Degree Courses
Bachelor study (BSc.)
Overview of study programmes
Mathematics
Financial Mathematics
Insurance Mathematics
Mathematics and Economics
Mathematical and Computer Modelling in Physics and Engineering
Mathematics and Computers in Practice
Business Administration
Computer Science
Universal study programme, the students profile through the choice of selected lectures and seminars
Physics
Applied Meteorology
Material Research and new Technologies
Vacuum and Cryogenic Techniques
Physical Foundations of Medical Techniques
Applied Chemical Physics
Applied Nuclear Physics
Computers in Physics and Technology
Photonics
Master study (MSc.)
Fields of Study and Study Programmes
Mathematics
Insurance and Financial Mathematics Applied probability, actuarial mathematics, theory of finance, banking and insurance, accounting, mathematical modelling and computational methods, financial management, risk theory, pensions.
Probability, Mathematical Statistics, Econometry
Probability: Application of probability theory to problems in natural sciences, technology and economy. The curriculum comprises the courses on advanced probability theory, stochastic analysis and differential equations, reliability theory, quality control.
Statistics: Theory of mathematical statistics and applications to biology, medicine and industry. Classical statistics, multivariate statistical analysis, nonparametric and sequential methods, robust methods, time series analysis.
Econometrics: Stochastic modelling of complex economic and socioeconomic phenomena, systems and processes including those from finance and insurance. Stochastic analysis, econometrics, stochastic optimization, time series analysis, implementation and verification of models.
Mathematics and Management Mathematical methods for management, quality management, quality control, statistics in industry, design of experiments, measurement and calibration.
Mathematical and Computer Modelling in Physics and Engineering Interdisciplinary study connecting applied mathematics and physics. Partial differential equations, continuum mechanics and thermodynamics, solidphase and fluid mechanics, plasma physics and optimization. Related numerical methods. Engineering applications.
Computational Mathematics Mathematical modelling using computational technique. Computational processes, algorithms, computational modelling, simulation, process control, solution of complex industrial problems, numerical analysis.
Mathematical Structures Algebra in computer science and natural sciences, discrete mathematics, dynamics, mathematical logic and set theory, Riemann geometry and harmonic analysis, topology and category theory
Mathematical Analysis Theory of functions of real and complex variable, measure and integral, functional analysis and topology, ordinary and partial differential equations, potential theory.
Computer Science
Algorithms and Computational Complexity Complexity theory in a broad sense and data structures. Computability theory and an introduction to recursion theory. Analysis of computational complexity of particular algorithms.
Nonprocedural Programming and Artificial Intelligence Logic, combinatorics and complexity theory. Artificial intelligence, methods, algorithms, and data representation. Nonprocedural programming languages, logic and functional programming, theory, implementation methods and applications. Neural nets, theory and applications.
Discrete Mathematics Studying discrete structures and processes. The main subjects are combinatorics, graph theory, algorithms, computational complexity, computational and discrete geometry and probabilistic and algebraic methods in discrete mathematics.
Data Engineering The main courses include database and information systems, information retrieval, and implementation techniques of database systems. Advanced courses cover problems of query languages and theory of relation databases. The database lectures are accompanied by possibilities to work with the products such as ORACLE, Informix, Progress, and MUMPS. Analysis and design methods of information systems are supported by P+ and LBMS.
Computing Systems Computer architecture, microprocessors, peripherals. Operating systems principles, communication and synchronisation. Local area networks, internetworking. Principles of compiler design and construction. Distributed operating systems, platforms, languages, and algorithms.
Computer Graphics Computational geometry Đ efficient algorithms and data structures, curves and surfaces for computer graphics. 2D drawing and clipping, halftoning methods, basic color science and color reproduction, coding, transformation and compression of raster images. 3D scene representation and rendering, shading, modern methods of digital image synthesis. Digital image processing and robotics. Graphical user interfaces.
Software Engineering Formal specification methods. Structured and object methods of analysis and design, data and process modelling, UpperCASE, LowerCASE. Software life cycle, prototyping, software maintenance, quality assurance. Software management, project planning and scheduling.
Computational and Formal Linguistic Theoretical background of CL and a formal description of language on all its levels (phonology, morphology, syntax, semantics, discourse). Methods of analysis (automata, grammars, statistical methods). Text corpora. Applications (error correction, information extraction, machine translation, speech). Languages: Czech, English, other according to interest.
Mathematical Optimization Theory and methods of optimization and applications of optimization methods. Solving operations research problems, in which it is necessary to find an optimal solution or decision with respect to a given optimality criterion or an appropriate compromise solution or decision in case of more optimality criteria involved.
