MATHEMATICS AND COMPUTER SCIENCE

2017—2018

**MATHEMATICS AND COMPUTER SCIENCE FACULTY**

**Undergraduate Program**** **

**DEPARTMENTAL ADVISING EXPERTS 2017–2018:** David Pollack, Han Li, Dan Licata

**Department/Program Description**

The Mathematics and Computer Science Department offers a major in mathematics and a major in computer science. We also participate in the Informatics and Modeling Certificate Program.

Each student’s course of study is designed to provide an introduction to the basic areas of mathematics or computer science and to provide the technical tools that will be useful later in the student’s career. The course of study is planned in consultation with the department’s advisory committees or the student’s faculty advisor.

### Mathematics

**Student Learning Goals**

The department has the following learning goals for mathematics majors:

- Develop a basic understanding of, and computational facility with, major objects of mathematical and applied interest, such as functions, vector spaces, and groups.
- Understand abstract mathematical reasoning, e.g., understand an abstract system of rules, find examples of objects that satisfy those rules, conjecture theorems from those examples, and prove those theorems.
- Understand some mathematical applications and ways to use mathematics in practice, and be able to make connections to topics outside of the strict course content.
- Students should be able to write about and speak about mathematics, clearly and elegantly.

**Admission to the Major**

Every student is welcome to major in Mathematics. Students are advised to finish calculus up to MATH222 and linear algebra (either MATH221 or MATH223) before making the decision.

**Major Requirements**

- A year of differential and integral calculus (typically MATH121 and MATH122)
- MATH221 or MATH223
- MATH222
- An elementary knowledge of algorithms and computer programming. (Successful completion of either COMP112 or COMP211 satisfies this requirement.)
- MATH261 and MATH225
- A coherent selection of four additional electives, chosen in consultation with an advisor from the department. Any
**MATH**course at the**200+**level can be used as an elective for the major.

**Notes:**

- Students who have completed a year of calculus in high school may place out of one or both of MATH121 and MATH122.
- An AP score of 4 or 5 on the AB calculus exam indicates the student should begin in MATH122.
- An AP score of 4 or 5 on the BC calculus exam indicates the student should consider beginning in any of MATH221, MATH222, or MATH223.
- Students may not earn credit for both MATH221 and MATH223.
- Students must complete either MATH228 or MATH261 by the end of their junior year.
- With advance approval from the departmental advisory committee, mild adjustments are allowed. For example, a Wesleyan course with substantial mathematical content but that is not listed in
**MATH**may be used toward the four-electives requirement. Please note, however, that both MATH225 and MATH261 must be taken at Wesleyan to complete the major, and substitutions for these courses will not be approved.

**Honors**

An undergraduate may achieve the BA with honors in mathematics via one of several routes:

- The honors thesis, written under the supervision of a faculty member under conditions monitored by the University Committee on Honors.
- A strong performance in a suitable sequence of courses, normally including some graduate courses, selected in consultation with a member of the department’s advisory committee. The candidate also is expected to prepare a public lecture on a topic chosen together with a faculty advisor.
- The comprehensive examination, offered by the department and/or by visiting consultants to select students nominated by the faculty.

**Language Requirement**

Undergraduate majors in mathematics are encouraged to study languages while at Wesleyan; majors who are considering graduate study in mathematics should note that graduate programs often require a reading knowledge of French, German, and/or Russian.

**BA/MA Program**

This program provides an attractive option for mathematics majors to enrich their course and research background. Students are advised to begin research by their junior year if they intend to pursue the BA/MA. Admission is competitive and based on GPA, faculty recommendations, and research experience. For more information, visit wesleyan.edu/grad/degree-programs/ba-ma.html. Advanced undergraduates may enroll in graduate (**500-**level) courses.

**Additional Information**

**Lectures.** The departmental colloquium series presents lectures on recent research by invited mathematicians and computer scientists from other institutions. Advanced undergraduates are encouraged to attend these colloquia and to participate in graduate seminars. The undergraduate Math Club hosts informal talks in mathematics, accessible to students at all levels.

