math ku phd thesis

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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2024-25 Entire Catalog

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• Enroll & Pay
• Prospective Students
• Current Students
• Degree Programs

Graduate Resources

These are resources that are useful for the math department's graduate students.

The qualifying exams archive has been moved to a Sharepoint folder, Math Department Resources . The exams are given in August and January. The quals sign-up sheet will be available around four weeks before the exams.

Also in our Sharepoint resource folder, we have a TeX thesis template that was developed by one of our former graduate students (Aaron Barrett, MA '22).  This may be a more useful thesis/dissertation template than the formatting guide posted at KU Libraries.  We have also posted the College's electronic thesis formatting self-checklist.

GTA evaluation forms are in Department Resources.

Resources for Math Graduate Students

• Math Graduate Scholarships
• Graduate Student Funding
• Summer Activities Report Form
• Electronic Thesis and Dissertation Process
• KU Mathematics Dissertation Collection
• KU Writing Center

Centre for Symmetry and Deformation

• Symmetry and Deformation
• Courses/theses

Courses and Theses

The center offers a range of courses of courses at the MSc and PhD level, which can be found below. We also run a number of reading seminars for PhD students and postdocs which are which you can find by clicking  this link.

Courses at the centre

The following is a list of all courses held by employees (within the centre's mathematical areas) at the centre from September 2018 to June 2019. The titles link to course descriptions at the UCPH course catalogue .

Block 1 Categories and Topology (CatTop) Algebraic Geometry 2 (AlgGeo2) Algebraic Topology (AlgTop) Analysis on Manifolds (AnMan)

Block 2 Experimental Mathematics (XM) Functional Analysis (FunkAn) Homological Algebra (HomAlg) Topics in Algebraic Topology (TopTop) Topics in Operator Algebras (TopOp)

Block 3 Algebraic Topology 1.5: Cohomology (AlgTop1.5) Introduction to Operator Algebras (IntoOpAlg)

Block 4 Algebraic Topology II (AlgTopII) Introduction to K-Theory (K-Theory)

PhD, MSc, and BSc theses:

For inspiration we list every bachelor project (see also the 2018 bachelor catalogue ), master project and master thesis written at the center.

• Past topology theses
• Past NCG theses

On these pages we also list PhD theses. We have many PhD students and highly prioritize the PhD program. Look here for  PhD stipends at the Centre for Symmetry and Deformation  and  general information about the PhD program in mathematics .

Further links

• Reading seminars at the center
• UCPH-Science course catalog
• UCPH-Science academic calendar
• The schedule group structure (A,B,C)

Core Faculty

Associated Faculty

Niels Grønbæk Lars Halvard Halle Henrik Holm Ian Kiming Fabien Pazuki Morten Risager Asger Törnquist

Kasper K.S. Andersen Alexander Berglund Bob Oliver

In memoriam: Uffe Haagerup

Postdocs/Junior Faculty

Past postdocs

• Enroll & Pay
• Prospective Students
• Faculty & Staff

Theses & Dissertations

Graduate students preparing to complete a thesis or dissertation should review the following dates and guidelines to ensure their work is properly submitted and meets all requirements for graduation.

Formatting & Submission

Formatting requirements.

All theses and dissertations completed at KU must meet KU's requirements for formatting specific components, including Title and Acceptance pages.

Submission Deadlines

Find out when to submit your thesis or dissertation and other required items.

How to Submit Your Thesis or Dissertation Electronically

The University of Kansas requires that all students whose degree programs require defense of a thesis or dissertation must publish their research in order to fulfill degree requirements.

Thesis and Dissertation Binding Instructions

Departments or advisors may require submission of a printed copy of every thesis and dissertation.

Additional Resources

Thesis and dissertation workshop.

This workshop covers what you need to know as soon as you get ready to submit your thesis or dissertation. It is offered once in the Fall and once in the Spring.

Previously Published Theses and Dissertations

We encourage you to explore theses and dissertations previously published by KU students from your department and related fields.

Embargoing Your Electronic Thesis or Dissertation

Embargo is the term used to describe a delayed release or delayed availability of scholarly work. Under certain circumstances, scholars may need to embargo their work.

Still Have Questions?

Remember to check with your department or school for questions about bibliographic style, including styles for citations, illustrations, tables, charts, etc.

• Enroll & Pay
• Prospective Students
• Current Students
• Degree Programs

Master of Arts in Mathematics

The Mathematics Department offers the degree of Master of Arts (MA) in Mathematics.  The MA entails study of advanced mathematics, typically culminating in a substantial thesis or final project demonstrating expertise in a particular area of mathematics. It is intended for students who plan to pursue a PhD subsequently, as well as those who intend to work outside academia. 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. The program offers three options for the MA degree, as detailed in the Degree Requirements section of the Graduate School Catalog. College-wide requirements for graduate students may be found in the  Graduate School Catalog .

