phd philosophy of mathematics

Mathematics

Doctor of philosophy in mathematics, get your phd in mathematics.

The PhD program places a strong emphasis on preparation for research and teaching.

Students must earn at least 72 semester hours of graduate credit and spend at least three years in residence at a graduate college, including at least one year at the University of Iowa.

They must complete specific courses designated as preparatory for the PhD qualifying examinations; pass the qualifying and comprehensive exams; and write a PhD thesis.

For a complete description of these requirements see the graduate student handbook .

Graduate student resources

• For students enrolled fall of 2020 and later
• For students enrolled prior to fall of 2020
• Teaching assistant handbook
• Graduate policies
• PhD qualifying and MS comprehensive exam archives
• Absence form

Create your academic path

You'll find degree overviews, requirements, course lists, academic plans, and more to help you plan your education and explore your possibilities.

Current course list

The MyUI Schedule displays registered courses for a particular session and is available to enrolled students. The list view includes course instructors, time and location, and features to drop courses or change sections.

University of Missouri

College of Arts and Science

Mathematics

Mathematics doctor of philosophy (phd).

Core and Advanced Graduate Courses: Definitions and Schedules

The 8000 level courses appearing in the tables below constitute the current working definition of the term core graduate course. A graduate course not appearing on this list will be called an advanced graduate course. Please refer to the full list of Mathematics courses 4000 and above .

To increase the predictability and the enrollment for core graduate courses, we will stay with the prescribed Fall/Spring schedule for core graduate courses. Advanced graduate courses may be offered on a less predictable basis.

Advanced graduate courses include three sub-classes:

• Topics courses (title includes the word "Topics"). Examples include 8102, 8302, ...
• Seminar courses (title includes the word "Seminar"). Examples include 9187, 9287, ...
• Other advanced courses (title does not include either of the words "Topics" or "Seminar") . Examples include 8642, 8680, 8629, 8670...

It is allowable to list a Topics or Seminar course multiple times on your program of study assuming it is not a duplicate course covering the same topic. Topics and Seminar courses generally have subtitles describing the topic, to list two courses on the program of study it is sufficient that the subtitles are (significantly) different even if the course numbers are the same . On your transcript, only the course number and title appear. On your program of study, you should try to list the course number, the abbreviated title and the subtitle. For example:

• 8102 Topics: Lie Algebras
• 9487 Seminar: Morse Theory

This is a professional research degree designed to prepare students for various advanced professional careers, including college teaching and research.

Year 0 courses include basic advanced undergraduate material, which incoming Ph.D. students are required to master before engaging in graduate coursework.  Well prepared incoming students can petition to skip some or all of the Year 0 courses.  The Director of Graduate Studies will administer an informal exam to see if the students are sufficiently ready to skip Year 0 courses.

Year 1 courses will train students to develop a common solid foundation on basic graduate mathematics. The Ph.D. student is required to pass all 6 courses, and to pass qualifying exams in Algebra and Real Analysis. The qualifying exams will be given in May of each year, shortly after finals week.  There will be an opportunity to retake a qualifying exam in August just before the beginning of the Fall semester.  The Analysis qualifying exams will be from topics from Real Analysis I and Real Analysis II.  The Algebra qualifying exams will be from topics from Algebra I and Algebra II. Extremely well prepared students, with the permission of their initial adviser and the Director of Graduate studies, may take one or both qualifying exams in August before they start their first semester.  If they pass, then with the permission of their initial adviser and the Director of Graduate studies they may skip the corresponding courses in Year 1.

Year 2 and above are the post-qual core courses.  Every Ph.D. student must complete at least six of the post-qual core courses.  (Note that the parity of the year is determined by the beginning of the AY.  For example, Spring 2016 occurs in the beginning of AY 2015, and so would be considered to be in an odd year.)

The graduate student must further complete a course of study approved by the doctoral program committee and pass a comprehensive examination. The active areas of research interest of the current members of the staff are: algebraic geometry, commutative algebra, number theory, representation theory, analysis (real, complex, functional and harmonic), analytic functions, applied mathematics, financial mathematics and mathematics of insurance, scattering theory, differential equations (ordinary and partial), differential geometry, dynamical systems, general relativity, mathematical physics, probabilistic analysis and topology.

Note: Effective at the start of Winter Semester 2007, there is NO foreign language proficiency requirement for the Mathematics PhD. However, a student's Doctoral Committee still retains the discretion to impose a foreign language proficiency requirement.

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Doctor of Philosophy (PhD)

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. Full program requirements can be found in the OSU Department of Mathematics Graduate Program Handbook [pdf]. 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 a 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 or analysis courses, and three additional courses from  Math 6602 ,  Math 6411 ,  Math 6451 , and 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.

PhD students are expect to complete their Breadth Requirements within their first two years from admission. Most students fulfill two breadth sequences  in their first year through qualifying courses and the third in their second 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

• Math 6111, 6112; Abstract Algebra
• Math 7121, 7122; Number Theory
• Math 7141, 7142; Algebraic Geometry
• Math 7161, 7162; Lie Groups
• Math 6211, 6212; Real Analysis
• Math 7211, 7212; Functional Analysis
• Math 7221, 7222; Ergodic Theory
• Math 6411, 6451; Differential Equations
• Math 7412, 7413; Ordinary Differential Equations
• Math 7452, 7453; Partial Differential Equations
• Math 6701, 6702; Differential Manifolds & Geometry
• Math 6801, 6802; Algebraic Topology
• Math 7711, 7721; Riemannian & Kahler Geometry
• Math 7851, 7852; Differential Topology
• 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   ([email protected])

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

French Department Translation Exam Coordinator: Matthew Lang   [email protected] https://frit.osu.edu/graduate/graduate-reading-proficiency-exam/french-reading-proficiency-exam

German Department Translation Exam Coordinator: Natascha Miller [email protected] http://germanic.osu.edu/german-reading-exam

Russian Department Translation Exam Coordinator: Larysa Stepanova [email protected] https://slavic.osu.edu/courses/transfer-credit-and-placement

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:

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.  

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.

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.

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. 

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.

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.

Applied Mathematics - Doctor of Philosophy (PhD)

Mathematics 3 (M3) Building on Waterloo's Campus

Conduct mathematics-based research and generate new knowledge in a multidisciplinary environment with the PhD in Applied Mathematics program.

At North America’s only dedicated Faculty of Mathematics and the #1 school in Canada for mathematics and computer science, you’ll connect theoretical advances and innovative mathematics to develop novel solutions to the pressing problems facing today’s world.   

Through a combination of coursework and original research, you’ll learn cutting-edge applications of mathematical theory in a broad range of fundamental and applied sciences, with five areas of research to choose from including control theory and dynamical systems, fluid mechanics, mathematical medicine and biology, mathematical physics, and scientific computation.  

With the competitive edge provided by mentorship through the Faculty’s connections around the world, you’ll be prepar ed to pursue a career in academia, government or industry.  

Research areas and degree options:

• Control and Dynamical Systems
• Fluid Mechanics
• Mathematical Medicine and Biology
• Mathematical Physics
• Scientific Computing

Program overview

Department/School : Applied Mathematics Faculty : Faculty of Mathematics Admit term(s) : Fall (September - December), Winter (January - April), Spring (May - August) Delivery mode : On-campus Program type : Doctoral, Research Length of program : 48 months (full-time) Registration option(s) : Full-time, Part-time Study option(s) : Thesis

Application Deadlines

• January 15 (for admission in September)
• June 1 (for admission in January of the following year)
• October 1 (for admission in May of the following year)

Key contacts

[email protected]

I see all these great scholars around me, like my supervisor Sue Ann Campbell. And like Anita Layton, Ghazal Geshnizjani, my committee members, and so many others in the department. I see their passion for what they do and their dedication to helping us grad students succeed. It’s very heartening. It motivates me to reach that level where I can give back in the same way. Maliha Ahmed, Applied Mathematics, PhD

Supervisors

• Review the  finding a supervisor resources

Admission requirements

• Minimum grade point average: 78% or its equivalent
• It is absolutely essential that the application for admission into the program contain evidence of potential for performing original research. This should be provided by successful completion of a Master's thesis in a mathematics-related discipline.
• In some circumstances a student enrolled in the MMath program (thesis) in Applied Mathematics may transfer to the PhD program without completing their MMath program

Degree requirements

• Review the   degree requirements   in the Graduate Studies Academic Calendar, including the courses that you can anticipate taking as part of completing the degree
• Check out   Waterloo's institutional thesis repository - UWspace   to see recent submissions from the department of Applied Mathematics graduate students

Application materials

• The SIF contains questions specific to your program, typically about why you want to enrol and your experience in that field. Review the  application documents web page  for more information about this requirement
• If a statement or letter is required by your program, review the  writing your personal statement resources  for helpful tips and tricks on completion
• Transcript(s)
• Three  references , normally from academic sources
• TOEFL 90 (writing 25, speaking 25), IELTS 7.0 (writing 6.5, speaking 6.5)

Tuition and fees

Visit the  graduate program tuition page  on the Finance website to determine the tuition and incidental fees per term for your program

Review living costs and housing

Review the   funding graduate school resources   for graduate students

• Department of Mathematics

Graduate Studies

• Mathematics, PhD

Doctor of Philosophy in Mathematics (PhD)

Requirements outline.

