1. Doctoral Training Programs
-
Pure Mathematics
-
Applied Mathematics
-
Didactics of Mathematics
Within these, the main research areas cover algebra, analysis, dynamical systems, geometry, discrete mathematics, and stochastics.
2. Structure of the Doctoral Training
The doctoral training is fundamentally based on the professional and trust-based relationship between the doctoral student and the supervisor(s).
2.1. Doctoral Credits
The total number of credits to be earned during the doctoral training: 240.
Before the official closure of each reporting period, the doctoral student prepares a report on the work completed for the Head of the Doctoral School, which is reviewed and signed by the supervisor.
At the Doctoral School of Mathematics, credits can be earned for study, research, and teaching activities as follows.
2.2. Study Credits
During the first two years of the program, the student must complete at least 5 courses, including at least 2 courses in the first year.
The Doctoral School announces the doctoral courses at the beginning of each semester. Each course (regardless of the number of hours, including reading courses) is worth 5 credits. The list of regularly announced doctoral courses can be found in Section 3, “Course Lists”.
Upon the doctoral student’s request, participation in online courses, summer or winter schools, or workshops can be accepted as a course. The request must be submitted to the Head of the Doctoral School with the support of the supervisor.
2.3. Research Credits
The completion of research credits is certified by the supervisor.
Achievements related to research seminars, conferences, and publications can be credited multiple times within a semester. For achievements related to publications, in the case of co-authorship with another doctoral student, the credits may be divided by the Doctoral School Council.
The condition for submitting the doctoral dissertation for review is at least two published or accepted publications (including at least one journal article) related to the topic of the doctoral dissertation, published or accepted in a peer-reviewed journal indexed in an international database and published in a world language.
2.4. Teaching Credits
Teaching exercise session for one semester, 2 credits per weekly teaching hour. A maximum of 8 credits can be earned per semester under this category.
The doctoral student must request the inclusion of teaching credits. The completion is verified and certified by the teaching coordinator of the Bolyai Institute.
3. Course Lists
3.1. Algebra
Course Name |
Course Code |
Group Theory |
MATD1xx |
Lattice Theory |
MATD1xx |
Semigroup Theory |
MATD1xx |
Universal Algebra |
MATD1xx |
Ordered Sets |
MATD1xx |
Clone Theory |
MATD1xx |
Finite Algebras |
MATD1xx |
Graph Homomorphism Problems |
MATD1xx |
3.2. Dynamical Systems
Course Name |
Course Code |
Partial Differential Equations |
MMNV23E |
Dynamical Systems |
MMNV24E |
Ordinary Differential Equations |
MATD2xx |
Functional Differential Equations |
MATD2xx |
Numerical Methods for Differential Equations |
MATD2xx |
Dynamical Models |
MATD2xx |
Nonlinear Dynamics |
MATD2xx |
3.3. Geometry
Course Name |
Course Code |
Algebraic Topology |
MMNM42E |
Combinatorics of Convex Polytopes |
MMNM43E |
Integral Geometry and Geometric Probability |
MATD3xx |
High-Dimensional Convex Geometry |
MATD3xx |
Brunn-Minkowski Theory of Convex Bodies |
MATD3xx |
Discrete and Combinatorial Geometry |
MATD3xx |
Stochastic Geometry |
MATD3xx |
Geometry of Vector Systems |
MATD3xx |
Selected Topics in Geometry |
MATD3xx |
3.4. Discrete Mathematics
Course Name |
Course Code |
Discrete Mathematics 2 |
MMNK51E |
Extremal Graph Theory |
MMNM55E |
Mathematical Cryptography |
MMNM56E |
Combinatorial Computational Models |
MMNM53E |
Counting Problems |
MATD4xx |
Algebraic and Random Methods in Combinatorics |
MATD4xx |
Finite Geometries, Codes, Cryptography |
MATD4xx |
Selected Topics in Graph Theory |
MATD4xx |
3.5. Analysis
Course Name |
Course Code |
Measure and Integration Theory |
MATD5xx |
Complex Analysis |
MATD5xx |
Functional Analysis |
MATD5xx |
Banach Algebras and Operator Theory |
MATD5xx |
Selected Topics in Functional Analysis |
MATD5xx |
Approximation Theory |
MATD5xx |
Potential Theory |
MATD5xx |
Fourier Series, Fourier Integrals I |
MATD5xx |
Fourier Series, Fourier Integrals II |
MATD5xx |
3.6. Stochastics
Course Name |
Course Code |
Probability Theory |
MMNK61E |
Stochastic Processes |
MMNV63E |
Financial and Risk Processes |
MMNV64E |
Statistical Analysis of Time Series |
MMNV61E |
Mathematical Statistics |
MMNV62E |
Markov Chains |
MATD6xx |
Branching Processes |
MATD6xx |
Selected Topics in Stochastics |
MATD6xx |
3.7. Didactics of Mathematics
Course Name |
Course Code |
Problem Solving in Mathematics and Mathematics Teaching |
MATD7xx |
Research Methodology and Applied Statistics |
MATD7xx |
Chapters from the Cultural History of Mathematics |
MATD7xx |
Elementary Combinatorics |
MATD7xx |
Digital Resources in Geometry Teaching |
MATD7xx |
Chapters from the Methodology of Teaching Higher Mathematics at Secondary and Tertiary Levels |
MATD7xx |
4. The Comprehensive Exam
During the doctoral training, at the end of the fourth semester, as a conclusion of the training and research phase and as a prerequisite for starting the research and dissertation phase, a comprehensive exam must be completed, which evaluates the academic and research progress.
