UB's doctoral program in mathematics aims toward generating career options for our students. Additionally, the program guides students toward being prepared for research by the end of third year of coursework.

As reported by the American Mathematical Society, *mathematician* was the #1 rated career in CareerCast’s Job Rated 2014 report. Individual who have demonstrated a high level of mathematical acumen by obtaining a PhD in mathematics are highly prized in both the academic and private sector job markets.

**The requirements below are for students admitted in Fall 2016 and later. The main steps in completing a PhD are:**

**(A) First Year's Coursework and Evaluation exams—**Successfully completing the first year's 6 core courses and passing at least 4 out of 6 evaluation exams attached to these courses. For students interested in pursuing research in pure mathematics the 6 core courses are in algebra, analysis and geometry/topology. For students interested in pursuing research in applied mathematics the 6 core courses are in analysis, numerical analysis and methods in applied mathematics.

**(B) Oral Examination and Advancing to Candidacy—**An oral examination covering material in advanced topics and research ideas in the student's chosen area of research. This oral examination is also the final requirement for advancement to candidacy and should be taken before the end of the student's third year.

**(C) PhD Thesis and Final Oral Examination—**Writing a dissertation and passing an oral defense.The dissertation must consist of original research of sufficient quality for publishing in a respectable mathematics journal.

After the 1st year's course work, the student will take more advanced courses at the 600 level and 700/800 level topics course. Entering their 3rd year, students will focus on their preferred area of research. Advancement to candidacy and dissertation work requires passing an oral exam. Students should pass their oral examination prior to the end of the 3rd year of the program.

In addition to these primary steps, the program offers a 1st year mentoring seminar meant to help students in their career development and management. Topics covered include: study mathematics; using LaTex; media in research mathematics; documenting your achievements; writing, editing and publishing mathematics; seminars, conferences and workshops; and, job options for PhD's in mathematics.

This mentoring seminar will also include faculty talks directed at graduate students, presenting their area of research.

Both the MA and the PhD degrees have residency requirements: one year for the MA and two years for the PhD.

The main steps in obtaining a PhD in mathematics are: (A) Satisfactory completion of first year's coursework and evaluation exams; (B) Passing oral exam in intended area of research and advancing to candidacy; and, (C) Writing the dissertation and successfully defending it in a final oral exam. The aspects of each step are more fully discussed on this page.

The course schedule outlined below is for students in the PhD program who are supported by a teaching assistantship and tuition fellowship. It is 9-credits per semester. For students who do not have support an additional 3-credit course is require so as to be a full time student.

**Learning mathematics is a shared enterprise.** Thus, all members of an entering doctoral class advance through the first year coursework as a cohort.

**Fall semester:**

- Pure Track.
- MTH 534, Basic Measure Theory.
- MTH 519, Introduction to Abstract Algebra.
- MTH 527, Introduction to Topology I.

- Applied Track.
- MTH 539, Methods of Applied Mathematics.
- MTH 537, Introduction to Numerical Analysis I.
- One of: MTH 534, Basic Measure Theory; or

MTH 519, Introduction to Abstract Algebra; or

MTH 527, Introduction to Topology I.

- MTH 539, Methods of Applied Mathematics.

**Spring semester offering:**

- Pure Track.
- MTH 625, Complex Variables.
- MTH 520, Advanced Linear Algebra.
- MTH 528, Introduction to Topology II.

- Applied Track.
- MTH 540, Methods of Applied Mathematics II.
- MTH 538, Introduction to Numerical Analysis II.
- One of: MTH 639 Fourier Analysis; or,

MTH 625, Complex Variables.

MTH 520, Advanced Linear Algebra.

MTH 528, Introduction to Topology II.

**Evaluation Exams:** Attached to each first year course is an evaluation exam. This exam will be given during the regularly scheduled final exam time. All first year evaluation exams are pass/fail. To continue in the PhD program a student needs to achieve at least 4-out-of-6 exam passes. To continue in the MA program a student needs to achieve at least a 3-out-of-6 exam passes. To be in good standing in any graduate program a student needs a GPA of B or above.