The Ph.D. degree is a research degree and the principal requirement is that a student writes an original research thesis. The thesis is produced under the supervision of a faculty member and is examined by a committee of three departmental faculty and an outside expert. To qualify to write a thesis, a candidate for a Ph.D. in mathematics first must pass three Preliminary Examinations. It is recommended that Ph.D. candidates discuss possible research opportunities with the Director of Graduate Studies and/or faculty members soon after they enter the Ph.D. Program. Entering students should outline an appropriate sequence of courses to learn the essential material for pursuing their research interests. After a student has passed the Preliminary Examinations they must choose an advisor from the Mathematics Department faculty. A candidate's thesis usually is developed and written with the guidance of this advisor who will later chair the thesis defense committee. The time required to obtain a Ph.D. degree varies a lot. The department does not support graduate students as Teaching Assistants for more than five academic years.

Ph.D. Degree Requirements

The requirements that must be satisfied for a candidate to receive a Ph.D. include:

• The candidate must pass Preliminary Examinations .
• The candidate must obtain a grade of B or better in at least 24 semester credit hours of courses in the Mathematics Ph.D. program. Students should take doctoral research classes MATH 8x98 (where “x” is the number of credit hours) while conducting thesis research. Students must register for. the course MATH 8x99 “Doctoral Dissertation” in the semester when they intend to graduate
• After passing all three Preliminary Examinations the candidate is subject to Annual Performance Review (APR). The APR evaluates research progress of the candidate. The APR is conducted in oral or written form by a committee consisting of at least two faculty members of the Mathematics Department. The APR committee is chaired by the candidate’s advisor. Candidates failing the APR are subject to termination from the Ph.D. program.
• The candidate must be in residence, and take 9 semester credit hours of courses, in two consecutive long semesters, Fall followed by Spring. Alternatively, the candidate must be in residence and take a full load in consecutive Spring, Summer, and Fall terms.
• The candidate must write a Doctoral Dissertation with the guidance of an advisor who is a regular faculty member of the Mathematics Department.
• The candidate must defend their Dissertation in a public examination by a thesis committee consisting of at least 4 members, three of whom are faculty members in the Mathematics Department and at least one member outside UH Mathematics Department.
• NSM Thesis and Dissertation General Guidelines and Instructions
• NSM Thesis and Dissertation Formatting Instructions
• NSM Thesis and Dissertation Submission Instructions
• NSM Checklist for Thesis and Dissertation Review
• NSM Deadlines & Academic Calendar : This link provides deadlines for the submission of Dissertations.
• *The Graduate Record Examination (GRE) is waived for the Ph.D program within the Department of Mathematics.
• International students can not exclusively register for online courses.

Course Selection:

• Information about courses may be found at this link .
• Students can discuss advisor selection process with the Director of Graduate Studies.
• The above is only an outline of the primary requirements for the degree. The Director of Graduate Studies and others can provide more detailed information about conditions. The college and the university may have further requirements as listed at College and websites.
• PhD students can take topics classes at Rice University, UT Health, UTMB, or Baylor College of Medicine. Students must submit the Inter-Institutional Course Registration Form to the Graduate Director for approval. Taking an outside class must be essential for the completion of graduate degree. Thus, students must obtain a prior approval of their PhD avisor (signature on the form).
• Course Selection Requests: Please contact the Director for Instructional Support and Coordination < [email protected] > for more information.

Teaching Opportunities for Ph.D. Students:

As a condition, a student should have experiences of teaching Calculus recitation class with reasonable teaching evaluation. For an international student, by Texas law, the student must pass the English SPEAK test or its equivalence.

All PhD applicants who submit their complete application before the appropriate deadline are automatically considered for Teaching Assistantship.

Please contact the Director for Instructional Support and Coordination for more information about course selection requests .

Preliminary Examinations:

The Preliminary Examination is the final step in assessing the student’s ability and appropriate mathematical background to undertake a program of supervised research and study leading to a Ph.D. in Mathematics. Students who have completed their Master's degree in Mathematics may often be ready to take the Preliminary Examination without further course study.

Preliminary Examinations are three-hour, closed book written examinations that are given in each of the topics listed below. The questions in the examination emphasize problem solving skills and mathematical ability as opposed to rote memorization.

Preliminary Examinations are usually offered twice a year: at the end of the Fall and Spring semesters.

Students who receive support from the Department of Mathematics are expected to pass the Preliminary Examination according to the rules below. For non-supported students, the University rules apply.

All students are supposed to pass three Preliminary Examinations before the beginning of their third year in the Ph.D. program.

The following rules apply:

1. Students must pass three Preliminary Examinations from the different topic groups listed below

2. At least one out of the three Preliminary Examinations must be a core sequence. Core sequences are:

Review information for the preliminary written examinations:

Additional problems from past preliminary exams:

All preliminary exams are based on the content of the corresponding course. Please contact the instructor who taught the corresponding course most recently to obtain the up-to-date information.

2024-25 Academic Catalog

Doctor of philosophy in mathematics, why study mathematics.

Because mathematics is a framework upon which humanity builds an understanding of the world.

Mission of the Graduate Program:

The mission of the Graduate Program of the Department of Mathematics is to prepare students for leadership roles in meeting the mathematical needs of our society and to produce professional mathematicians for positions in universities, colleges, industry, governmental agencies, and research centers.

Doctor of Philosophy in Mathematics:

The Mathematics Department offers the degree of Doctor of Philosophy (Ph.D.) in Mathematics.  The Ph.D. program provides broad and deep expertise in mathematics, culminating in a dissertation that includes significant original work.  It is intended for students with a strong mathematical background who plan a career in research in academia or industry. A broad range of specialties is possible; research interests of department faculty include algebra, analysis, combinatorics, control theory, dynamical systems, geometry, numerical analysis, partial differential equations, probability, and statistics. There are two tracks: Pure Mathematics and Applied Mathematics. The requirements for each track are listed in the section Degree Requirements. College-wide requirements for graduate students may be found in the  Graduate School Catalog .

Admission to Graduate Studies

An applicant seeking to pursue graduate study in the College may be admitted as either a degree-seeking or non-degree seeking student. Policies and procedures of Graduate Studies govern the process of Graduate admission. These may be found in the Graduate Studies section of the online catalog.

Please consult the Departments & Programs section of the online catalog for information regarding program-specific admissions criteria and requirements. Special admissions requirements pertain to Interdisciplinary Studies degrees, which may be found in the Graduate Studies section of the online catalog.

Admission to the Ph.D. in Mathematics

The minimum prerequisites for admission are:

• an undergraduate degree from an accredited institution with a program of study in mathematics;
• a record of achievement that shows strong promise of success in graduate school, including a 3.0 cumulative grade-point average in undergraduate studies and a 3.0 grade-point average in mathematics (department requirement);
• course work in abstract algebra, linear algebra, and advanced calculus or introduction to analysis (comparable to KU courses  MATH 500 , MATH 558 , and MATH 590 ).

It is beneficial to have preparation in probability/statistics (comparable to  MATH 627 / MATH 628 ) and/or numerical analysis (comparable to  MATH 581 ). Although not required, it is also helpful to have taken introductory courses in complex analysis ( comparable to  MATH 646 ), partial  differential equations (comparable to  MATH 647 ), geometry (comparable to MATH 660 / MATH 661 ),  and/or topology.