Mathematical Economics Various applications of mathematical methods in economics both on microeconomic and on macroeconomic level. Mathematical approach to building appropriate mathematical models of real economic structures and situations.
Physics
Astronomy and Astrophysics Theoretical lectures on fundamental astronomy, celestial mechanics, astrophysics (stellar interior and atmospheres, interstellar matter), solar system astrophysics, solar physics, galactic and extragalactic astronomy, relativistic astrophysics and cosmology. Practical training in photometry, spectroscopy, positional and ephemeris astronomy, computing orbits.
Geophysics New methods in the theory of seismic wave propagation, physics of earthquakes and ground motions, structural studies with possible applications to the oil and coal prospecting. Geodynamics and physical geodesy concentrate on convective processes and physical parameters of the globe, with a close relation to gravimetry, geothermics, and geomagnetism. For details, see http://karel.troja.mff.cuni.cz.
Meteorology and Climatology Dynamical and synoptic meteorology, numerical modelling of atmospheric processes, meteorological forecasts. Air pollution problems, spreading and modelling airpollution, atmospheric chemistry. Boundary layer meteorology, atmospheric turbulence. Climatology, modelling of climate, climatic changes, statistical methods in climatology. Stratospheric ozone.
Theoretical Physics Classical and modern physics, mathematics, mathematical modelling. Special lectures on modern quantum mechanics and field theory, astrophysics and cosmology, condensed matter theory, mathematical physics, computer physics.
Solid State Physics Structure and microphysical interpretation of properties of condensed matter as a base for electronics and material science and optoelectronics. Lectures on theoretical and experimental physics of semiconductors, metals, superconductors, magnetic and dielectric materials, and ionic crystals.
Optics and Optoelectronics Quantum optics, nonlinear optical properties of matter, coherence and statistic properties of light, lasers, methods of optical communications and information processing, material research, fundamentals of semiconductor and optoelectronic elements and structures, integrated optics and photonics. Mathematical modelling and computational physics.
Physical Electronics and Vacuum Physics Motion and interactions of the electrically charged particles in a vacuum, gases, solid materials and on the boundaries between two materials. Mutual interactions of neutral atoms or molecules with each other and with the surface of the condensed matter. Physics of the surfaces and thin films, plasma physics, vacuum physics, computer modelling.
Physics of Molecular and Biological Structures Biophysics, chemical physics, polymer physics. Lectures on experimental physics, theoretical physics and mathematical modelling. Basic education in chemistry and biology.
Nuclear and Subnuclear Physics Fundamental elements of matter, elementary particles and their interactions. Properties of atomic nuclei, their structure and reactions. Laws governing forces acting between the nucleons. Experimental subnuclear and nuclear physics, quantum theory of particles and nuclei.
Teachers Education for secondary schools, upper grade
Mathematics and Descriptive Geometry Mathematics: Courses and training on theory of mathematical education.
Descriptive Geometry: courses and training on descriptive geometry, application of computers in descriptive geometry.
Mathematics and Physics Mathematics: Courses and training on theory of mathematical education.
Physics: Courses and training on theory of physics education, school physics experiments, computers in physics education.
Mathematics and Computer Science Mathematics: Courses and training on theory of mathematical education.
Computer Science: Courses and training on computer science.
Physics and Computer Science Physics: Courses and training on theory of physics education, school physics experiments, computers in physics education.
Computer Science: Courses and training on computer science.
Teachers Education for secondary schools, lower grade
Mathematics and Physics Mathematics: Courses and training on theory of mathematical education.
Physics: Courses and training on theory of physics education, school physics experiments, computers in physics education.
Extended Teachers Courses for Graduate Students
Mathematics Mathematical courses for graduate students who are not graduated in mathematics.
Physics Courses in physics for graduate students who are not graduated in physics.
Computer Science Courses in computer science for graduate students who are not graduated in computer science.
Descriptive Geometry Courses in descriptive geometry for graduate students who are not graduated in descriptive geometry.
Logic Courses in logic for graduate students who are not graduated in logic.
Doctoral study (PhD.)
Fields of Study and Study Programmes
PhD. Programmes  Computer Science
I1 Theoretical computer science These studies include the theory of computational complexity, formal languages, computer geometry, logic programming and artificial intelligence. Applications and problems of methodology and didactics are also included
I2 Software systems Distributed systems, operating systems, software engineering, realtime systems, computer graphics, neural networks. Objectoriented paradigm in programming languages, distributed systems and databases. Programming languages (theory of translation, typing models, semantics models).
