The pure mathematics major in the Department of Mathematics graduate program offers studies leading to doctoral and master's degrees in mathematics. Research faculty members represent the fields of number theory, arithmetic geometry, algebraic geometry, geometric topology, algebraic topology, harmonic analysis, and complex analysis.
The pure mathematics major is designed to provide both a solid preparation in core areas of pure mathematics (algebra, real and complex analysis, and geometry/topology), and specific expertise in one or more areas of active research.
All students enrolled in the pure mathematics graduate major who have taken 36 graduate credit hours of graduate courses, and have passed the three core course sequences, are eligible for a master's degree. This includes students who are intending to continue on as a doctoral candidate as well as students who opt to leave the program with a master's.
A student who passes all requirements for Ph.D. candidacy is automatically eligible for a master's degree (M.S.).
Students who wish to leave after passing the core course sequences are eligible for a masters' degree provided that they satisfy the following requirements:
Checklist for master's degree:
- Students must take 36 credit hours of graduate courses at Florida State University, of which 30 hours of letter graded courses.
- Students must maintain a 3.0 cumulative grade point average.
- Courses must be chosen in conference with the director of pure math, or with an advisor chosen by the student with the agreement of the director and department chair. Normally the required courses consist of the core course sequences.
Ph.D. students must qualify in all two out of three core subjects:
- Algebra,
- Topology, and
- Analysis.
In order to gain or retain TA status, students are expected to follow the guidelines about timely progress. Qualification consists of either getting A or A- in Parts I and II of a course sequence, or passing a corresponding qualifier exam on the same material.
The qualifier exams are offered twice each academic year at the beginning of the fall and spring semesters (in August and January). Students who arrive with advanced background in the form of previous graduate coursework may take a qualifier exam to exempt all or part of a course sequence.
The qualifier exams in pure mathematics are in the three core areas of analysis, algebra and topology. Each exam is based on material covered in the first two semesters of each sequence. In particular, the Analysis qualifier exam corresponds to a year of coursework, one semester in Complex Analysis and one (the first) semester of Real Analysis.
Topics currently covered by the qualifying examinations in pure mathematics:
The topics are periodically revised; students are advised to consult with the area director for up-to-date information.
After qualification, the first step toward a Ph.D. is to choose a major professor and pass a candidacy examination. This is usually done by the end of the students' third year of graduate study. To gain or retain TA status, a graduate student is expected to finish their candidacy by the beginning of their seventh semester of graduate study. Once a candidate, the student continues work with the major professor on an original research problem. Typically by the end of their fifth or sixth year of study, they write and defend a Ph.D. thesis, thus earning their doctoral degree.
- Major professor and committee: the major professor and committee must be declared and officially appointed by the department chair at least a month before the candidacy exam takes place. The major professor is the main advisor to the student, and will guide the student toward a Ph.D. thesis and degree. The rest of the committee should consist of at least three mathematics faculty members and one faculty member from an outside department. All members of the committee must have doctoral directive status. This outside member, also called the university representative, need not have any relationship with or specialized knowledge concerning the dissertation topic. The primary function of the outside committee member is to oversee the proceedings and verify that these took place in accordance with university rules.
- Courses: in order to achieve candidacy, the student must have finished all course requirements and passed three qualifier exams. Students should be enrolled in at least one graded course (a core course or a topics course) per semester until candidacy is achieved. During the semester of candidacy, students register for 2-4 credit hours of a DIS course with their major professor (MAT6908).
- Format: the student, major professor, and committee agree on a format for the candidacy exam. Typically, this will include one or two seminar-level presentations. Sometimes the student will also be asked to write a presentation demonstrating his mastery of the material agreed to previously. Assigned written material must be submitted in a timely manner, at least two weeks before the candidacy exam.
- Final approval: the committee officially approves the student's work, and puts the approval in writing usually after a final oral candidacy examination.
- Official candidacy: immediately upon passing the candidacy exam, the student must register for MAT8964 for 0 credit hours. Only then is a student officially considered a Ph.D. candidate. Students maintaining timely progress must achieve candidacy by the end of their third year in residence.
According to university rules, a student may only keep his/her status as Ph.D. candidates for a length of five years after candidacy, after which a renewal process is necessary to reinstate a student's candidacy
Course Requirements
All pure math graduate students must pass (or exempt, see below) the following three core course sequences:
- Algebra: Groups, rings, and vector spaces (GRV) I & II (MAS5307 & MAS5308), Abstract Algebra I (MAS5311);
- Topology: Topology I & II (MTG5326 & MTG5327), Algebraic topology (Top IIIa) (MTG5346) or Differential topology (Top IIIb) (MTG5932);
- Analysis: Measure and integration I & II (MAA5616 & MAA5617); Theory of functions of a complex variable I (MAA5406).
The course layout for the first two years consists of three courses in each core sequence in algebra, topology and analysis. The analysis course layout consists of two semesters of Real Analysis (Measure & Integration) followed by one semester of Complex Analysis. In the second year the student adds intermediate-level courses.
Elective — or topics‐courses — are more advanced courses that are taken by students who have progressed past the first four semesters.