The prerequisite for taking the comprehensive exam is earning at least 90 credits during the training and research phase (first four semesters) and completing all study credits required by the training plan of the Doctoral School. Exceptions are made for students preparing individually for the doctoral degree, whose student status is established upon successful completion of the comprehensive exam.
The comprehensive exam consists of two main parts: in one part, the candidate’s theoretical knowledge is assessed (theoretical part), and in the other part, the candidate reports on their scientific progress (dissertation part).
In the first, theoretical part of the comprehensive exam, the candidate takes an exam in two subjects, with two topics from one subject and one topic from the other. The list of subjects and topics is provided in Section 5, “Subjects of the Comprehensive Exam”. The theoretical exam may include a written component.
In the second, dissertation part of the comprehensive exam, the candidate gives a presentation on their literature review, reports on their research results, presents their research plan for the second phase of the doctoral training, and outlines the schedule for completing the dissertation and publishing the results.
Before the exam, the supervisor evaluates the candidate in writing, addressed to the chair of the examination committee and sent to the secretary of the Doctoral School.
5. Subjects of the Comprehensive Exam
5.1. Universal Algebra and Lattice Theory
Topics
-
Classical Algebraic Structures
-
Universal Algebra
-
Clones
-
Finite Algebra
-
Lattice Theory
-
Coordinate Systems in Lattice Theory
5.2. Group and Semigroup Theory
Topics
-
Finite Groups and Fields
-
Group Theory
-
Semigroup Theory
-
Regular Semigroups
-
Universal Algebraic Studies of Semigroup Classes
5.3. Functional Analysis
Topics
-
Functional Analysis
-
Measure and Integration Theory
-
Topological Vector Spaces
-
Banach Algebras
-
Operator Theory
5.4. Classical Analysis
Topics
-
Elements of Real Function Theory
-
Complex Analysis
-
Fourier Series
-
Fourier Integrals
-
Orthogonal Series
5.5. Constructive Analysis
Topics
-
Approximation with Trigonometric and Algebraic Polynomials
-
Function Spaces and Approximation Operators
-
Orthogonal Polynomials
-
Potential Theory and Applications
5.6. Differential Equations
Topics
-
Basics of Ordinary Differential Equations
-
Basics of Partial Differential Equations
-
Dynamical Systems
-
Stability Theory
-
Functional Differential Equations
-
Partial Differential Equations in Function Spaces
5.7. Geometry
Topics
-
Convex Geometry
-
Discrete and Combinatorial Geometry
-
Combinatorics of Polytopes
-
Classical Integral Geometry
-
Stochastic Geometry
-
Algebraic Topology
-
Differential Geometry and Topology
5.8. Discrete Mathematics
Topics
-
Graph Theory
-
Set Systems
-
Block Systems and Codes
-
Counting Problems
-
Complexity Theory
-
Combinatorial Methods in Geometry
-
Cryptography and Coding Theory
5.9. Stochastics
Topics
-
Basics and Strong Laws of Probability Theory
-
Limit Theorems
-
Chapters from Mathematical Statistics
-
Stochastic Processes with Discrete State Space
-
Stochastic Processes with Continuous State Space
5.10. Didactics of Mathematics
-
Goals of Mathematics Teaching, Learning Theories
-
Teaching Euclidean Geometry and Combinatorics
-
Teaching Elements of Algebra and Calculus
-
Teaching Elements of Probability and Statistics
-
History and Philosophy of Mathematics in Mathematics Education
6. Credit Distribution and Sample Curriculum in Tabular Form
6.1. Credit Table
Weekly Regular Consultation with Supervisor |
10 credits |
Processing of Scientific Literature |
10 credits |
Survey of Research Problems |
5 credits |
Research Project Report |
5 credits |
Participation in Research Seminar (2 hours per week) |
3 credits |
Presentation at Research Seminar |
3 credits |
Presentation at Hungarian Conference |
3 credits |
Presentation at International (Foreign Language) Conference |
5 credits |
Technical Report |
10 credits |
Submission of Publication to Peer-Reviewed Journal in a World Language |
15 credits |
Final Acceptance of Publication in Peer-Reviewed Journal in a World Language |
15 credits |
Other Accepted Scientific Publication |
10 credits |
6.2. Sample Curriculum (Recommended)
Semester | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Total |
---|---|---|---|---|---|---|---|---|---|
Studies |
|||||||||
Courses (credits) |
10 |
5 |
5 |
5 |
25 |
||||
Teaching (credits) |
4 |
4 |
4 |
4 |
4 |
4 |
24 |
||
Research (can be taken once per semester) |
|||||||||
Processing of Scientific Literature; Technical Report; Weekly Regular Consultation with Supervisor (10 credits) |
20 |
20 |
20 |
20 |
80 |
||||
Survey of Research Problems; Research Project Report (5 credits) |
5 |
10 |
10 |
25 |
|||||
Research (can be taken multiple times per semester) |
|||||||||
Participation in Research Seminar (2 hours per week); Presentation at Research Seminar; Presentation at Hungarian Conference (3 credits) |
3 |
9 |
9 |
21 |
|||||
Presentation at International (Foreign Language) Conference (5 credits) |
5 |
5 |
10 |
||||||
Submission of Publication to Peer-Reviewed Journal in a World Language (15 credits) |
15 |
15 |
30 |
||||||
Final Acceptance of Publication in Peer-Reviewed Journal in a World Language (15 credits) |
15 |
15 |
|||||||
Other Accepted Scientific Publication (10 credits) |
10 |
10 |
|||||||
Total (credits) |
34 |
29 |
29 |
32 |
28 |
28 |
30 |
30 |
240 |