**COMPUTER SCIENCE**

**Student Learning Goals**

- Understanding abstraction: At its heart, Computer Science is the study of abstractions for the purpose of understanding computation, and as such students must learn appropriate levels of abstraction for solving computational problems. All courses in the curriculum contribute to this goal.
- Programming: Students must learn how to program in a high-level language, as such programming is the primary tool in Computer Science. This is typically how students are first exposed to the field, and our majors achieve this goal in the freshman or sophomore year by taking the gateway sequence COMP 211—212.
- Analysis: Students must learn how to reason about computation; this includes analyzing algorithms and proving properties such as correctness and complexity and requires an understanding of appropriate mathematical tools. The courses that focus primarily on this goal are COMP 312 (Design and analysis of algorithms) and COMP 321 (Design of programming languages).
- Creation: Students must learn how to create original computational structures; this requires an understanding of fundamental techniques in algorithm and data structure design and an ability to combine established techniques in novel ways. All courses in the curriculum contribute to this goal.
- Limits: Students must understand not only how to analyze and create computational structures, but also the limits of computation itself; this requires an understanding of the mathematical foundations and formalisms of Computer Science. This goal is primarily addressed in COMP 301 (Automata theory and formal languages).

**Admission to the Major**

To declare the computer science major, a student must have

- earned a C or higher in COMP211;
- either earned a C or higher in COMP212 or be enrolled in COMP212 and be earning a grade of C or higher based on completed work; and
- either earned a C or higher in MATH228 or MATH261 or be enrolled in MATH228 or MATH261 and be earning a grade of C or higher based on completed work.

**Note:** The MATH228 or MATH261 requirement applies to students declaring the COMP major after June 30, 2016.

**Major Requirements**

To complete the computer science major, a student must complete the following courses:

Course | Course Title | Credit Hours |
---|---|---|

Required Courses | ||

COMP211 | Computer Science I | 1 |

COMP212 | Computer Science II | 1 |

COMP301 | Automata Theory and Formal Languages | 1 |

COMP312 | Algorithms and Complexity | 1 |

COMP321 | Design of Programming Languages | 1 |

COMP331 | Computer Structure and Organization (or COMP231 if taken before 2015-2016) | 1 |

MATH228 | Discrete Mathematics | 1 |

or MATH261 | Abstract Algebra: Groups, Rings, and Fields | |

MATH221 | Vectors and Matrices | 1 |

or MATH223 | Linear Algebra | |

Select two additional electives | 2 |

**Notes:**

**Honors**

An undergraduate may achieve the BA with honors in computer science via the following route:

- The honors thesis, written under the supervision of a faculty member under conditions monitored by the University Committee on Honors.

**Related Programs or Certificates**

**Informatics and Modeling Certificate.** The department is an active participant in the Informatics and Modeling Certificate (wesleyan.edu/imcp). The certificate provides a framework to guide students in developing analytical skills based on the following two pathways:

- Computational Science and Quantitative World Modeling (CSM): wesleyan.edu/imcp/csm.html
- Integrative Genomic Sciences (IGS): wesleyan.edu/imcp/igs.html

The CSM pathway introduces students to modeling techniques and provides students with a foundation in the quantitative simulation, evaluation, and prediction of natural and social phenomena. The IGS pathway introduces students to the interdisciplinary field of bioinformatics and its relationships to molecular genomics, evolution, structural biology, and bioethics. The department offers courses that support both pathways, such as COMP211 and COMP212, and also offers special interdisciplinary courses for the IGS pathway, such as COMP327 and COMP350. The certificate requirements are described in the links for the two pathways.

**BA/MA Program**

This program provides an attractive option for mathematic majors to enrich their course and research background. Students are advised to begin research by their junior year if they intend to pursue the BA/MA. Admission is competitive and based on GPA, faculty recommendations, and research experience. For more information, visit wesleyan.edu/grad/degree-programs/ba-ma.html. Advanced undergraduates may enroll in graduate (**500-**level) courses.

**Additional Information**

**Lectures.** The departmental colloquium series presents lectures on recent research by invited mathematicians and computer scientists from other institutions. Advanced undergraduates are encouraged to attend these colloquia and to participate in graduate seminars. The undergraduate Math Club hosts informal talks in mathematics; accessible to students at all levels.

**Graduate Program**

**General Introduction**

#### Doctor of Philosophy in Mathematics

The Mathematics and Computer Science Department’s graduate programs include a PhD program in mathematics and MA programs in mathematics and in computer science. The research emphasis at Wesleyan at the doctoral level is in pure mathematics and theoretical computer science. One of the distinctive features of our department is the close interaction between the computer science faculty and the mathematics faculty, particularly those in logic and discrete mathematics.

Among possible fields of specialization for PhD candidates are algebraic geometry, algebraic topology, analysis of algorithms, arithmetic geometry, categorical algebra, combinatorics, complex analysis, computational logic, data mining, elliptic curves, ergodic theory, fundamental groups, Galios theory, geometric analysis, graph theory, homological algebra, Kleinian groups and discrete groups, knot theory, logic programming, mathematical physics, model theory, model-theoretic algebra, number theory, operator algebras, probability theory, proof theory, topological dynamics, and topological groups.