**Deficiency:** Students whose performance at the end of their 1st year is judged to be significantly insufficient by the Graduate Director will be dismissed from the program before the beginning of their 2nd year. Students who are marginally below the mark (e.g., pass 2 out of 4 exams or better at PhD level) and/or are marginally below the required B-GPA level, so that they can still advance with their original cohort, have an opportunity to retake the relevant exams in the final exam week of the Fall and Spring semesters in their 2^{nd} year. If the student passes these “make ups’’ (i.e., pass 4-out-of-6 in total for PhD and 3-out-of-6 for MA), then the student will be allowed to advance through the program along with their original cohort. If not, then the student will be dismissed from the program.

After the first year's course work, the student will take more advanced courses at the 600 level and 700/800 level topics course. Students also typically arrange individual reading courses with professors and participate in area seminars.

Entering their third year, students will be focusing on their preferred area of research and the faculty with whom they would like to work. Students will be required to form an oral examination committee of two or three faculty members chaired by a potential thesis advisor.

Students will work with their committee to prepare a syllabus outlining topics to be covered in the oral examination including a bibliography of books and/or articles. Typically the topics to be covered in the oral examination should be at the level of 600 to 800 level courses and include material that the student learned individually.

The syllabus must be approved by the Graduate Director’s office and the student’s committee members. Students should pass their oral examination prior to the end of the third year of the program.

The final departmental steps in attaining the degree is completion of a dissertation that must consist of original research of sufficient quality for publishing in a respectable mathematics journal. It is not unusual for the mathematics in a single dissertation to generate two or three published manuscripts.

Jenny Russell

Assistant to the Graduate Director

Department of Mathematics

227 Mathematics Building, Buffalo, NY 14260-2900

Phone: 716-645-8782; Fax: 716-645-5039

Email: jennyr@buffalo.edu

**For those students admitted to the program in 2015, the prior requirements remain in effect.**

The main steps in completing a PhD are: passing qualifying examinations; and, writing a dissertation.

The qualifying examinations are taken in several parts. During the first year of full-time study, the student must pass the First Qualifying Examination, an exam on basic material from undergraduate algebra and analysis. During the second year, the student must pass a more advanced, but quite flexible Second Qualifying Examination based on courses at the 600 level and above. By the end of the third year, the student must pass another exam, the nature of which will vary from student to student, and depends primarily on the student's area of study and thesis advisor.

The dissertation must consist of original research of sufficient quality to be published in a respectable mathematics journal. Upon completion of the second qualifying exam, the student will choose (in consultation with the director of graduate studies) a doctoral committee, the chair of which will direct the thesis research. Upon completion of the thesis, the student must pass a final oral examination administered by the department.

Both the MA and the PhD degrees have residency requirements: one year for the MA and two years for the PhD.

The week before classes begin in August, all new M.A. and Ph.D. students must take the First Qualifying Examination*.* The syllabus for this exam is based on undergraduate analysis and algebra (including linear algebra). This exam is given before classes begin to enable the student and the director of graduate studies to refer to its results while deciding the most appropriate courses for the student.

The main steps in obtaining a PhD are passing the qualifying examinations, writing a thesis, and passing a final oral examination on this thesis. The departmental regulations concerning each of these are given below. The regulations are interpreted by the graduate studies committee which, on written petition from a student, may permit deviations from the rules, provided there are exceptional circumstances. In addition to the departmental regulations, there are university requirements which must also be satisfied.

**Admission with Advanced Standing**

At the time of admission to UB's Graduate School, the director of graduate studies may decide that certain students have advanced standing of one or two semesters of graduate work, depending on UB Graduate School requirements. This will be done after examining the graduate records of the students and taking account of his previous courses, the institutions where he studied, his proficiency in English (TOEFL), etc. It will be clear from what follows that such students will have to fulfill various requirements more quickly than normally admitted students.

