## Overview

The program of studies for a Math Department PhD student is divided into two main parts: **Pre-** and **Post-Candidacy**. Before taking the **Candidacy Exam**, students need to fulfill numerous requirements which ensure solid preparation in core mathematical areas as well in their chosen specialization. These include passing the **Qualifying Requirements** as well as fulfilling the **Breadth** and **Foreign Language** requirements. The **Candidacy Exam** is usually taken sometime during the third year, marking the end of the preparatory period and the beginning of research leading ultimately to the **PhD Dissertation**. Time from admission to graduation usually averages around 6 years but may vary greatly depending on many factors such as initial preparation level, individual academic progress, complexity of chosen specializations, and strategic thesis and job decisions. The **OSU Graduate School** has requirements as well, which can be found in the Graduate School Handbook.

## Headstart Training

Newly admitted graduate students are required to arrive **4** **weeks** prior to the beginning of the Autumn semester in order to participate in our **Headstart Training** teaching preparation program. This is prerequisite for holding a GTA appointment. Further academic preparation activities are scheduled during this time as well. Exceptions can be made only for students who are offered university fellowships and typically involve a one-year deferment of the teaching portion only. Monetary compensation is provided to all Headstart participants.

## Qualifying Requirements

For the **Theoretical Track**, there are four Qualifying Requirements, corresponding to the content of the four courses Math 6111, Math 6112, Math 6211, and Math 6212 in *Abstract Algebra* and *Real Analysis*. Each requirement can be passed by either receiving a grade of A or A- in the respective course, or by receiving a passing grade on a respective (separate) qualifying examination. The Math 6111 and 6211 courses are offered every Autumn Semester and the Math 6112 and 6212 courses are offered every Spring Semester. The four examinations, one for each course, are offered each August (typically in the week before the start of classes) and are open to both incoming and continuing students. Each exam is two hours in length and covers roughly the material of the respective courses.

All four requirements need to be fulfilled by the end of the third semester of study (not including summer). Thus a student has four attempts to fulfill, for example, the 6111-requirement (twice by taking the course, and twice by exam) and three attempts to fulfill the 6112-requirement (once by taking the course, and twice by exam). The 6211 and 6212 requirements are analogous.

Interested students may substitute *one *of these four requirements with an approved 6000-level year-long course sequence with A or A- grades. This includes all regular full-year 6000-level sequences, namely, Math 6221-6222, 6251-6252, 6411-6451, 6501-6502, 6601-6602, 6701-6702, and 6801-6802. (For example, the 6112 requirement can be substituted by taking the Math 6411-6451 Differential Equations sequence with A or A- grades in each course). Students should consult the GSC Chair about the use of the 6001-6004 Logic courses.

Qualifying requirements for the **Applied Track** combine a mandatory Scientific Computing (Math 6601) course, one of the algebra and analysis courses, three additional courses from Math 6602, Math 6411, and Math 6451, as well as the algebra and analysis courses.

Passing the Qualifying Requirements also entails an increase in stipend, assuming otherwise satisfactory academic progress (see Financial Support). The outcome of initial exams and coursework will also inform future advising.

Syllabi and exam materials can be viewed at https://math.osu.edu/grad/current/phd/quals.

## Advisor and Breadth Requirements

Upon attaining *Regular PhD *status,* *students are matched with faculty members who guide them to potential dissertation research ideas. The primary task of this **Dissertation Advisor** is to facilitate their advisee’s development as a mathematician.

The course work required for admission to **Candidacy** is referred to as our **Breadth Requirements**. The purpose of these requirements is to ensure that graduates master not only their eventual field of specialization, but also develop the breadth, versatility, and maturity expected from mathematicians working in academic professions that traditionally require a **PhD**. The requirements are as follows:

**Course Sequences:**Complete a year-long sequence from each of**three different mathematical areas**(see below)- All courses must be passed with a grade of
**B+**or higher - Course sequences used for the qualifying requirement (such as, for example, 6111-6112) may also be used for the breadth requirement. However, a passed qualifying exam does not count towards a breadth requirement.

Nearly all **PhD** students complete their **Breadth Requirements** within their first **two years** from admission. General expectation is that students fulfill **two breadth sequences** within **one year** from passing their **Quals** and the **third one** in the **following year**. Timely completion of breadth requirements may influence stipend level and summer support.