I3 Mathematical linguistics Theoretical description of language as a formal system (generative procedures and declarative description models of discourse patterns), computational processing, morphological and syntactic analysis.
PhD. Programmes  Mathematics
M1 Mathematical logic Mathematical logic, set theory, recursion theory, Peano arithmetic and complexity theory, applied mathematical logic and applied set theory, the philosophy and history of mathematics.
M2 Algebra This area consists of the following subareas: Binary systems, groups and group representations, universal algebra, rings, modules and homological algebra, commutative algebra, lattices and Boolean algebras. Applications, methodology and didactic problems also belong to this area.
M3 Theory of numbers Includes the following specialisation: elementary and combinatorial theory of numbers, analytical theory of numbers, probabilistic theory of numbers, geometry of numbers, diophantine equations and approximations
M4 Combinatorics and discrete mathematics Classical combinatorial theory, block designs, configurations, coding theory and combinatorial enumeration. Graph theory, graph algorithms and representations of graphs and partially ordered sets. Discrete and computational geometry. Computational complexity issues in discrete mathematics (complexity of graph, combinatorial and geometrical algorithms and problems). Probabilistic methods in combinatorics and graph theory. Graph algorithms and representations of graphs and partially ordered sets. Discrete and computational geometry. Computational complexity issues in discrete mathematics (complexity of graph, combinatorial and geometrical algorithms and problems). Probabilistic methods in combinatorics and graph theory.
M5 Geometry and global analysis The foundations of algebraic and differential topology, global analysis of manifolds, Riemannian geometry, Lie groups and Lie algebras and their representations, kinematics and robotics (geometrical foundations), differential operators and harmonic analysis, geometrical methods in physics.
M6 Topology and general structures Topological structures (combined with algebraic ones), their relation to set theory, analysis, computer science and combinatorics. Category theory related to algebras and topological spaces and to computer science (theory of automata).
M7 Theory of functions and functional analysis Functions of real and complex variable, theory of measure and integral, special functions, Fourier and harmonic analysis, topological vector spaces, Banach spaces, operator theory, calculus of variations and functional analysis.
M8 Differential and integral equations, potential theory Ordinary and partial differential equations and their applications in sciences. Qualitative properties of weak solutions of partial differential equations (both linear and nonlinear) and their systems. Classical and modern potential theory and its applications. Some branches of the calculus of variations and functional analysis connected with differential equations.
M9 Probability and mathematical statistics Probability theory, random processes, time series, robust methods, linear models, nonparametric methods, asymptotic statistics, sequential methods, changepoint problem, bootstrap, stochastic analysis
M10 Econometry Deterministic and stochastic optimization, econometrics, mathematical economics, mathematical and stochastic modelling in economics, finance and insurance, applications of probability and mathematical statistics in economics and finance.
M11 Scientific and technical calculations Discretisation of boundary value and evolution problems, finite element techniques, solution of large scale algebraic systems, multilevel methods, optimization
M12 Operational research Operational research is studied as a mathematical discipline having its application in technical and economical practice. It uses a combination of methods of algebra, analysis, control theory, linear and nonlinear optimization, probability theory and mathematical statistics. Both deterministic and stochastic problems are studied.
M13 Financial and insurance mathematics (actuarial mathematics) Mathematical modelling in finance and insurance. Risk theory. Statistical methods in insurance, banking and finance. Advanced financial software. Social insurance and pensions. Demography.
PhD. Programmes  Physics
F1 Theoretical physics, astronomy and astrophysics Research in various branches of theoretical physics and astrophysics, particularly in nuclear and atomic physics, statistical physics, solid state physics, plasma physics, the mathematical foundations of quantum mechanics, classical general relativity, relativistic astrophysics, the mathematical and physical cosmology, theoretical astronomy and astrophysics. Within observational astronomy the research is mainly concentrated on galactic astronomy, in particular on binary systems.
F2 Physics of plasma and ionised media Experimental and theoretical investigation of the low and high temperature plasmas as well as the study of the space plasma processes. This study involves experimental technics, methodics and industrial applications of plasmas. A significant part of the study programme is devoted to interdisciplinary topics such as the plasma chemistry of physics of ionised media.
F3 Physics of condensed matter and material research Including the whole spectrum of subjects which the title suggests. In the first three terms students are given advanced lectures on quantum mechanics, statistical physics, theory of solid state physics and modern experimental methods given by top experts in the field. In the scientific work for thesis, supervision is given by specialists with excellent expertise in a particular field.
