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

  1. Classical Algebraic Structures

  2. Universal Algebra

  3. Clones

  4. Finite Algebra

  5. Lattice Theory

  6. Coordinate Systems in Lattice Theory

5.2. Group and Semigroup Theory

Topics

  1. Finite Groups and Fields

  2. Group Theory

  3. Semigroup Theory

  4. Regular Semigroups

  5. Universal Algebraic Studies of Semigroup Classes

5.3. Functional Analysis

Topics

  1. Functional Analysis

  2. Measure and Integration Theory

  3. Topological Vector Spaces

  4. Banach Algebras

  5. Operator Theory

5.4. Classical Analysis

Topics

  1. Elements of Real Function Theory

  2. Complex Analysis

  3. Fourier Series

  4. Fourier Integrals

  5. Orthogonal Series

5.5. Constructive Analysis

Topics

  1. Approximation with Trigonometric and Algebraic Polynomials

  2. Function Spaces and Approximation Operators

  3. Orthogonal Polynomials

  4. Potential Theory and Applications

5.6. Differential Equations

Topics

  1. Basics of Ordinary Differential Equations

  2. Basics of Partial Differential Equations

  3. Dynamical Systems

  4. Stability Theory

  5. Functional Differential Equations

  6. Partial Differential Equations in Function Spaces

5.7. Geometry

Topics

  1. Convex Geometry

  2. Discrete and Combinatorial Geometry

  3. Combinatorics of Polytopes

  4. Classical Integral Geometry

  5. Stochastic Geometry

  6. Algebraic Topology

  7. Differential Geometry and Topology

5.8. Discrete Mathematics

Topics

  1. Graph Theory

  2. Set Systems

  3. Block Systems and Codes

  4. Counting Problems

  5. Complexity Theory

  6. Combinatorial Methods in Geometry

  7. Cryptography and Coding Theory

5.9. Stochastics

Topics

  1. Basics and Strong Laws of Probability Theory

  2. Limit Theorems

  3. Chapters from Mathematical Statistics

  4. Stochastic Processes with Discrete State Space

  5. Stochastic Processes with Continuous State Space

5.10. Didactics of Mathematics

  1. Goals of Mathematics Teaching, Learning Theories

  2. Teaching Euclidean Geometry and Combinatorics

  3. Teaching Elements of Algebra and Calculus

  4. Teaching Elements of Probability and Statistics

  5. 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