**Definition of Total Semesters of Graduate Work**

The sum of the semesters of graduate work as defined by (i) and (ii) below yields the total semesters of graduate work which will simply be called "semesters of graduate work".

(i) A student admitted with graduate coursework may credited with one or two semesters of graduate work, according to Graduate School requirements.

(ii) For every semester at SUNYAB that a student is registered for fewer than nine credit hours, the credit hours are to be totaled and divided by nine. The result, rounded down to the next integer, will also be counted as semesters of graduate work. In no event will a student be said to have completed more than two semesters of academic work in one calendar year.

**Deficiency**

A student is considered to have a deficiency if in the first semester as a graduate student at UB, the student officially enrolls in, and completes, Math 519 (introductory algebra) or Math 531 (introductory real variables). The student should base her/his decision on whether to take these courses on advice from the director of graduate studies and on evaluation of the student's knowledge in algebra and analysis by the relevant area committees.

**First Qualifying Examination**

The First Qualifying Exam is a three-and-a-half-hour written examination based on a syllabus covering introductory real variables at the level of MTH 431-432, introductory abstract algebra at about the level of MTH 419, and linear algebra at about the level of MTH 420. The examination is given twice a year, during the week prior to the beginning of each semester.

The purpose of the first examination is to assist the director of graduate studies and the student in deciding soon after the student's entry into the UB Graduate School, whether or not the student will be admitted to the the PhD program in mathematics.

Normally, to remain in the PhD program, a student is required to pass this examination within the first two years of graduate work. A student who entered with a deficiency is not required to pass this examination until the first opportunity after completiing two semesters of graduate work. See the *Syllabus for the First Qualifying Examination* (Revised 04/25/13) attached as a pdf, below.

**Second Qualifying Exam**

This consists of two three-hour area examinations, selected by each student from the following four choices: ALGEBRA; ANALYSIS; GEOMETRY/TOPOLOGY; and DIFFERENTIAL EQUATIONS. It is the purpose of the second qualifying examination to insure that each student has a rudimentary command of at least two "core" areas of mathematics.

To remain in the PhD program a student is required to obtain a grade of A or B for one of the area examinations no later than the beginning of his fourth semester of graduate work and an average of at least B for both of the area exams no later than the beginning of his *fifth* semester. Students may repeat the examinations, within the time limit, without penalty and are encouraged to take at least one of the examinations as early as possible. See *Information on the Second Quaifying Examination,* attached as a pdf, below.

**Doctoral Committee**

During the semester in which he completes the Second Qualifying Examination, each student will select a major professor, who is a member of the graduate faculty, in consultation with the director of graduate studies. The latter and the major professor will then choose the student's doctoral committee, consisting of at least three members of the faculty with the major professor as chair.

**Admission to Candidacy**

The student's doctoral committee will set the requirements for admission to candidacy. These are subject to the approval of the director of graduate studies and may include, but are not restricted to, any of the following: an oral examination on "research level" material, a project, a series of lectures on "research level" mathematics, or a written qualifying examination in another department. These requirements must be satisfied by the end of the *sixth semester* of graduate work.

**Language Requirements**

There are no language requirements.

**Additional Course Work**

Before the final oral exam, each student should pass, with a grade of **A, B ,** or **S**, *two* one-semester graduate course in subjects other than those of his or her second qualifying exam. These courses are to be approved by the director of graduate studies. Each PhD student must complete 72-credit hours from: (a) selected 500 level mathematics courses; (b) 600-800 level Mathematics courses, with the exception of thesis guidance, seminar courses, and other courses of this nature; (c) courses designated by his/her major professor.

**PhD Thesis and Final Oral Examination**

The final departmental steps in attaining the degree of Doctor of Philosophy are:

1. Completion of a thesis satisfactory to the major professor and the student's doctoral committee;

2. Approval by the UB Graduate School that the student proceed to examination on his/her thesis at a final oral examination;

3. Submission of the thesis to each member of the doctoral committee at least three weeks prior to the final oral examination;

4. Passing the final oral examination.