The course layout is:
qual = Ph.D. qualifier course; inter = intermediate level; elec = elective course option
Four-semester grid
Course layout for the first two years.
First Fall
- Measure and Integration I (qual)
- GRV I (qual)
- Topology I (qual)
First Spring
- Measure and Integration II (qual)
- GRV II (qual)
- Topology II (qual)
Second Fall
- Complex Analysis I
- GRV III
- Topology IIIa (Algebraic Topology) or Topology IIIb (Differential Topology) — offered in alternate years
Second Spring
- Functional Analysis or Fourier Analysis (inter)
- Algebraic Geometry or Number Theory (inter)
- Differential Geometry or other Geometry/Topology course (inter)
Current and Future Course Offerings
In the table below you can find the planned course offerings for the current and future semesters. The courses beyond Spring 2020 might be subject to slight variations.
Electives (or advanced topics) will usually be taken by students past their second year, who have more flexible schedules. The actual elective courses offerings are decided on a year-by-year basis, and they reflect the individual faculty member's research interests.
The highlighted portions depict the progression from core courses (first three semesters) to the more advanced ones.
Core courses
Algebra core sequence
The algebra core sequence consists of three courses: Groups, Rings and Vector Spaces I & II, and Abstract Algebra I. (The latter is often nicknamed GRV III.)
This sequence of courses covers basic material on classical algebraic structures such as categories, groups, rings, modules, including standard results as as the Sylow theorems, the classification of modules over a PID, factorization in integral domains, linear algebra over fields and more general rings, Galois theory, and basic notions in homological algebra, as well as multilinear algebra (symmetric and alternating algebras), Kähler differentials, the de Rham complex. The three semesters fit into a seamless continuum.
GRV I & II make up the material for the qualifier exam in Algebra. The specific topics are listed in detail at the Algebra qualifier page.
Analysis core sequence
The Analysis core sequence consists of two semesters of Real Analysis (under the name of Measure & Integration I and II) followed by one semester of Complex Analysis.
Real Analysis covers various aspects of modern measure and integration theory; in particular, passing to the limit under the integral, double vs iterated integration, Riesz representation theorem, foundations of operator theory, and locally convex topological spaces. Complex Analysis covers the fundamentals of Analysis in the complex domain in one variable, including complex differentiability, power series, complex integration, and classification of singularities and residues; it also covers more advanced topics, such as Möbius transformations, the Riemann and the Open mapping theorems.
Measure & Integration I and Measure & Integration II make up the material for the qualifier exam in Analysis. Exam topics are listed at the Real Analysis qualifier page.
Topology core sequence
The Topology core sequence consists of three courses: Topology I & II, and one of Topology IIIa (Algebraic Topology—MTG5346) or Topology IIIb (Differential Topology—MTG5932) (they alternate, see table above).
Topology I covers point set topology, in particular basic notions such as bases, limits, continuous functions, fundamental separability and countability properties, connectedness and compactness, and important results including the Tietze extension theorem, Urysohn’s lemma, and various metrization theorems. Topology II focuses on homotopy, homotopy equivalence, CW structures, and a detailed discussion of the fundamental group with an emphasis on its functorial properties. Fundamental results such as 2-dimensional Brouwer fixed point theorem, Borsuk-Ulam theorem, and Van Kampen’s theorem are discussed as well. Covering spaces are discussed in detail too, with an emphasis on relationships between fundamental group and covering spaces. Topology IIIa covers Algebraic Topology, in particular simplicial homology, singular homology, and cohomology. Topology IIIb covers Differential Topology with an emphasis on differentiable structures, the inverse function theorem, differentiable maps, tangent vectors and tangent bundle, differential forms and cotangent bundle, exterior differentiation, and flows and Lie derivative.
Topology I & II make up the material for the qualifier exam in Topology. The specific topics are listed in detail at the Topology qualifier page.
Intermediate topics courses
Intermediate topics courses are designed to complement the core courses with more advanced, but not super-specialized material. Each of these courses is (roughly) offered every other year.
Algebra intermediate topics
- Number Theory
- Commutative Algebra and Algebraic Geometry
Analysis intermediate topics
- Partial Differential Equations
- Functional Analysis
Topology intermediate topics
- Geometry of fiber bundles
- Symplectic Geometry
Advanced topics courses
Advanced topics courses are meant to provide an in-depth presentation of more specialistic research areas or to provide an introduction to new mathematical research areas.
- Advanced Algebraic Geometry
- Ergodic Theory
- Computer Algebra
- Algebraic Foundations for Topological Data Analysis
Entering graduate students are expected to have taken at least two semesters of undergraduate courses in each of the subjects abstract algebra and real analysis. Semester long courses in general topology and complex analysis are also strongly suggested.
Some students arrive with extra mathematical background, for example, masters level courses, a masters degree in mathematics, or expertise in a specialized area as exemplified in a research project or published paper. Such students have the possibility to get a head start toward qualification and candidacy.
For admissions please find the relevant informations at our admissions page. You will then be directed to the University Application Form. When filling out the form, the choose the mathematics major.