The Mathematics Department currently does not require the general or subject Graduate Record Examination (GRE), although applicants may submit scores if they wish.  International students whose native language is not English must fulfill English language requirements specified by university policies.

Applicants must submit a graduate application online , including the following required materials:

• Transcript from each college or university the applicant has attended (an official transcript must be sent upon acceptance and completion of degree).
• Applicant’s résumé/curriculum vitae.
• A list of the textbooks used in mathematics courses beyond calculus.
• A statement of purpose indicating the applicant’s mathematical preferences and interests.
• 3 letters of reference.
• International applicants must fulfill the University's requirements for English proficiency .

Incomplete applications will not be considered. The minimum admission requirements do not guarantee admission. The Department of Mathematics evaluates candidates and makes recommendations to the Office of Graduate Studies regarding admission. The number of students admitted to the program changes from year to year, and admissions are competitive based on all application materials.

There are no additional application forms for financial support. Students are considered for support based on merit. Most Ph.D. students accepted by the program receive an offer of financial support in the form of a Graduate Teaching Assistantship. The number of GTAs available is limited. Further information about applications and admissions is available from the  Department of Mathematics .

Contact the department:

Michelle Morrison Graduate Program Coordinator Department of Mathematics 433 Snow Hall [email protected]

Ph.D. Degree Requirements

The department requires the student to meet the following requirements before taking the comprehensive examination.

• Pass two written qualifying examinations: one exam in either algebra or analysis and a second exam in either numerical analysis or probability/statistics.  Both qualifying examinations must be completed by the beginning of the student’s fifth semester.
• Complete the required qualifying exam coursework: MATH 727 (Probability), MATH 765 (Analysis I), MATH 781 (Numerical Analysis I), MATH 791 (Abstract Algebra I).  Passing a qualifying exam exempts a student from the corresponding course. This coursework must be completed before the preliminary examination.
• Pass a preliminary examination in an area close to the focus of the eventual doctoral dissertation. The preliminary examination must be completed by the beginning of the student’s eighth semester.
• To meet the Research Skills requirement, students must complete an introductory programming language course approved by the Graduate Committee. The course may have been taken at KU or at another university, either as a graduate or undergraduate. Students may meet the Research Skills requirement by passing EECS 138 or EECS 168 .  Alternatively, students may complete a computing project approved by their advisor and the Graduate Committee demonstrating competence in either a programming language or the use of specialized software that supports the student’s research.
• The Responsible Scholarship requirement must be met by completing the departmental training in responsible scholarship for mathematicians.  The training is offered every spring semester.  Students must have passed the qualifying exams and be working with an advisor in order to participate.
• Complete the course requirements for a track in either Pure Mathematics or Applied Mathematics, as outlined below.

Note:  Contact your department or program for more information about the qualifying exam coursework requirement, the research skills and responsible scholarship, and the current requirements for doctoral students. Current policies on Doctoral Research Skills and Responsible Scholarship are listed in the Graduate Studies section of the online catalog and in the KU Policy Library.

Pure Mathematics

This track requires:

In addition, the pure-track student must complete four additional MATH courses at the 800 level or above before the final examination. MATH 896 , MATH 899 , MATH 993 and MATH 999 may not be used to satisfy this requirement. MATH 990 may be used to satisfy this requirement only with Graduate Committee approval. Courses outside Mathematics may be used to satisfy this requirement only with Graduate Committee approval.

Applied Mathematics

In addition, the applied-track student must complete four additional MATH courses at the 800 level or above before the final examination. MATH 896 , MATH 899 , MATH 993 and MATH 999 may not be used to satisfy this requirement. MATH 990 may be used to satisfy this requirement only with Graduate Committee approval. Courses outside Mathematics may be used to satisfy this requirement only with Graduate Committee approval.

Examination Preparation

Normally the work required to prepare a student for the oral comprehensive examination (and to do research) includes one or more semesters of advanced courses, directed readings, and seminars. In the oral comprehensive examination, a student must show proficiency in the chosen area of mathematics. Precise areas of responsibility on this examination are discussed in detail with the advisory committee (the student’s advisor and two other members of the department’s Graduate Faculty).

In addition to meeting general requirements, the Ph.D. candidate in mathematics must complete a minimum of 28 credit hours of mathematics coursework (this number includes 1 credit hour of MATH 999 ). The minimum amount of credit hours is possible only if a student passes all Ph.D. qualifying exams in lieu of the preparatory coursework ( MATH 727 , MATH 765 , MATH 781 , MATH 791 ). The program routinely takes 12 semesters to complete when factoring in research and milestone exams. A typical student completes 72 or more credit hours when enrolled full-time.

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University of Notre Dame

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MSIM PROGRAM FOR PHILOSOPHY PhD STUDENTS, MAMP PROGRAM FOR MATHEMATICS PhD STUDENTS, AND JOINT MATH/PHILOSOPHY PhD PROGRAM

In collaboration with the Philosophy Department, the Mathematics Department at Notre Dame offers several joint programs for students interested in Mathematical Logic as well as Philosophy. The acronym MSIM stands for Master of Science in Interdisciplinary Mathematics, and this degree is given by the Mathematics Department. The program is available also to students from fields besides Philosophy. See this link: https://math.nd.edu/graduate/msim-degree/ for more information about joint Mathematics/Philosophy graduate degrees at Notre Dame. If you are a PhD student at Notre Dame interested in the MSIM degree with your primary area of interest not in Mathematical logic and Philosophy, the MAMP (Master of the Arts in Mathematical Philosophy) program may be of interest. See https://philosophy.nd.edu/graduate-program/mathematical-philosophy-minor/ for additional information about MAMP. This page gives instructions for how to apply for the MSIM, MAMP, or the joint PhD program in Mathematical Logic and Philosophy. Please contact the Math or Philosophy DGS if you have additional questions.

Admission to either of these degree programs requires the approval of both the Mathematics and Philosophy Departments. Similarly, any extension of the deadlines discussed below need the approval of the Mathematics and Philosophy Departments. Approval by the Philosophy Department requires primary approval by the Logic Group within that department, and final certification by the Philosophy DGS. Approval by the Math department requires approval by the Logic Group within that department in consultation with the Math DGS and if necessary, the graduate committee. As these degrees are additional degrees beyond the student’s Ph.D. program, they are not funded separately. We expect that the students earning these degrees will be exceptional.

A student in the joint PhD program will have to find a Mathematics adviser and a Philosophy adviser. The student will write a single PhD thesis, but it may have separate parts with a Math or Philosophy focus.

Philosophy Primary

An essential criterion for admission to the MSIM or Joint Degree for a Philosophy graduate student by the mathematics department is the approval of a mathematics department faculty member who agrees to oversee the student’s work. This will normally require that the student has become integrated into that faculty member’s research group, and has proposed a viable area for research. It is the student’s responsibility to find their own advisor. Given that, the path towards admission to the MSIM or Joint Degree is as follows:

• Year 1 and 2 (coursework): In addition to Philosophy coursework, the student takes the Mathematics department’s Logic Sequence Math 60510 and Math 60520 in Year 1. The student is also required to take two additional basic courses in mathematics. Basic Algebra I and II is a common choice, but other choices are possible. These courses should be completed in the first two years. S/he also, in this period, plays an active role in some part of the Mathematics department’s on-going research seminars, lectures, etc. 
• Year 2: The student finds a faculty member willing to supervise some advanced work in that faculty member’s area. This might be over the first summer, during the second year, or during the second summer. The student should become well-integrated into the research group of the intended supervisor and take topics courses in Logic.
• Application to the MSIM program is made prior to the start of Year 3, and we encourage applicants interested in continuing to do the joint PhD program to apply well before the start of year 3. The application will include a description of the courses to be taken for the degree, and of the proposed Master’s thesis topic, both of which will have been designed in consultation with the proposed advisor along with letters of support from their advisors.  It is expected that the student’s work in Mathematics classes outside Logic will be above average and similarly with their work in philosophy.  
• If the student is admitted to the MSIM program, s/he will work during Year 3 with the Mathematics advisor on their thesis topic.  
• Students interested in pursuing the joint PhD program, which is called Philosophy and Mathematics, should take the oral exam in Mathematics by the beginning of Year 4. This oral is understood to be similar to the one taken by students pursuing a Ph.D. in Mathematical Logic. Passing this oral exam is required to earn the Joint Ph.D. 
• While working towards the MSIM degree, a student interested in pursuing the joint PhD program must express an intention to apply by the end of May of Year 3 and apply by the beginning  of Year 4. An application consists of a description of the courses to be taken for the degree, a research proposal, hopefully some completed research and letters of support from their advisors. If the decision at that stage is that the student needs further work, then the student may submit a revised application during Year 4.  In any event, if a philosophy student is to be accepted to the joint PhD program, this must happen by the first day of classes in the 5th year.
• If the student in the MSIM program is not admitted to the Joint PhD program, s/he will normally finish the requirements for, and be awarded, the MSIM degree on route to completion of the Philosophy PhD.  In this case we expect the MSIM thesis to be completed by the end of Year 4.  A public defense of the Master’s thesis is expected. The defense should happen by early in Year 5. 
• If the student is admitted to the Joint PhD program in Philosophy and Mathematics, s/he need not complete the requirements for the MSIM degree. Research completed in pursuit of the MSIM thesis might be incorporated into the research for the joint PhD dissertation.  We do expect this degree to be completed within 6 years. 