**All Breadth Requirements** must be **completed** by the time of the **Candidacy Exam**.

### Breadth Requirements Chart

**Algebra **(Course Number, Sequence Title)

- Math 6111,6112; Abstract Algebra
- Math 7121, 7122; Number Theory
- Math 7141, 7142; Algebraic Geometry
- Math 7161, 7162; Lie Groups

**Analysis **(Course Number, Sequence Title)

- Math 6211, 6212; Real Analysis
- Math 7211, 7212; Functional Analysis
- Math 7221, 7222; Ergodic Theory

**Differential Equations** (Course Number, Sequence Title)

- Math 6411, 6451; Differential Equations
- Math 7412, 7413; Ordinary Differential Equations
- Math 7452, 7453; Partial Differential Equations

**Topology / Geometry** (Course Number, Sequence Title)

- Math 6701, 6702; Differential Manifolds & Geometry
- Math 6801, 6802; Algebraic Topology
- Math 7711, 7721; Riemannian & Kahler Geometry
- Math 7851, 7852; Differential Topology

**Alternative Topics** (Course Number, Sequence Title)

- Any Two Math 6001-6004; Advanced Mathematical Logic
- Math 6221, 6222; Complex Analysis
- Math 6251, 6252; Theory of Probability
- Math 6501, 6502; Combinatorics & Graph Theory
- Math 6601, 6602; Numerical Methods in Scientific Computing
- Math 7611, 7612; Computational Partial Differential Equations
- Math 7651, 7652; Applied Complex Variables and Asymptotics

## Foreign Language Requirement

The foreign language requirement ensures the ability to read (with the aid of a dictionary) one foreign language chosen from among **French**,** German**, or **Russian. **It can be fulfilled in one of the following two ways:

**Class:** Students with little or no prior knowledge of the chosen language can fulfill their language requirement by passing one of the following classes with a grade of **B** or better:

**French 6571****German 6101**or**German 6102****Russian 6171**or**Russian 6172**

**Exam: **Alternatively, a student may pass a translation exam in one of the languages above.

To schedule the exam, please start by contacting the **Mathematics Department Language Coordinator:**

Dr. Andrzej Derdzinski (derdzinski.1@osu.edu)

To find out dates and information on the exams, please see below:

**French Department Translation Exam Coordinator:**

Sonya Afanasyeva afanasyeva.1@osu.edu

https://frit.osu.edu/graduate/graduate-reading-proficiency-exam/french-reading-proficiency-exam

**German Department Translation Exam Coordinator:**

May Mergenthaler Mergenthaler.4@osu.edu

http://germanic.osu.edu/german-reading-exam

**Russian Department Translation Exam Coordinator:**

Larysa Stepanova stepanova.1@osu.edu

## Specialization & Advisor

An important candidacy requirement is the choice of a **dissertation specialization** and a **Dissertation Advisor**. The diligent, timely, and careful pursuit of a future research direction is likely the most important responsibility of a prospective **PhD candidate**. The student should be fully invested in the choice of specialization, which will impact his/her future academic trajectory more than anything else. There are currently 65+ regular mathematics faculty on the Columbus campus, plus over 20 additional faculty on the branch campuses, who can supervise doctoral dissertations. Consult our current **Graduate Faculty List** for names, contacts and specializations. Under special circumstances, students can also be advised by faculty outside of the department. The advisor pool in our department is thus as large as that of any department in the country.

There are numerous opportunities for students to get to know potential advisors. This includes having them as teachers in introductory classes, attending the **Invitations to Mathematics** lecture series, regular research seminars, and colloquia (see **Events**), or self-development through academic advisors, peers, and publicly available research information. After narrowing down possible specializations, students typically sample faculty and topics by taking numerous reading courses (**MATH 6193**) on special topics with a few prospective advisors. These provide introductions to future research areas that are too specialized to be covered in regular courses. The one-on-one teaching of a reading course may also serve as a preview of the advisor / advisee interaction in future thesis work.

The choice of thesis advisor usually evolves out of this process. After student and faculty agree on the thesis advising, the student reports the change from the **Preliminary Academic Advisor** to the chosen **Dissertation Advisor** to the **Math Graduate Office **using the form located outside the **Grad Office**.

## Master of Science (MS) Degree

Information on how students admitted into the PhD program can earn the MS Degree can be found at https://math.osu.edu/grad/current/ms.

## Candidacy Examination

For a graduate student to become an official **PhD Candidate,** he/she has to **pass** the **Candidacy Exam**. The **Candidacy Exam** evaluates the validity and scope of the dissertation proposal, and serves as a forum for critique and guidance towards the successful completion of dissertation research. This exam is regulated by the university's Graduate School and permission from the department to take the exam is subject to the following requirements (for more detailed information on Candidacy see https://gradsch.osu.edu/handbook/all#7-0). These concern the composition of the committee, the written, and the oral portion of the examination:

**Committee**: The committee consists of four regular faculty with graduate P-status, including the advisor of the candidate who serves as the chair. Other committee members can be from other Ohio State departments but have to have graduate P-status in their programs. Additional members, beyond these four, can be added by petition and according to Graduate School rules.

**Written Portion**: The written portion consists of a 10-15 page dissertation proposal in which goals, scope, methods, and background of the planned research is outlined. The document has to contain mathematically rigorous statements, needs to be type-set along the usual publishing standards in the field (e.g., LaTeX), and should contain a substantial bibliography that includes all pertinent publications the intended research will be based on. The proposal of the written portion should be submitted to the committee at least ten days before the presentation and oral portion.