F4 Physics of molecular and biological structures Physical processes in complex organic molecules, macromolecular and biological systems. Quantum theory of electronic structure, energy and electric charge transfer, molecular interactions and thermodynamics in molecular and supramolecular systems. Physical diagnostic methods for the investigation of structure, function, dynamics and interactions in molecular and biological structures on the molecular, polymer, membrane, cellular, tissues and whole organisms levels. Influence of external physical fields.
F5 Physical electronics and vacuum physics Investigation of heteroepitaxial thin films growth, reconstruction of surfaces induced by absorbed molecules, study of the structure dependent reactivity of thin film model catalysts, study of electron interactions in the surface regions of solids, study of charge transport in metal/insulator thin film systems, study of surfaces and thin film growth using scanning tunnelling microscopy and spectroscopy. In the field of vacuum physics, processes governing the behaviour of ultra and extrahigh vacuum systems, especially mechanically stimulated desorption and thermal stimulated effusion of dissolved gases at very low concentration are investigated.
F6 Quantum optics and optoelectronics Semiconductors optoelectronics, semiconducting materials for photon detectors, ultrafast processes in semiconductors and semiconductor nanostructures, picosecond and femtosecond spectroscopy, femto/picosecond nonlinear optics, ultrafast optical nonlinearities in semiconductor nanostructures, porous silicon, semiconductor nanocrystals in glass, magnetooptical spectroscopy of low dimensional semiconductor quantum structures, magnetooptical spectroscopy and magnetometry for magnetic multilayers with resolution to a single atomic monolayer, theoretical investigations of guided waves, correlation induced changes of spectra, chiral media, optical sensors, photonic components.
F7 Geophysics All branches of the physics of the Earth. Studies of seismology are oriented to new methods in the theory of seismic wave propagation, the physics of earthquakes and ground motions, and structural studies (with possible applications to oil and coal prospecting). Geodynamics and physical geodesy include studies of convective processes and the physical parameters of the globe, with a close relation to gravimetry, geothermics and geomagnetism. Research in physics of the upper atmosphere, SunEarth relations etc.
F8 Meteorology and climatology Dynamic and synoptic meteorology, boundary layer meteorology, micrometeorology, meteorological forecasts, atmospheric chemistry, airpollution problems, atmospheric effects of aerosols, stratospheric and tropospheric ozone, radiation in the atmosphere, atmospheric optics, acoustics and electricity, statistical methods in climatology, modelling of the climate, antropogenous effects on climatic conditions.
F9 Subnuclear physics The subnuclear structure of matter. Quantum field theory. Theory and models of interactions of elementary particles. The standard model: electroweak interactions and quantum chromodynamics. Grand unification and physics beyond the standard model. Particle physics in accelerators. Detectors and electronics. Data processing. Team collaboration in highenergy experiments.
F10 Nuclear physics Nuclear structure and spectroscopy – theoretical descriptions, experimental determination of nuclear properties, statistical aspects of fermion systems. Nuclear reactions – mechanism, cross sections. Subnuclear degrees of freedom – mesonic and quark degrees. Applied nuclear physics – nuclear material analysis methods, nuclear energy, nuclear therapy.
F11 Mathematical and computer modelling Mathematical and computer modelling in science and technology with an emphasis on nonlinear physics, mathematics and computational physics including corresponding numerical methods: nonlinear differential equations, calculus of variation and control theory, nonlinear continuum mechanics and thermodynamics of solids and fluids, particle modelling – the Monte Carlo and molecular dynamics methods, calculations of molecules and solids, dynamics of excitations and corresponding visualisation.
PhD. Programmes  General Problems
D1 General problems of mathematics and computer science This aims to prepare toplevel secondary school teachers who are aspiring to write textbooks or to play an important role in education. It consists mostly of elementary mathematics, the history of mathematics and of the theory of teaching mathematics at secondary and specialised schools.
D2 General problems of physics Research in three main branches: philosophy and methodology of physics, the history of physics and the teaching of physics. The research work in all of these branches is supposed to be based on a good technical knowledge of the physics involved. Typical problems concerning the teaching of physics include the development of experimental equipment for the demonstration of different physical phenomena, computer simulations for demonstrative purposes etc.
Admission Procedure
Bachelor or master studies
To be admitted to bachelor or master studies, an applicant has to fill out application form, present her/his secondary schoolleaving certificate (this certificate must be equivalent to Czech schoolleaving certificate) and to pass satisfactory an admission exam. For the study in Czech language an admission Czech knowledge exam is necessary.
Generally, application form must be sent to the Study and student affairs department of the faculty in the second half of February. Exams then take place in the middle of June or at the end of August. For exact deadlines please contact Study and student affairs department.
Application forms can be obtained from Study and student affairs department of the faculty. This department also can give more detailed information about admission procedure.