Mathematics Primary

An essential criterion for admission to the MAMP or Joint PhD for a mathematics graduate student by the Philosophy Department is the recommendation of a Philosophy Department faculty member who agrees to oversee the student’s work. This will normally require that the student has articulated a viable area for research and demonstrated to the satisfaction of the faculty member relevant competence to undertake a research project in the area. It is the student’s responsibility to find their own advisor. The joing PhD program for graduate students in the Department of Mathematics is called the PdD in Mathematics and Philosophy.

• Years 1 and 2: The student should enroll in approximately 1 research seminar in the Philosophy Department each semester. [Generally, four philosophy seminar courses with a heavy writing component are needed for a student applying to the MAMP or joint PhD program.] Knowing that the application to the MAMP includes the submission of a sample of philosophical writing, it is wise to make sure to take seminars with substantial writing components so that by the time of application the student will have experienced several episodes of writing and rewriting in light of instructor feedback. Students should check with the instructor whether the course has a substantial writing component.  Most seminars will fit this description, but some Logic seminars will not. Note that all seminars taken prior to application to the MAMP will be retroactively counted towards fulfillment of the MAMP (and/or Joint PhD.) degree. For students interested in MAMP or the joint PhD program,the Mathematics department can delay two of the required basic courses to the second year to allow students time to complete their philosophy seminars.
• An application to MAMP or the joint PhD program should consist of a transcript, the written work from Philosophy seminars, and a research paper, together with faculty evaluations.
• Year 2 or 3: Application to the MAMP should be made at the end of year 2 or during year 3. Students with a definite expectation to eventually apply to the Joint PhD program are strongly advised to apply before the start of year 3. Students with no plan to apply to the Joint PhD program should in any event apply to the MAMP before the end of their 3rd year of study, so that the commencement of their MAMP thesis research does not disrupt the timely completion of their Mathematics dissertation research.
• Applications for the Joint PhD in Mathematics and Philosophy. will only be considered for students in their first four years of graduate study, and admission to the Joint PhD  must happen before the start of the student’s 5th year. A student applying to the Joint PhD program need not complete their MAMP thesis; in many cases the work going towards that thesis will be further elaborated as the Philosophy component of their PhD dissertation.
• A student who does not apply to the Joint PhD in Mathematics and Philosophy should submit their MAMP thesis by the end of their 4th year.
• A student admitted to the Joint PhD program in Mathematics and Philosophy should have a research plan suited for completion of the degree by the end of their 6th year.

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• Doctor of Philosophy in Mathematics (PhD)
• Graduate School
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Go to programs search

Mathematicians use theoretical and computational methods to solve a wide range of problems from the most abstract to the very applied. UBC's mathematics graduate students work in many branches of pure and applied mathematics. The PhD program trains students to operate as research mathematicians. The focus of the program is on substantial mathematical research leading to the PhD dissertation. Students also develop their skills in presenting and teaching mathematics and its applications.

For specific program requirements, please refer to the departmental program website

What makes the program unique?

UBC has one of the largest and most vigorous departments of mathematics in Canada. Our faculty routinely win national and international awards for their research and teaching achievements. We have an engaged and sociable cohort of graduate students who are essential members of a broad selection of active research groups. Each group holds a variety of seminars and events that allow graduate students, postdoctoral fellows, visitors and faculty to enjoy regular interaction.

UBC is the headquarters for the Pacific Institute of Mathematical Sciences (PIMS). PIMS hosts a plethora of mathematical events such as conferences and summer schools, greatly enriching the scientific environment in the quantitative sciences at UBC. Our mathematics students are also regular participants at the nearby Banff International Research Station for Mathematical Innovation and Discovery. Finally, our Institute for Applied Mathematics provides options for interdisciplinary studies for PhD students who wish to work in applied and computational mathematics.

UBC's math program has a high reputation and there are many renowned professors in the department. This was a selling point of the UBC math graduate program for me.

Pardis Semnani

Quick Facts

Program enquiries, admission information & requirements, 1) check eligibility, minimum academic requirements.

The Faculty of Graduate and Postdoctoral Studies establishes the minimum admission requirements common to all applicants, usually a minimum overall average in the B+ range (76% at UBC). The graduate program that you are applying to may have additional requirements. Please review the specific requirements for applicants with credentials from institutions in:

• Canada or the United States
• International countries other than the United States

Each program may set higher academic minimum requirements. Please review the program website carefully to understand the program requirements. Meeting the minimum requirements does not guarantee admission as it is a competitive process.

English Language Test

Applicants from a university outside Canada in which English is not the primary language of instruction must provide results of an English language proficiency examination as part of their application. Tests must have been taken within the last 24 months at the time of submission of your application.

Minimum requirements for the two most common English language proficiency tests to apply to this program are listed below:

TOEFL: Test of English as a Foreign Language - internet-based

Overall score requirement : 100

IELTS: International English Language Testing System

Overall score requirement : 7.0

Other Test Scores

Some programs require additional test scores such as the Graduate Record Examination (GRE) or the Graduate Management Test (GMAT). The requirements for this program are:

The GRE is not required.

2) Meet Deadlines

3) prepare application, transcripts.

All applicants have to submit transcripts from all past post-secondary study. Document submission requirements depend on whether your institution of study is within Canada or outside of Canada.

Letters of Reference

A minimum of three references are required for application to graduate programs at UBC. References should be requested from individuals who are prepared to provide a report on your academic ability and qualifications.

Statement of Interest

Many programs require a statement of interest , sometimes called a "statement of intent", "description of research interests" or something similar.

• Supervision

Students in research-based programs usually require a faculty member to function as their thesis supervisor. Please follow the instructions provided by each program whether applicants should contact faculty members.

Instructions regarding thesis supervisor contact for Doctor of Philosophy in Mathematics (PhD)

Citizenship verification.

Permanent Residents of Canada must provide a clear photocopy of both sides of the Permanent Resident card.

4) Apply Online

All applicants must complete an online application form and pay the application fee to be considered for admission to UBC.

Tuition & Financial Support

Financial Support

Applicants to UBC have access to a variety of funding options, including merit-based (i.e. based on your academic performance) and need-based (i.e. based on your financial situation) opportunities.

Program Funding Packages

All full-time students who begin a UBC-Vancouver PhD Mathematics program in September 2018 or later will be provided with a funding package of at least $24,256 for each of the first four years of their PhD. The funding package may consist of any combination of internal or external awards, teaching-related work, research assistantships, and graduate academic assistantships.

Average Funding

• 52 students received Teaching Assistantships. Average TA funding based on 52 students was $13,784.
• 48 students received Research Assistantships. Average RA funding based on 48 students was $11,580.
• 3 students received Academic Assistantships. Average AA funding based on 3 students was $1,814.
• 54 students received internal awards. Average internal award funding based on 54 students was $13,279.
• 4 students received external awards. Average external award funding based on 4 students was $27,083.

Scholarships & awards (merit-based funding)

All applicants are encouraged to review the awards listing to identify potential opportunities to fund their graduate education. The database lists merit-based scholarships and awards and allows for filtering by various criteria, such as domestic vs. international or degree level.

Graduate Research Assistantships (GRA)

Many professors are able to provide Research Assistantships (GRA) from their research grants to support full-time graduate students studying under their supervision. The duties constitute part of the student's graduate degree requirements. A Graduate Research Assistantship is considered a form of fellowship for a period of graduate study and is therefore not covered by a collective agreement. Stipends vary widely, and are dependent on the field of study and the type of research grant from which the assistantship is being funded.

