Core Coursework

Math 6000: Communicating Mathematics

(2 units, Course Code: C-05, k-factor: 1)

The communicating mathematics course will focus on communicating mathematics precisely and effectively. The two facets of this focus will be written communication and oral communication. Students will become familiar with LaTex and Beamer, and will prepare an article based on a topic of their choosing (preferably, something they have encountered in one of their courses and would like to learn a little more about). They will also prepare a Beamer presentation and give a 15 min. conference math talk. There will be reading assignments for this course that deal with writing and presenting mathematics. Possible resources include the following:

• How to Write Mathematics, by Paul Halmos
• MAA Mathematical Communication website

2 hour class, once per week in the Fall (combines well with Teaching practicum).

Proposed catalog description: Seminar focusing on effective and precise written and oral communication of mathematics. Students will become familiar with scientific word processing and presentation programs such as LaTex and Beamer, and will prepare a professional article and presentation.

Math 6018: Graduate Analysis

(4 units, Course Code: C-02, k-factor: 1)

Catalog Description: A rigorous development of the calculus of vector valued functions of several variables, including a theoretical development of the derivative and its applications, the Inverse and Implicit Function Theorems, a development of Riemann Integration, Fubini's Theorem, differential forms, change of variables in integration, and Stokes' Theorem. Additional topics may include but are not limited to: abstract measure theory, metric spaces, or an introduction to manifolds and differential geometry.

Offered every Fall?

Math 6016: Graduate Algebra

(4 units, Course Code: C-02, k-factor: 1)

Proposed catalog description: Continuation of {whatever the last Abstract Algebra course is}. Review of basic ring theory focusing on irreducibility and associated ideals, followed by fields and field extensions, Galois groups, solvability of groups and solvability of polynomials by radicals, and the Fundamental Theorem of Galois Theory. Prerequisite: The equivalent of Math 546.

Here is a longer description, courtesy of Gary: This course is a continuation of our Math 546 course and starts with a review of rings, including the topics of PID, prime and maximal ideals, quotient ring, and a simple/transcendental algebraic element. Then, the irreducibility results/criteria of Gauss/Eisenstein along with the irreducibility of the pth cyclotomic polynomial, and simple and multiple algebraic field extensions leading to the existence of splitting fields, and of finite fields, are covered. Later, the uniqueness (of splitting fields) along with the concept of an automorphism of a field extension is given, and the Galois group of a polynomial is introduced. The solvability of groups and the insolvability of the symmetric group on n letters for n bigger than 4 is covered, and radical extensions, and the solvability (of a polynomial) by radicals is discussed. The necessary condition that the Galois group (of a polynomial) be solvable (in the group sense) in order for the polynomial to be solvable by radicals is provided, and followed by an example of a specific quintic polynomial which is not solvable by radicals. Time permitting, the "Fundamental Theorem of Galois Theory" and the characterization theory which states that the Galois group being solvable is both necessary (mentioned above) and sufficient for solvability by radicals are explained.

Culminating Experience

Math 6971 - 6972: Thesis I/II

(3 units each, Course Code: S-24, k-factor: 0.667)

Student enrolls in Thesis I by finding an advisor and filling out a memo request to the Graduate Coordinator as one has done to enroll in 695. A requirement of Thesis I is to produce a proposal and to find a thesis committee, this proposal is submitted to the MA committee for approval. A prerequisite for Thesis I is a GPA of 3.25 or greater. A prerequisite for Thesis II is successful approval of the thesis proposal. Thesis I is graded as CR once the proposal is accepted, and Thesis II is graded as CR once the thesis has been successfully defended and the graduate school accepts a completed thesis. Otherwise, the possible grades for this course are: CR, RP.

Catalog Description for Math 6971, Thesis Project I. Thesis preparation. Requirements: assemble thesis committee and submit thesis proposal to the MA Committee. Prerequisites: Consent of thesis advisor, and advancement to candidacy. A written proposal must be submitted to the MA coordinator no later than the 14th week of the semester preceding enrollment in Math 6971.

Catalog Description for Math 6972, Thesis Project II. Continuation of Math 6971. Requirements: successful completion and defense of the thesis. Prerequisites: Math 6971, advancement to candidacy, and approval of thesis proposal produced in Math 6971 by the MA Committee.

Comprehensive Exam Preparation

(1 unit each, Course Code: S-25, k-factor: 0.5)

Taken in the expected final semester of the program. One must apply to enroll in this class by, perhaps, week 14 of the semester prior. This application is a proposal for a 3-person committee (with signatures). The student will enroll in one of each of these courses for each faculty member on their committee. For more information on the exam and its content, see the "Culminating Experience" page. This application will also have the dates, times, and locations of the Algebra and Analysis comprehensive exam which the student will also take. Although this may not be repeated for credit, the student may retake any failed exam a maximum of one time, comprising two total attempts at any one exam. Only those failed exams must be retaken. A grade of CR is given upon successful completion of all of the exams, while it remains as RP in the event any exam is not passed.

Math 6916: Algebra Comprehensive Exam Preparation

Catalog Description: Preparation for the Algebra Comprehensive Exam. Prerequisites: To be taken in the final semester of the program, advancement to candidacy, consent of instructor to prepare, administer, and grade the exam. Graded as CR when the exam is successfully passed, and RP otherwise. The exam may only be retaken once in the event it is not passed.

Math 6918: Analysis Comprehensive Exam Preparation

Catalog Description: Preparation for the Analysis Comprehensive Exam. Prerequisites: To be taken in the final semester of the program, advancement to candidacy, consent of instructor to prepare, administer, and grade the exam. Graded as CR when the exam is successfully passed, and RP otherwise. The exam may only be retaken once in the event it is not passed.

Math 6900: Elective Comprehensive Exam Preparation

Catalog Description: Preparation for the Elective Comprehensive Exam. Prerequisites: To be taken in the final semester of the program, advancement to candidacy, consent of instructor to prepare, administer, and grade the exam. This exam is administered by the student's faculty committee, which consists of the instructors of Math 6906, Math 6908, and Math 6900. A written proposal must be submitted to the MA coordinator no later than the 14th week of the semester preceding enrollment in Math 6900. This proposal must contain the date, time, and location of all comprehensive exams, it must also list the parameters of the Elective Comprehensive Exam, and must have the approval of the instructors of Math 6906, Math 6908, and Math 6900 before submission to the MA coordinator for approval by the MA Committee. Graded as CR when the exam is successfully passed, and RP otherwise. The exam may only be retaken once in the event it is not passed.