Job Placement

After graduation, our MA students usually obtain jobs in government or the private sector (e.g., financial, insurance, telecommunication), as well as teaching jobs at junior colleges.

• Course Offerings

What are the prerequisites for admission?

The prerequisites for admission to our graduate program 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; and

Note:  Please use the following  GPA calculator  to determine your GPA if your degree-granting institution uses a grade/point system different from the one used in the U.S. To assist the Admissions Committee, you can list your converted GPA on your CV and the online application. Graduate Admissions confirms GPA estimates using this same GPA calculator. 

coursework in abstract algebra, linear algebra, and advanced calculus or introduction to analysis (comparable to KU's MATH 500, 558, and 590.)

It is beneficial to have some preparation in probability/statistics (comparable to MATH 627/628) and/or numerical analysis (comparable to MATH 581). Although not required, we also recommend that applicants take introductory courses in complex analysis (comparable to MATH 646), differential equations (comparable to MATH 647), and/or topology. You can view a description of any of these classes by searching for them here:  https://catalog.ku.edu/liberal-arts-sciences/math/#courseinventory .

Please also note that the Mathematics Department currently does not require the general or subject Graduate Record Examination (GRE). 

Does the department provide financial assistance?

Only our PhD students are guaranteed funding. If funding is an issue, we recommend looking for funded Master's programs or applying to our PhD program, which offers 10-15 fully-funded spots each academic year.

How much is tuition?

If helpful, you can estimate tuition costs using  KU’s tuition calculator . Be sure to select “Nonresident”, if you are not a Kansas resident. You will then select “Graduate Student” in “Liberal Arts & Sciences.” MA students pursuing Option 3 need 36 semester hours to graduate.

What materials are needed to apply?

The following materials must be submitted online:

Applicant's résumé/curriculum vitae, (please do  NOT  include your GRE score)

Please use the following  GPA calculator  to determine your GPA if your degree-granting institution uses a grade/point system different from the one used in the U.S. To assist the Admissions Committee, you can list your converted GPA on your CV and the online application. Graduate Admissions confirms GPA estimates using this same GPA calculator. 

Statement of Purpose

In your personal statement you should indicate your mathematical background and interests and how these make you a good fit for the program. It is helpful for you to mention a professor or professors in our department whose research you find interesting. 

Copies of official transcripts from each university or college you have attended.

Please do not upload grade reports from a student-accessed online portal. Transcripts should be issued from your university's Registrar's office, these usually have a seal/signature on them to denotes they are official.

Official transcripts (sent directly to KU from the Registrar) are not required until after you have been admitted into the Ph.D. program.

Three letters of recommendation

Contact information for your letter writers will be submitted in your application. Letter writers will then receive a request to submit a letter on your behalf electronically.

List of Textbooks

A list of textbooks used in the applicant's mathematics courses beyond calculus (please include authors)

When are applications due?

To begin in the spring semester, submit an application by October 1. For fall or summer, submit an application by May 1.

When will I be notified of an admission decision?

Application review begins immediately following the deadline. Please aim to have all materials submitted by the deadline (this includes letters of recommendation).

What are your English proficiency requirements?

You can read our English proficiency requirements for admission at  this link .

• Department of Mathematical Sciences
• Study programmes
• PhD program
• PhD calls - how to apply

The usual entry level for our (3 years) PhD program is a Master's Degree. Besides having the appropriate entry level degree, to become a PhD student at our department you need first of all: 

• a stipend to cover your living expenses (your salary) and the fees ( taxameter ) to the university (currently DKK 80,000 per year),  to be admitted to the PhD school at the Faculty of Sciences,
• The department offers a number of PhD stipends every year. Many of them are tied to external grants within specific research areas, but we aim to have a few stipends that are not tied to any external funding.

We have calls for applications to all our available stipends twice a year, with deadline November 15th and April 1st, respectively. The calls will be announced here  in September and February.

If you have obtained financial support from other sources to do a PhD in Copenhagen, then you only need to worry about item 2 above, but we recommend that you also in this case submit an application to one of our PhD calls (indicating that you already have financial support).

The applications for the PhD calls will be evaluated by the research committee (Forskningsudvalg) of the department.  

If you have been offered a stipend through one of our PhD calls, being admitted to the PhD school is  usually  just a matter of form (see After accepting a PhD stipend ). 

Applying for a PhD

• After accepting a PhD stipend  
• Mini-guide on how to fill the online application form