**Presentation**: The candidate is required to describe his/her proposal in a short presentation of approximately 30 minutes to the committee immediately before the start of the oral portion of the examination. The details of the format are determined by the advisor, including whether the presentation should be public and questioning during the presentation.

**Oral Portion**: Following the presentation there is a two-hour oral examination by the committee. This time has to be completely dedicated to the questioning by the committee and is not allowed to contain further presentations. The questions can focus on the proposal itself and the validity and relevance of the research questions, but can also include skill and knowledge examination of the needed mathematical background, as well as test familiarity with prior research.

**Timeline**: An application for candidacy must be submitted via gradforms.osu.edu at least three weeks before the oral examination. The candidacy examination can be taken at any time during business hours when the university is open -- including summer terms and breaks. The final date on which a candidacy exam can be counted as being within any given semester is the day before the first day of the following semester (for example, a "Spring" exam can be scheduled up until the day before Summer Term begins). All committee members' approval signatures must also be submitted in GradForms prior to the first day of the following term.

**Other Requirements**: All pre-candidacy requirements of the Mathematics Department, as well as all credit and residency requirements of the Graduate School, have to be fulfilled by the end of the term prior to taking the exam. Candidates also need to be enrolled for at least 3 hours at the graduate level during the term of the exam (note: if you schedule your exam in summer, you will need to enroll in 4 total credit hours in order for your summer tuition waiver to apply). *Foreign language classes do not count toward the 3 graduate credits required to take a candidacy examination.*

Following the exam, the **Report on Candidacy **form must be approved on GradForms by all committee members.

## Post-Candidacy & Dissertation Research

After the **Candidacy Exam**, **PhD** students spend most of their time on research related to their dissertation, under the close supervision of their **Dissertation Advisor**.

There are some requirements during this time which **PhD Candidates** must abide by:

**Three-Hour Enrollment: Post-Candidacy**students are expected to enroll for exactly**3****credit hours**every Autumn and Spring semester. In most cases, this should be 3 credits of**MATH 8999**with their**Dissertation Advisor**. A 3 credit hour course may be substituted for the 8999 hours with permission of the advisor. Additional credit hours for enrollment are**not included**in the**Graduate Associateship**tuition waiver. A student may request to have tuition covered by the department for academically essential courses by a petition to the**Graduate Studies Committee**. In all other cases, tuition has to be paid for by the student or an external resource.**Fellowship**recipients will have different guidelines on this matter.**Continuous Enrollment:**Students who have passed their candidacy examination are required to be enrolled during**every**Autumn and Spring Semester. There are only exceptions for Summer and formally petitioned**Leaves of Absence**. For detailed rules on leaves, see**Section VII.7 of the Graduate Handbook.**

## Final Defense & Graduation

How long one takes to graduate may vary greatly depending upon initial preparation, chosen specialization, difficulty and scope of the research problem, diligence of the candidate, results required to be competitive in the chosen area or job market, and other factors. Requirements to be eligible for graduation include:

**Time Limits:**The university allows a maximum of**five years**from**passing the candidacy examination**until submission of the**final copy**of the**dissertation**. The Math Department however, has the expectation that you can accomplish this in**three years or less**. Continuation in the program is contingent on timely and satisfactory progress towards completing a dissertation as determined by the**Graduate Studies Committee.****Credit Hours:**Students are required to have accumulated**80 graduate credit hours**of mathematics courses by the time of graduation. It is possible to substitute some of these with graduate credits from courses outside of the mathematics department, if approved by the advisor and the**Graduate Studies Committee**. In addition, university rules require that**50**of these credit hours have to be beyond the**Master's degree**.

Once the **Dissertation Advisor** deems the **Dissertation Doctoral Draft** complete, the candidate needs to assemble a **Final Oral Exam Committee**. The committee consists of the **Dissertation Advisor** and **two** additional regular category P level faculty members, who will review the draft. The doctoral candidate must submit the **Application for Final Exam** form via GradForms no later than **two weeks prior** to the proposed final oral examination date. The approval of the draft is followed by the **two-hour** **Final Oral Examination** (dissertation defense) conducted before the dissertation committee members listed, plus a non-Math representative assigned by the **Graduate School**. See the **PhD Dissertations** link on the Department website for samples of past approved Dissertations.

The department supports the search for academic jobs in several ways. Before graduation, the department provides travel support for students, helps with letters, and circulates job opportunities. After graduation, many former students with can find employment as lecturers with the department while they are looking for permanent jobs, if interim employment is needed.

## Program Time Expectations

- The qualifying requirements should be fulfilled by the middle of the second year. The graduate studies committee may decide on conditional continuation at either regular or probationary level in close cases.
- International students should be classroom teaching certified by ESL's Spoken English Program by the beginning of the second year.
- Students are expected to pass the candidacy examination
*before*the Autumn Semester of their fourth year. - Students are expected to graduate by the end of their sixth year. In cases where this is not possible, the graduate studies committee can be petitioned for a seventh year of financial support at a reduced stipend level.