Graduate Teaching Assistantships (GTA)

Graduate programs may have Teaching Assistantships available for registered full-time graduate students. Full teaching assistantships involve 12 hours work per week in preparation, lecturing, or laboratory instruction although many graduate programs offer partial TA appointments at less than 12 hours per week. Teaching assistantship rates are set by collective bargaining between the University and the Teaching Assistants' Union .

Graduate Academic Assistantships (GAA)

Academic Assistantships are employment opportunities to perform work that is relevant to the university or to an individual faculty member, but not to support the student’s graduate research and thesis. Wages are considered regular earnings and when paid monthly, include vacation pay.

Financial aid (need-based funding)

Canadian and US applicants may qualify for governmental loans to finance their studies. Please review eligibility and types of loans .

All students may be able to access private sector or bank loans.

Foreign government scholarships

Many foreign governments provide support to their citizens in pursuing education abroad. International applicants should check the various governmental resources in their home country, such as the Department of Education, for available scholarships.

Working while studying

The possibility to pursue work to supplement income may depend on the demands the program has on students. It should be carefully weighed if work leads to prolonged program durations or whether work placements can be meaningfully embedded into a program.

International students enrolled as full-time students with a valid study permit can work on campus for unlimited hours and work off-campus for no more than 20 hours a week.

A good starting point to explore student jobs is the UBC Work Learn program or a Co-Op placement .

Tax credits and RRSP withdrawals

Students with taxable income in Canada may be able to claim federal or provincial tax credits.

Canadian residents with RRSP accounts may be able to use the Lifelong Learning Plan (LLP) which allows students to withdraw amounts from their registered retirement savings plan (RRSPs) to finance full-time training or education for themselves or their partner.

Please review Filing taxes in Canada on the student services website for more information.

Cost Estimator

Applicants have access to the cost estimator to develop a financial plan that takes into account various income sources and expenses.

Career Outcomes

88 students graduated between 2005 and 2013: 1 is in a non-salaried situation; for 19 we have no data (based on research conducted between Feb-May 2016). For the remaining 68 graduates:

Sample Employers in Higher Education

Sample employers outside higher education, sample job titles outside higher education, phd career outcome survey, career options.

A great majority of our PhD graduates move on to postdoctoral fellowships and faculty positions at universities and research institutes in North America and around the world. However, a significant fraction of students move into careers in industry. Students considering non-academic careers are encouraged to complete an industrial internship (for instance through the Mitacs Accelerate program - headquartered at UBC) during their studies.

Enrolment, Duration & Other Stats

These statistics show data for the Doctor of Philosophy in Mathematics (PhD). Data are separated for each degree program combination. You may view data for other degree options in the respective program profile.

ENROLMENT DATA

Completion Rates & Times

Upcoming doctoral exams, tuesday, 17 september 2024 - 2:00pm - room 203.

• Research Supervisors

Advice and insights from UBC Faculty on reaching out to supervisors

These videos contain some general advice from faculty across UBC on finding and reaching out to a supervisor. They are not program specific.

This list shows faculty members with full supervisory privileges who are affiliated with this program. It is not a comprehensive list of all potential supervisors as faculty from other programs or faculty members without full supervisory privileges can request approvals to supervise graduate students in this program.

• Adem, Alejandro (Cohomology of finite groups, orbifolds, stringy topology, algebra, sporadic simple group, group actions, arithmetic groups, K-theory, homotopy theory, spaces of homomorphisms)
• Alacaoglu, Ahmet
• Angel, Omer (Probability theory, percolation, random graphs, random walks, particle processes, scaling limits)
• Bachmann, Sven (Mathematics and statistics; Mathematical Analysis; quantum phenomena; Mathematical physics; Quantum statistical physics; Topological states of matter)
• Balmforth, Neil (Fluid mechanics, nonlinear dynamics and applied partial differential equations)
• Behrend, Kai (Moduli spaces, Gromov-Witten invariants, string theory, Donaldson-Thomas invariants, Euler characteristics, categorification)
• Bennett, Michael (Number Theory, Diophantine Approximation and Classical Analysis)
• Bryan, Jim (Algebraic and differential geometry; Algebraic geometry, moduli spaces, enumerative invariants related to theoretical physics.)
• Cautis, Sabin (Mathematics and statistics; Geometry)
• Chau, Albert (Differential Geometry and Partial Differential Equations)
• Chen, Jingyi (Algebraic and differential geometry; Differential Geometry, Partial Differential Equations)
• Colliander, James (hamiltonian dynamical systems; partial differential equations; harmonic analysis)
• Coombs, Daniel (Mathematical biology; Cellular immunology; Complex physical systems; Epidemiology (except nutritional and veterinary epidemiology); Cell Signaling and Infectious and Immune Diseases; Cell biophysics; Disease models; Epidemiology; Immune cell signalling; Mathematics)
• Cytrynbaum, Eric (Bacterial cell division, Microtubule and cellular organization, Wave propagation in excitable media)
• Dao Duc, Khanh (Genomics; Mathematical biology; Neurocognitive patterns and neural networks; Agricultural spatial analysis and modelling; combine mathematical,computational and statistical tools to study fundamental biological processes; regulation and determinants of gene expression and translation; Machine Learning for Biological Imaging and Microscopy; Database development and management; Biological and Artificial Neural Networks for geometric representation)
• Doebeli, Michael Walter (Mathematical ecology and evolution, evolution of diversity, adaptive speciation, evolution of cooperation, game theory, experimental evolution in microorganisms)
• Feng, James (Chemical engineering; Mathematics and statistics; Biophysics; Complex fluids; Fluid mechanics; Mathematical biology)
• Fraser, Ailana (Differential Geometry, Geometric Analysis)
• Friedlander, Michael (numerical optimization, numerical linear algebra, scientific computing, Scientific computing)
• Frigaard, Ian (Fluid mechanics (visco-plastic fluids))
• Ghioca, Dragos (Drinfeld modules, isotrivial semiabelian varieties, Lehmer inequality)
• Gordon, Julia Yulia (Representation theory of p-adic groups and motivic integration; Trace Formula and its applications)
• Gustafson, Stephen James (Mathematics and statistics; Mathematical Analysis; Differential Equation; Global and Non-Linear Analysis; Mathematical physics; Nonlinear partial differential equations; Nonlinear waves; Topological solitons)
• Hauert, Christoph (Mathematics and statistics; Modelization and Simulation; Evolution and Phylogenesis; Biological Behavior; dynamical systems; evolution; game theory; social dilemmas; stochastic processes)
• Hermon, Jonathan (probability theory; Markov chains and the cutoff phenomenon; particle systems; percolation)

Doctoral Citations

Sample Thesis Submissions

• On problems of regularity and existence for critical drift elliptic equations and Navier-Stokes equations
• Symmetry-breaking bifurcations in compartmental-reaction diffusion systems with comparable diffusivities
• The polynomial method over finite rings and fields
• Global well-posedness and localized patterns of several reaction-diffusion systems involving advection
• Topics in arithmetic combinatorics
• Free boundary minimal submanifolds in geodesic balls of simply connected space forms
• On a completion of cohomological functors generalizing Tate cohomology
• Distribution of integral points on varieties
• Effective and explicit S-unit equations with many terms
• Classifying space for commutativity and unordered flag manifolds
• Finite-size scaling of a few statistical physics models in high dimensions
• Residual supersingular Iwasawa theory and μ-invariants for Zₚ²-extensions
• Numerical methods for biological flows laden with deformable capsules and solid particles
• The construction of blow-up solutions for some evolution equations
• Topics in discrete analysis

Related Programs

Same specialization.

• Master of Science in Mathematics (MSc)

At the UBC Okanagan Campus

Further information, specialization.

Mathematicians use theoretical and computational methods to solve a wide range of problems from the most abstract to the very applied. UBC's mathematics graduate students work in many branches of pure and applied mathematics.

UBC Calendar

Program website, faculty overview, academic unit, program identifier, classification, social media channels, supervisor search.

Departments/Programs may update graduate degree program details through the Faculty & Staff portal. To update contact details for application inquiries, please use this form .

Nicholas Richardson

Having grown up outside of Toronto and completed my undergrad and master's degree at the University of Waterloo, I was ready to change the scenery and go study somewhere else. I joke that is it the farthest I could move without leaving Canada, but more truthfully it was the campus that felt "right...

