Total for Business Emphasis: 24 units

#### Economics Emphasis

Course | Title | Units |
---|---|---|

ECON 101 | Introduction to Microeconomic Analysis | 3 |

ECON 301 | Intermediate Microeconomic Theory | 3 |

ECON 302 | Intermediate Macroeconomic Theory | 3 |

ECON 312 | Introduction to Econometrics | 3 |

ECON 615 | Mathematical Economics | 3 |

ECON 630 | Econometric Theory | 3 |

ECON 725 | Applied Data Analysis in Economics | 3 |

Elective units selected with approval of adviser: 3

Total units for Economics Emphasis: 24 units

#### Science Emphasis

Course | Title | Units |
---|---|---|

MATH 400 | Numerical Analysis | 3 |

MATH 430 | Operations Research | 3 |

MATH 460 | Mathematical Modeling | 3 |

MATH 490 | Mathematics Seminar | 3 |

Units selected on advisement from a coherent collection of courses in areas of science that use statistical methods. Under advisement, courses from other colleges may be selected. 12

Total for Science Emphasis: 24 units

## Minor in Mathematics

At least 12 units of courses counted toward the minor, including at least 6 upper division units, must be completed at SF State.

Courses are 3 units unless otherwise indicated. On-line course descriptions are available.

Course | Title | Units |
---|---|---|

MATH 226-228 | Calculus I-III (4 units each) | 12 |

Upper division electives on advisement 12

Total for a Minor in Mathematics: 24 units

## Master of Arts in Mathematics

### Admission to Program

In addition to the general requirements for admission, applicants to the master's program must have a 3.0 grade point average in the following three courses, or their equivalent: MATH 325, MATH 335, and MATH 370. Applicants who fail to satisfy this requirement but who are qualified in all other respects may be admitted on the condition that they bring their grades in these courses up to the 3.0 average during their first two semesters of graduate study (these three courses, however, may not be counted as electives toward the M.A. degree).

**Written English Proficiency Requirement**

All students in graduate programs at SF State must demonstrate Level One (entry) and Level Two (exit) writing proficiency in accordance with University, departmental and or programmatic guidelines.

**Level One:**

- Prior to admission: Minimum score of 4.0 on the Analytical Writing Analysis (AWA) on the GRE test.
- Conditional Admission: Applicants who do not satisfy Level I prior to admission must pass SCI 614 or 729 with a grade of B-minus or better not later than the second semester. (Students should note that SCI 614 can be taken only through the College of Extended Learning and may not count as units toward the degree. MATH 729 is only offered in spring semesters.)

**Level Two:** Satisfactory completion of MATH 898.

Upper division courses acceptable on an Advancement to Candidacy form will be determined by the student with approval of the graduate adviser.

Courses are 3 units unless otherwise indicated. On-line course descriptions are available.

### Program

Course | Title | Units |
---|---|---|

MATH 710 | Analysis I | 3 |

MATH 850 | Algebra | 3 |

Additional units selected from graduate courses other than MATH 898, or MATH 899. 6 units.

**Upper division/graduate mathematics or related courses** (18 units)

At least 3 unpaired graduate units in mathematics must be included ^{1} and at most 9 units may be selected from unpaired upper division courses. Student must complete either a thesis ^{2} with oral defense or take the comprehensive examinations and write an expository paper.

Minimum total: 30 units

### Culminating Experience

Candidates for the M.A. in Mathematics must complete a Culminating Experience. Two options are available.

**Thesis Option.** Students may choose to write a thesis and present an oral defense. Students considering the thesis option should contact the department chair or graduate adviser for further details. A master's thesis should contain new theorems or algorithms, a novel application, or an original approach to an established result. The resulting manuscript must be prepared according to university guidelines following a style similar to that used by the *Notices of the American Mathematical Society*. Each MA thesis has a principal adviser and two additional readers. The expected time to completion for this paper is one academic year. Thesis guidelines for students and advisers are available from the Mathematics Graduate Adviser.

**Comprehensive Examination/Expository Paper Option**. Students selecting this option take two written examinations and write an expository paper. Students must take two examinations selected from algebra, analysis, and statistics. Written examinations are administered during the last two weeks of each semester. Examinations last two and a half hours, and a student takes no more than one examination per day. Departmental syllabi for the examinations are available at least four months in advance of each administration. Each examination requires students to integrate material from several undergraduate and graduate courses, to demonstrate their ability to write short proofs in correct mathematical English, and to demonstrate the falsity of propositions by counter-examples. Students who fail an examination may repeat it at least once, with additional attempts requiring the written approval of the graduate adviser.

The expository paper is completed in two stages. First, students must complete a departmental proposal form, including: the title and abstract of the proposed paper, the what-why-how aspects of the research in question, a brief preliminary bibliography, and the approval of the proposal by a committee consisting of a faculty adviser and two additional readers from the Mathematics faculty. Once students have an approved proposal, they may begin work on the project under the guidance of the faculty adviser. Completion of the paper is subject to signed approval by all members of the committee.

Further information about these options can be obtained from the department web site: http://math.sfsu.edu.

**footnotes:**

- MATH 730 must be included in this category unless the student has completed an undergraduate complex analysis course with a grade of B or higher.
- MATH 898 Master's Thesis, if chosen will earn 3 units toward this requirement.