Graduate students at Wesleyan enjoy small classes and close interactions with faculty and fellow graduate students. Graduate students normally register for three classes a semester and are expected to attend departmental colloquia and at least one regular seminar. The number of graduate students ranges from 18 to 22, with an entering class of three to six each year. There have always been both male and female students, graduates of small colleges and large universities, and U. S. and international students, including, in recent years, students from Bulgaria, Chile, China, Germany, India, Iran, and Sri Lanka. All of the department’s recent PhD recipients have obtained faculty positions. Some of these have subsequently moved to mathematical careers in industry and government.

The doctor of philosophy degree demands breadth of knowledge, an intense specialization in one field, a substantial original contribution to the field of specialization, and a high degree of expository skill.

Five years are usually needed to complete all requirements for the PhD degree, and two years of residence are required. It is not necessary to obtain the MA degree en route to the PhD degree. Students may choose to obtain the MA in computer science and the PhD in mathematics. Any program leading to the PhD degree must be planned in consultation with the departmental Graduate Education Committee.

**Courses**

At least 16 one-semester courses are required for the PhD degree. Several of the courses are to be in the student’s field of specialization, but at least three one-semester courses are to be taken in each of the three areas: algebra, analysis, and topology. First-year students are expected to take the three two-semester sequences in these areas. However, students interested in computer science may replace coursework in one of these areas with coursework in computer science, with the permission of the departmental Graduate Education Committee. One of the 16 courses must be in the area of logic or discrete mathematics, as construed by the departmental Graduate Education Committee.

**Language Requirement**

Students must pass reading examinations in either French, German, or Russian. It is strongly recommended that PhD candidates have or acquire a knowledge sufficient for reading the mathematical literature in all three of these languages. Knowledge of one of these three languages is required.

**Progress and Qualifying Exams**

**General preliminary examinations. **The general preliminary examinations occur in the summer after the candidate’s first year of graduate study and cover algebra, analysis, and topology (or computer science, in the case of students including this option among their three first-year subjects).

**Special preliminary examination. **For a graduate student to become an official PhD candidate as recognized by the department, the student has to pass the Special Preliminary Examination, an oral examination that must be passed by the end of the student’s third year of graduate work. The student’s Examination Committee determines the subject matter content of the Special Preliminary Examination. This committee is chaired by the student’s dissertation advisor and must include at least two additional faculty members of the department. The Special Preliminary Examination will be based primarily, but perhaps not exclusively, on the student’s field or specialization. Specific details of the form and content of the examination shall be determined by the Examination Committee at the time the subject matter content is discussed.

**Teaching**

After passing the preliminary examinations, most PhD candidates teach one course per year, typically of 20 students, supervised by senior faculty.

**Thesis/Dissertation/Defense**

**Dissertation.**The dissertation, to be written by the PhD candidate under the counsel and encouragement of the thesis advisor, must contain a substantial original contribution to the field of specialization of the candidate and must meet standards of quality as exemplified by the current research journals in mathematics.**Selection of dissertation advisor.**A graduate student should select a dissertation advisor by the end of the student’s second year of graduate work.**Defense of dissertation.**The final examination is an oral presentation of the dissertation in which the candidate is to exhibit an expert command of the thesis and related topics and a high degree of expository skill.

**General Introduction**

**Master of Arts**

The requirements for the master of arts degree are designed to ensure a basic knowledge and the capacity for sustained, independent scholarly study.

**Courses**

Six one-semester graduate courses in addition to the research units MATH549 and MATH550 or COMP549 and COMP550 are required for the MA degree. The choice of courses will be made in consultation with the departmental Graduate Education Committee.

**Thesis/Dissertation/Defense**

The thesis is a written report of a topic requiring an independent search and study of the mathematical literature. Performance is judged largely on scholarly organization of existing knowledge and on expository skill, but some indications of original insight are expected.

In the final examination, an oral presentation of the MA thesis, the candidate is to exhibit an expert command of the chosen specialty and a high degree of expository skill. The oral presentation may include an oral exam on the material in the first-year courses. A faculty committee evaluates the candidate’s performance. Three semesters of full-time study beyond an undergraduate degree are usually needed to complete all requirements for the MA degree. Any program leading to the MA degree must be planned in consultation with the departmental Graduate Education Committee.

#### ADDITIONAL INFORMATION

For additional information, please visit wesleyan.edu/mathcs/graduate/.