Gabriel Currier

I quite like the kind of math that people do here, and enjoy working with my supervisors. The campus is also a beautiful place and the graduate student community is pretty laid back and friendly.

Nathan Lawrence

Many factors contributed to my choice of UBC for graduate school. I was attracted to Vancouver’s geographical similarities to Portland in the pacific northwest. Also, I have family in the area. However, most importantly, I was intrigued and inspired by my professors and advisors to take on the...

Considering Vancouver as your next home?

This city won’t disappoint. It has it all: sea, parks, mountains, beaches and all four seasons, including beautiful summers and mild, wet winters with snow.

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Doctor of Philosophy Program

Besides satisfying the general regulations of the Graduate School for the degree of Doctor of Philosophy, the student must comply with the requirements briefly outlined below. For complete details about these requirements see section IV of the Graduate Handbook .

Pass four Qualifying Examinations . The exams are based on material that is covered in the courses listed and on material from undergraduate prerequisites. Credit for passing a similar examination at another university cannot be transferred. See sections IV and VI of the Graduate Handbook for more information.

Advanced Topics Examinations. A student becomes eligible to take the Advanced Topics Examination after passing the Qualifying Examinations.

Plan of Study. The plan of study should be submitted electronically to the Graduate School through myPurdue by each student preparing to hold their Advanced Topics. 

Preliminary Examination. The preliminary examination for most students will only require the completion of a form for the Graduate School. An oral or written examination may be required by the student's advisory committee for admission to candidacy. Graduate School regulations require that at least two sessions (including summer sessions) must elapse between the preliminary examination and the thesis defense.

Admission to Candidacy. To be admitted to candidacy for the Ph.D. degree, the student must have fulfilled the requirements above which are detailed in section IV of the Graduate Handbook .

Dissertation. A thesis must be submitted in final form presenting new results of sufficient importance to merit publication.

Recommendation for the Ph.D. Degree. If the requirements are met within the time limits detailed in section IV of the Graduate Handbook , the candidate will be recommended to the faculty to receive the degree of Doctor of Philosophy.

For information about financial support and research in absentia see section IV of the Graduate Handbook .

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Doctor of Philosophy (DPhil)

What is a dphil.

A DPhil is Oxford's name for a PhD - a higher research degree which allows you to make an original contribution to mathematics in the form of a thesis. A DPhil takes three to four years to complete. During your DPhil, you will be supervised by at least one academic, although some students will have more than one supervisor (particularly if they are working across disciplines). Unlike CDT courses (and PhDs in other countries), you will begin to do research straight away and there is no prescribed taught component. 

As part of your study toward a DPhil in Mathematics at Oxford, you will also be required to complete broadening and skills training and deliver class teaching to undergraduates, to enhance your broader mathematical knowledge and develop your career. You are very welcome to attend seminars and there may also be journal clubs or seminar series specific to your area of study. 

If you enjoy doing mathematics, and would like to be part of a lively and world-class research institute, take a look at our research groups to see if they align with your own interests. 

How to apply

All applications should be submitted online through the University's Graduate Application Form . Before you apply, check that you can meet the entry requirements , and read the   University of Oxford's graduate application guide .

Key Deadlines

Application deadlines for the DPhil in Mathematics:

• 8th January 2025
• 4th March 2025

Please apply by the 8th January deadline to be considered for available University-administered or Departmental scholarships. 

Martingale Foundation Postgraduate Scholarships

The Martingale Foundation awards fully funded Scholarships for postgraduate degrees in the mathematical sciences at research universities in the UK. 

Tuition fees and research expenses are fully covered, and Scholars receive a tax free living wage stipend. Martingale Scholars also receive access to leadership and career develop through a multi-year programme of training and support. Visit the Martingale website for more information.  

Applications for the 2025 academic year are open until 27 October 2024.  

Why do a PhD?

Research interests:  group theory, representation theory and algebraic aspects of geometry.

Who's who in Algebra

Find out more about the group

Combinatorics

Research interests: extremal combinatorics, graph theory, and combinatorial number theory.

Who's who in Combinatorics

Functional Analysis

Research interests: operator theory, including unbounded operators, and abstract differential equations.

Who's who in functional analysis

Research interests: algebraic geometry,  geometric representation theory , and differential geometry.

Who's who in Geometry

History of Mathematics

Research interests: history of algebra (19th and 20th century), history of modern algebra, and Soviet mathematics. 

Research interests: analytic topology,  geometric stability theory, and the model theory of p-adic fields and diophantine geometry.

Who's who in Logic

Machine Learning and Data Science

Machine Learning and Data science are being developed using wide ranging mathematical techniques. Our particular research expertise include: applied and computational harmonic analysis, networks, optimisation, random matrix theory, rough paths, topological data analysis, and the application of these methods.

Who's who in machine learning and data science

Mathematical & Computational Finance

Research interests: behavioural finance, financial big data, high dimensional numerical methods, stochastic analysis.

Who's who in Mathematical and Computational Finance

Mathematical Biology

Research interests:  cancer modelling, collective behaviour, gene regulatory networks, multiscale modelling, pattern formation, and sperm dynamics.

Who's who in Mathematical Biology

Mathematical Physics

Research interests: gauge and gravity theories (quantum field theories), string theory, twistor theory, Calabi-Yau manifolds, quantum computation and cryptography.

Who's who in Mathematical Physics

Number Theory

Research interests: analytic number theory, arithmetic geometry, prime number distribution, and Diophantine geometry.

Who's who in Number Theory

Numerical Analysis

Research interests: complexity in optimisation, symmetric cone programming, numerical solutions of PDEs. 

Who's who in Numerical Analysis

Oxford Centre for Industrial and Applied Mathematics

Research interests: energy, industry, geoscience, networks, finance, methodologies.

Who's who in OCIAM

Oxford Centre for Nonlinear Partial Differential Equations

Research interests: geometric analysis, inverse problems, nonlinear hyperbolic systems, specific PDE systems.

Who's who in OxPDE

Stochastic Analysis

Research interests:  rough path theory, Schramm-Loewner evolution, mathematical population genetics, financial mathematics, self-interacting random processes.

Research interests: geometric group theory, 3-manifold topology and knot theory, K-theory, algebraic topology.

Who's who in Topology

Ph.D. Program

Degree requirements.

In outline, to earn the PhD in either Mathematics or Applied Mathematics, the candidate must meet the following requirements.

• Take at least 4 courses, 2 or more of which are graduate courses offered by the Department of Mathematics
• Pass the six-hour written Preliminary Examination covering calculus, real analysis, complex analysis, linear algebra, and abstract algebra; students must pass the prelim before the start of their second year in the program (within three semesters of starting the program)
• Pass a three-hour, oral Qualifying Examination emphasizing, but not exclusively restricted to, the area of specialization. The Qualifying Examination must be attempted within two years of entering the program
• Complete a seminar, giving a talk of at least one-hour duration
• Write a dissertation embodying the results of original research and acceptable to a properly constituted dissertation committee
• Meet the University residence requirement of two years or four semesters

Detailed Regulations

The detailed regulations of the Ph.D. program are the following:

Course Requirements

During the first year of the Ph.D. program, the student must enroll in at least 4 courses. At least 2 of these must be graduate courses offered by the Department of Mathematics. Exceptions can be granted by the Vice-Chair for Graduate Studies.

Preliminary Examination

The Preliminary Examination consists of 6 hours (total) of written work given over a two-day period (3 hours/day). Exam questions are given in calculus, real analysis, complex analysis, linear algebra, and abstract algebra. The Preliminary Examination is offered twice a year during the first week of the fall and spring semesters.

Qualifying Examination

To arrange the Qualifying Examination, a student must first settle on an area of concentration, and a prospective Dissertation Advisor (Dissertation Chair), someone who agrees to supervise the dissertation if the examination is passed. With the aid of the prospective advisor, the student forms an examination committee of 4 members.  All committee members can be faculty in the Mathematics Department and the chair must be in the Mathematics Department. The QE chair and Dissertation Chair cannot be the same person; therefore, t he Math member least likely to serve as the dissertation advisor should be selected as chair of the qualifying exam committee . The syllabus of the examination is to be worked out jointly by the committee and the student, but before final approval, it is to be circulated to all faculty members of the appropriate research sections. The Qualifying Examination must cover material falling in at least 3 subject areas and these must be listed on the application to take the examination. Moreover, the material covered must fall within more than one section of the department. Sample syllabi can be reviewed online or in 910 Evans Hall. The student must attempt the Qualifying Examination within twenty-five months of entering the PhD program. If a student does not pass on the first attempt, then, on the recommendation of the student's examining committee, and subject to the approval of the Graduate Division, the student may repeat the examination once. The examining committee must be the same, and the re-examination must be held within thirty months of the student's entrance into the PhD program. For a student to pass the Qualifying Examination, at least one identified member of the subject area group must be willing to accept the candidate as a dissertation student.

