### General

### Program Description

The program gives a comprehensive knowledge of several areas in mathematics and introduces the students to doing research in theoretical and/or applied mathematics. Besides purely theoretical courses, many courses are application-oriented. Courses are offered in algebra, number theory, real and complex analysis, topology, geometry, probability theory and statistics, discrete mathematics and operations research but also in such interdisciplinary subjects as bioinformatics and theoretical computer science. The students may also choose from high-level application-oriented courses, which present state of the art issues of the given area, like complex systems, financial mathematics, etc.

The program is aimed at students who have at least a BSc degree in mathematics or a related field (physics, computer science, engineering, etc.) In the latter case, a certain number (65) of mathematical credits is required from earlier studies.

**Strength of program**

One of the main features of the program is the great variety of courses, covering several areas of mathematics. Our graduates will have a broad knowledge of many areas of mathematics. Besides offering an introduction and basic foundation in many areas, some of the subjects lead to up-to-date research results.

Most of the teachers of the program have international teaching experience and they regularly give classes also at foreign universities, including North American institutions. Young mathematicians, bringing in freshness and new momentum, are also involved in the program. Our instructors all have scientific degrees and a good research record. Examples show that graduating from our program is a very good starting point for doctoral or (at a later stage) postdoctoral studies.

Of particular interest is the fact that many researchers in the internationally renowned Hungarian school of combinatorics have started their careers at our university and many of them still have a position at the Institute of Mathematics. For example, the Wolf Prize and Kyoto Prize winner Prof. László Lovász is a professor at our university. Recent Abel prize winner, Prof. Endre Szemerédi is also a graduate of our school. But one could also recall the Ostrowski Prize of Prof. Miklós Laczkovich (professor of our university), the Gödel Prize of Prof. László Babai (former professor), the Coxeter Prize of Prof. Balázs Szegedy (a graduate of our university), etc.

**Structure**

**Basic courses**

- Analysis
- Basic algebra (reading course)
- Basic geometry (reading course)
- Complex functions
- Differential geometry I
- Geometry III
- Introduction to topology
- Probability and statistics
- Reading course in analysis
- Set theory (introductory)

**Core courses - Algebra and number theory**

- Groups and representations
- Number theory 2
- Rings and algebras

**Core courses - Analysis**

- Function series
- Fourier integral
- Functional analysis II
- Topics in analysis

**Core courses - Geometry**

- Algebraic topology (basic material)
- Combinatorial geometry
- Differential geometry II
- Differential topology (basic material)
- Topics in differential geometry

**Core courses - Stochastics**

- Discrete parameter martingales
- Markov chains in discrete and continuous time
- Multivariate statistical methods
- Statistical computing 1

**Core courses - Discrete mathematics**

- Algorithms I
- Discrete mathematics
- Mathematical logic

**Core courses - Operations research**

- Continuous optimization
- Discrete optimization

**Differentiated courses - Algebra**

- Commutative algebra
- Current topics in algebra
- Topics in group theory
- Topics in ring theory
- Universal algebra and lattice theory

**Differentiated courses - Number theory**

- Combinatorial number theory
- Exponential sums in number theory
- Multiplicative number theory

**Differentiated courses - Analysis**

- Chapters of complex function theory
- Complex manifolds
- Descriptive set theory
- Discrete dynamical systems
- Dynamical systems
- Dynamical systems and differential equations
- Dynamics in one complex variable
- Ergodic theory
- Geometric measure theory
- Nonlinear functional analysis and its applications
- Operator semigroups
- Partial differential equations
- Representations of Banach-*-algebras and abstract harmonic analysis
- Riemann surfaces
- Seminar in complex analysis
- Special functions
- Topological vector spaces and Banach-algebras
- Unbounded operators of Hilbert spaces

**Differentiated courses - Geometry**

- Algebraic and differential topology
- Convex geometry
- Differential topology problem solving
- Discrete geometry
- Finite geometries
- Geometric foundations of 3D graphics
- Geometric modeling
- Lie groups and symmetric spaces
- Riemannian geometry
- Supplementary chapters of topology I. – Topology of singularities. (special material)
- Supplementary chapters of topology II – Low dimensional manifolds

**Differentiated courses - Stochastics**

- Analysis of time series
- Cryptography
- Introduction to information theory
- Statistical computing 2
- Statistical hypothesis testing
- Stochastic processes with independent increments, limit theorems

**Differentiated courses - Discrete Mathematics**

- Applied discrete mathematics seminar
- Codes and symmetric structures
- Complexity theory
- Complexity theory seminar
- Data mining
- Design, analysis, and implementation of algorithms and data structures I
- Design, analysis, and implementation of algorithms and data structures II
- Discrete mathematics II
- Geometric algorithms
- Graph theory seminar
- Mathematics of networks and the WWW
- Selected topics in graph theory
- Set theory I
- Set theory II

**Differentiated courses - Operations research**

- Applications of operations research
- Business economics
- Approximation algorithms
- Combinatorial algorithms I
- Combinatorial algorithms II
- Combinatorial structures and algorithms
- Computational methods in operations research
- Game theory
- Graph theory
- Graph theory tutorial
- Integer programming I
- Integer programming II
- Inventory management
- Investments analysis
- LEMON library: solving optimization problems in C++
- Linear optimization
- Macroeconomics and the theory of economic equilibrium
- Manufacturing process management
- Market analysis
- Matroid theory
- Microeconomy
- Multiple objective optimizations
- Nonlinear optimization
- Operations research project
- Polyhedral combinatorics
- Scheduling theory
- Stochastic optimization
- Stochastic optimization practice
- Structures in combinatorial optimization

**Career opportunities**

Our graduates will be able to apply for Ph.D. studies either at Eötvös Loránd University or anywhere in the world. Many students will, however, continue their career immediately in the industrial research and development, often at high tech industries in telecommunication, financial institutions or insurance companies or in software development of such research giants as Google.

**Job examples**

- University professor
- Research mathematician in a research institute
- System analyst in a financial institution (bank, investment, insurance)
- High tech industry
- Teacher of mathematics

**Admission requirements**

**Entry requirements**

BSc degree in Mathematics or in a related field (Physics, Computer Science, Engineering, etc.) in the latter case at least 65 credits in Mathematics is required.

**Language requirements**

- A minimum level of language proficiency (oral) (A1-C2): B2
- A minimum level of language proficiency (written) (A1-C2): B2

**Further comments:**

English language proficiency test (almost any test is accepted). Note, however, that it is the applicant’s responsibility to make sure that he/she will be able to follow the English language lectures and fulfill the requirements. No extra language courses will be organized by the university for free.

**Documents to submit with the application**

- Online application form
- Bachelor-level degree
- Transcript of records
- CV
- Motivation letter
- Letter of recommendation
- Copy of the main pages of the passport
- Passport photo
- Medical certificate
- Certificate of clean criminal records
- Copy of application fee transfer
- Language certificate

**Further comments (if needed)**

BSc degree in mathematics or in a related field (physics, computer science, engineering, etc.); in the latter case, at least 65 credits in mathematics. Applicants still pursuing their BSc studies should obtain the diploma, or at least a certificate of being entitled to the diploma, before starting their MSc studies

**Application procedure**

The application starts with the online application system. Students need to register in the system, fill in the online application form, upload the required documents and follow the instructions during the application process.

*Last updated October 2019*

### About the School

Eötvös Loránd University (ELTE) - in terms of student and staff number, quality of education and research, and its international relations - is a prestigious institution of public higher education in ... Read More