Doctor of Philosophy in Mathematics

• Fall January 10

International students may need to surpass the Graduate School’s minimum English language proficiency exam scores for this program. If the graduate program has unique score requirements, they will be detailed below. Otherwise, please refer to the Graduate School’s minimum score guidelines.

• 540 TOEFL Minimum score for admission
• 75 TOEFLI Minimum score for admission
• 6.5 IELTS Minimum score for admission
• 105 Duolingo Minimum score for admission

Degree Description:

PhD in Mathematics This degree is awarded in recognition of distinctive scholarship and original contributions to knowledge in Mathematics. The PhD program is especially designed to prepare the student for teaching at the graduate level and doing mathematical research in academic, industrial and business settings. Students studying various fields within the realm of pure mathematics would be included in this PhD program.

PhD in Mathematics (Applied Mathematics Option) The specialization of modern academic disciplines provides both a challenge to those who wish to do research at the interface of mathematics and its areas of application and many opportunities to make valuable contributions. The Applied Mathematics Option allows students from a range of backgrounds to pursue a traditional applied mathematics program, while retaining the option to thoroughly learn an area of application. Entering students may not necessarily have a bachelor’s degree in Mathematics. However, they will be required to demonstrate a grasp of the core areas of advanced calculus and linear algebra at the level of a bachelor’s degree in Mathematics. They will then be given great latitude to take specialized courses in Mathematics and their area of application.

PhD in Mathematics with Education Emphasis The degree of PhD in Mathematics with Education Emphasis is awarded in recognition of scholarship and original contributions to the teaching and learning of mathematics. The main difference from the other PhD choices is in the research focus. The requirements for this PhD include competence in core mathematics, as well as study in the research methodologies applicable to research in mathematics education.

Admission Requirements:

Students should have taken upper-level analysis and linear algebra courses before applying, and have the equivalent of an undergraduate degree in mathematics, statistics, or a related field. Additional upper-level courses like abstract algebra, functional or complex analysis, optimization, applied mathematics, topology, etc. will be taken into account.

For international students, either a TOEFL, IETLS or Duolingo score is required. The minimum score for admission is listed above. The minimum score required for assistantship consideration is TOEFL score of 100, IELTS of 7, or Duolingo of 130. Exceptions can be found here:  https://gradschool.wsu.edu/international-requirements/

Students applying to the PhD Mathematics program will be automatically considered for an assistantship position.

The application will require:

• Unofficial transcripts from all previous institutions.
• GPA above a 3.0 on a 4.0 scale.
• Email contact information for 3 references.
• Personal statement. The personal statement is your chance to highlight relevant experiences and discuss your future research interests.
• GRE scores are not required but encouraged.

Career Opportunities:

Academia (tenure-track positions at universities and colleges worldwide), Bio-statistics (health and pharmaceutical companies), Risk analysis (financial and insurance companies, investment management), Research (state or federal government, software development), Mathematics education (publishing, consulting and developing of educational software).

Career Placements:

Graduates from the Mathematics PhD program begin careers in both academia and government or industry. A few examples of career placements in academia from the past few years include:

• Postdoc, USDA Northwest Climate Hub and WSU Center for Sustaining Agriculture
• Assistant Professor, Central Washington University
• Assistant Professor, Marian University (WI)

A few examples of career placements in government and industry from the past few years include:

• Regulatory Data Scientist, Office of Regulatory Innovation and Assistance, Washington State Government
• Quantitative Model Validation Analyst, First Tech Federal Credit Union (OR)
• Data Scientist, Emsi Labor Market Analytics (ID)

Contact Information:

• Graduate Coordinator [email protected]

Department of Philosophy

• Graduate Program >

Philosophy and Mathematics

The Departments of Philosophy and of Mathematics together offer a joint Ph.D. degree in Philosophy and Mathematics. Students in this program submit a single dissertation prepared under the supervision of members of both departments. Students enrolled in the Philosophy PhD program at Notre Dame can apply to the joint-degree program (typically during their third year of study). The joint-degree program does not accept applications from students outside of Notre Dame or admit students to the University. 

The joint Ph.D. program continues a  long tradition  of the advanced study of formal logic at the University of Notre Dame.

Tim Bays  (Philosophy) Jc Beall (Philosophy) Patricia Blanchette  (Philosophy) Peter Cholak  (Mathematics) Natasha Dobrinen  (Mathematics) Curtis Franks  (Philosophy) Joel David Hamkins  (Philosophy and Mathematics) Julia Knight  (Mathematics) Anand Pillay  (Mathematics and Philosophy) Nicholas Ramsey  (Mathematics) Sergei Starchenko  (Mathematics)

Philosophy students interested in the joint program begin preparing to apply immediately upon beginning their studies at Notre Dame. In addition to philosophy coursework, they take at least the two semester logic sequence, and sometimes other courses, in the Mathematics Department in their first year. They also take part in the Mathematics Department's research seminars. By the second year, they have taken several courses in the Mathematics Department and become integrated into the research group of a Mathematics faculty member whom they intend to be their mathematics supervisor.

Most students will then choose to apply to the Master of Science in Interdisciplinary Mathematics (MSIM). This application describes a list of courses and a research project that the student and his or her mathematics supervisor propose. While working towards the MSIM, the student decides either to complete the degree as initially described or to expand the project into a joint program dissertation. In the latter case, the student applies to the joint-degree program.

Requirements

Students in the joint program are required to take only 27 credits in Philosophy. Joint Program students need to take only two of the three seminars in the history of philosophy that are required for the regular Philosophy PhD. program, and the 3 credit seminar "Intermediate Logic" is not required. Students are required to take the two course logic sequence, the two course algebra sequence, and 9 additional credits in the Mathematics Department.

Joint program students take the Mathematics Department's oral candidacy exam instead of the Philosophy Department's oral exam. This consists of both the basic and advanced exams in logic and the basic exam in one other area.

Matteo Bianchetti, MSIM: "Infinite Time Computation: Strong and Weak Infinite Time Turing Machines" (C. Franks, J. Knight); Ph.D. in Philosophy: "Geometric representations in mathematical problem-solving. Intuition and creativity" (C. Franks), Current Placement: non-academic

Paul Trần-Hoàng , MSIM: "Model-Theoretic Galois Cohomology" (A. Pillay, C. Franks); Ph.D. in Philosophy: "Model-Theoretic Approaches to Theoretical Equivalence and Reduction" (T. Bays, C. Franks). Current placement: Visiting Assistant Professor, Vassar College

Graham Leach-Krouse , Joint Ph.D.: "Conceptions of Absolute Provability" (T. Bays, M. Detlefsen, P. Larson). Current placement: Associate Professor, Department of Philosophy, Kansas State University.

Chris Porter , Joint Ph.D.: "Mathematical and Philosophical Perspectives on Algorithmic Randomness" (P. Cholak, M. Detlefsen, C. Franks). Current placement: Assistant Professor, Department of Mathematics and Computer Science, Drake University.

Sean Walsh , Joint Ph.D.:   "Arithmetical Knowledge and Arithmetical Definability: Four Studies" (P. Cholak, M. Detlefsen). Current placement: Associate Professor, Logic and Philosophy of Science, UCLA

Andy Arana , Joint PhD: "Arithmetical Investigations: A Study of Models of Arithmetic and Purity of Methods"  (M. Detlefsen, J. Knight). Current placement: Professor of Philosophy, University of Lorraine and Archives Henri-Poincaré.

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Home / Programs / Doctor of Philosophy (Mathematics)

Doctor of Philosophy (Mathematics)

The Doctor of Philosophy in Mathematics Program is intended to encourage the development of mathematics in the Philippines through the production of highly trained mathematicians whose research works contribute to the development of new knowledge. Corollary to this main objective is the upgrading of mathematics teaching in colleges and universities.

The PhD Math program is a research-oriented program that

• prepares students for careers in the academe, industry and government that depend on advanced mathematics;
• aims to produce graduates with the knowledge, skills, perspective and understanding to be capable of self-directed and independent scientific work; and
• is planned to develop in the student fundamental understanding of certain basic fields of mathematics, a deep understanding of a major field of interest, and the ability to undertake advanced mathematical research, as evidenced by the completion and defense of a doctoral dissertation, the major requirement of the PhD program.

Applications for admission to the program are processed by the College of Science (more information here ). Students can apply for admission during the 1st semester or 2nd semester.

Aside from the general requirements for admission set forth by the College of Science, applicants of the PhD Math program must have either a BS or MS degree in mathematics, or its equivalent from a recognized institution of higher learning, and a high degree of intellectual capacity and aptitude for advanced study and research in mathematics.

For more inquiries, please send an email to [email protected] .

Students are required to accomplish the following:

• completion of a prescribed program of study consisting of at least forty-five (45) units of formal graduate courses in the case of students admitted into the program with only a BS degree in the discipline or a master’s degree in an unrelated discipline, or twenty-four (24) units of formal graduate courses in the case of students admitted into the PhD program with an MS degree in the discipline;
• maintenance of a CWAG of 1.75 or better at the end of each academic year until completion of the program of study;
• passing of the qualifying examination based on the core courses;
• passing of the candidacy examination after completion of all course work in the student’s program of study;
• completion of one (1) unit of a graduate seminar during the student’s course work and participation in a Graduate Research Colloquium of the College at least once every two years by giving a seminar on the progress of the dissertation work;
• completion of a doctoral dissertation based on an independent and original research;
• successful defense of the doctoral dissertation in a public doctoral examination;
• submission of a publication or an acceptance letter from a reputable, refereed scientific journal as defined by the Institute and approved by the Dissertation Committee; and
• submission of bound paper copies and a digital copy of the approved doctoral dissertation based on the approved College of Science format.

The maximum residence of student under the PhD Math program is six (6) years for those with MS degrees and eight (8) years for those with BS degrees only.

Core Courses

Other Required Courses

* courses in mathematics and allied fields, with at least twelve (12) of the fifteen (15) units chosen from advanced mathematics courses.

Registration Process

Refer to the Graduate Student Guide from the UPD College of Science website.

Program Advisers

For students admitted to the PhD Math program, please contact your program advisers listed below for your registration concerns.

• Aurelio de los Reyes V ( [email protected] )
• Romar dela Cruz ( [email protected] )
• University of the Philippines System
• University of the Philippines Diliman
• College of Science

Affiliations

• UP Diliman Mathematics Foundation, Inc.
• Mathematical Society of the Philippines
• Southeast Asian Mathematical Society
• International Mathematical Union

Department of Philosophy

Dietrich college of humanities and social sciences.

Philosophy and History of Mathematics

Logic and mathematics are tools for almost all members of the Department, but they are also objects of investigation. As tools they provide means of rigorously capturing aspects of experience; as objects of investigation they are examined as to their internal coherence, their philosophical justification, and their adequacy for particular purposes. Since ancient times, there has been an intimate connection between philosophical and mathematical thought, a relationship that can be seen in the philosophical reflections of Plato, Descartes, Leibniz, and Kant. Subtle interactions between philosophy and mathematics can also be seen in the development of mathematics in the 19th century, i.e., in the revolutionary conceptual advances made by Dirichlet, Riemann, Dedekind and others, as well as in the similarly dramatic changes in logic, brought about in large part by Boole, Frege, Peano, Peirce, and Schröder. Complemented by the continuing evolution of the sciences, for example in the work of Hertz, Mach, and Einstein, these developments form the background for the emergence of early analytic philosophy and modern mathematical logic. This historical background reshapes in significant ways the contemporary discussion in the philosophy of mathematics. The subject is deeply influenced by results of meta-mathematical investigations, but most importantly, for the work in this department, also by mathematical practice. The incompleteness theorems are considered, rightly, to be gems of work in mathematical logic of the last century. They are also viewed as having an enormous philosophical significance; that seems to be right, but only relative to a precise concept of "formal system". The latter is defined using the notion of computability. On the one hand, one can understand then the seemingly conflicting views of Gödel and Turing on mathematical knowledge and the capacities of the human mind. On the other hand, one can go back and see what was supposed to be captured by "formal theories" and uncover the dramatic transformation of mathematics in the 19th century. The "structuralist" view of mathematics one finds in Dedekind's "foundational" work really emerged out of his concrete work in algebraic number theory. His broad perspective on mathematics impacted that view, which was of course also influenced by Gauss, Dirichlet, and Riemann. Sieg has taken that view as a starting-point and joined it with a quasi-constructive perspective of accessible domains to arrive at an articulation of a "reductive structuralism". This position resolves a number of traditional epistemological and ontological problems. With Awodey and a number of other colleagues, Sieg is editing and translating the marvelous and most significant philosophical essays of Hilbert’s collaborator in the proof theoretic enterprise, Paul Bernays. Sieg's own perspective is developed in his book Hilbert’s Programs and Beyond that was published by Oxford University Press. The philosophy of mathematics has traditionally been concerned with questions of justification and correctness. But as of late a number of researchers have aimed to characterize more general methodological goals and values that influence the decisions that mathematicians make in their everyday practice, for example, when posing questions, formulating definitions, writing proofs, and structuring theories. Using insights from proof theory, formal verification, and the history of mathematics, Avigad has worked to develop more robust accounts of mathematical concepts, methods, and understanding. Awodey is exploring connections between category theory and structuralism, especially in light of the new Univalent Foundations program. Univalent Foundations includes a new foundational axiom, the Univalence Axiom, according to which isomorphic structures can be identified. This new principle has obvious philosophical consequences, and will doubtless challenge philosophers of mathematics to adjust their views. Philosophers interested in structuralism have begun to recognize the importance of these recent developments, with e.g. several leading philosophers of physics including Ladymann (Bristol) and Halvorson (Princeton) actively engaged in philosophical research on Univalent Foundations. In addition to the comprehensive book Homotopy Type Theory , Awodey has written a survey article on the topic of Univalence and structuralism to be published in the journal Philosophia Mathematica which will serve to introduce the subject to a wider audience of philosophers of mathematics. The history of mathematics can be a useful aid in developing a robust and informative philosophy of mathematics. Sieg has explored the logicist and structuralist perspectives that are found in the seminal foundational writings of Dedekind. These perspectives emerged out of the concrete mathematical work Dedekind did in algebraic number theory, and they stand in sharp contrast with those of his contemporary Leopold Kronecker. In work with Rebecca Morris, respectively with Dirk Schlimm, Sieg has analyzed as central features in Dedekind’s work the introduction of abstract concepts (structures) and the use of structure preserving mappings between different structures (of the same kind). The concept of abstraction involved here is not the classical one, found for example in Kant’s "Logik", but rather one that is exposed in the writings on logic of the contemporaneous and very influential Göttingen philosopher Hermann Lotze. Avigad has explored methodological aspects of Dedekind's work, especially the development of his theory of ideal divisors, in an effort to understand the way that such methodological considerations interact with philosophical views. With Rebecca Morris, he has studied the history of Dirichlet's seminal theorem on primes in an arithmetic progression, which illuminates the methodological forces that shaped the development of the modern function concept, as well as some of the issues that Frege had to address with his treatment of functions. Avigad has also considered the philosophical views of Kurt Gödel vis-a-vis the meta-mathematical tradition, and has tried to characterize and explain an uneasy tension between Gödel's views and Carnap's. This concrete work on the dramatic shift in the evolution of mathematics during the 19th century, where some speak of a "transformation of the subject" others of a "revolution," has deep impact on the philosophy of mathematics - a subject that as recently as twenty years ago was pre-occupied with the "Grundlagenstreit" between Brouwer and Hilbert in the 1920s. Both Hilbert and Brouwer are deeply connected to the broad issues alluded to above. Sieg, with colleagues W. Ewald, M. Hallett, and U. Majer, has been working on editing unpublished lecture notes of Hilbert's from the 1890s to the 1930s. They open up a completely novel and fresh perspective on the evolution of Hilbert's foundational thinking and the emergence of proof theory, but that means also on the origins of analytic philosophy.

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