Doctoral studies
To be admitted to doctoral studies, an applicant has to present her/his MSc. diploma (or equivalent) and to pass satisfactory an admission exam.
Before the admission interview students choose one of the fields of study as his/her main area of concentration (see below). During the admission procedure the student chooses her/his advisor and a broad dissertation theme.
Exams take place at the end of June or at the beginning at July, in September and in special cases (and only for foreign students) in February.
More information about admission procedure, fields of study, possible advisors etc. as well as application forms can be obtained from Study and student affairs department of the faculty.
About the Faculty & University
Charles University, the oldest university in Central Europe, was founded on April 7th, 1348 by Charles IV, the Holy Roman Emperor and King of Bohemia, with all medieval faculties  Theology, Law, Medicine and Arts. One of its first rectors was religious reformer Magister Jan Hus (burnt in 1415). The defeat of the Bohemian Estates in 1620 and the consequences of the Thirty Years War meant the forced recatholicisation. The CharlesFerdinand University was created out (1654) of the Arts Faculty of Charles University (the only faculty surviving after the period of Hussite Wars) and the Jesuit Clementine Academy. This university, despite the repression, acted as a stimulus for Czech national consciousness. The gradual introduction of enlightened reforms culminated at the end of the 18th century when nonCatholics were granted the right to study and German replaced Latin as the language of instruction.
The upsurge in Czech national life, science and political activity forced the Vienna government in 1882 to divide the university into two institutions, one German and one Czech. These operated alongside each other even in the Czechoslovak state (since 1918) to the Nazi occupation. On November 17th., 1939, all Czech institutions of higher education were closed by force. The German Prague university had been proclaimed a university of the Reich, and, as such, it was abolished in 1945.
After the liberation in 1945 the university life was soon to be disrupted and distorted by the effects of the Communist putsch in February 1948. The putsch meant purges, the end of academic freedoms and harsh repression of all disagreement expressions with the official ideology. Only after the revolution which began on 17th., November 1989, was there real change in a situation. The appointment of new representatives of a free academic community was legalised in January 1990.
Mathematics and physics were taught traditionally in the frame of the Philosophical Faculty (Faculty of Arts). In 1920 separated from this faculty the Faculty of Natural Sciences and the Faculty of Mathematics and Physics has been an individual faculty since 1952. Among the scholars who worked at and/or cooperated with the university and its faculties when staying in Prague were Tycho Brahe, Johannes Kepler, Bernard Bolzano, Ernst Mach, Albert Einstein, Rudolf Carnap and Jaroslav Heyrovský.
A characteristic feature of the Faculty of Mathematics and Physics is the close interconnection of teaching activities and research work. The students benefit from being taught by those at the forefront of scholarship and research in their fields. A part of the research has an applied aspect. Teaching is provided by more than 300 teachers. The scientific reputation of the Faculty is reflected in the high success rate of domestic and international grant proposals: in 1997 faculty experts have participated in 239 projects of Czech grant agencies and 67 international projects. The total of all grant funds in 1997 was 1,790,000 US$. Together with coworkers from the Academy of Sciences of the Czech Republic, the committed staff of the Faculty supervised in 1997 549 postgraduate students in 29 fields of study and 52 PhD theses were successfully defended.
The Faculty represents three sections: the School of Computer Science, the School of Mathematics and the School of Physics.
School of Computer Science
The School of Computer Science provides students with a full grounding in computer science. It is one of the faculty's strongest academic body and its education programmes are highly regarded. It offers a comprehensive curriculum from introductorylevel courses to graduate seminars focusing on critical research areas. Students find a unique set of opportunities available to study for the Bachelor’s degree, the Master's degree or the Doctoral degree. The MSc programme thoroughly prepares and qualifies students for advanced careers within the computer industry, and the PhD programme directly involves students in some of the most exciting and significant research being conducted anywhere in Czech Republic. The School of Computer Science has developed an outstanding research programme supported by national as well as EC grants.
School of Mathematics
Teaching mathematics connected with research has a long tradition at Charles University. Probably its most famous mathematician was Bernad Bolzano, and from this century it is Eduard Čech, the famous topologist and geometer.
There are about 90 academicians and 10 additional staff members employed by the mathematical part of the faculty. They provide mathematical classes both for its own students and students of other faculties of Charles University.
School of Physics
The scientific activity of the School of Physics covers (and is the only school in the Czech Republic to cover) the full range of modern physics. Disciplines with a long tradition, such as physical electronics and vacuum physics, astronomy, meteorology and geophysics are cultivated. On the base of a small group studying "atomic physics" in the thirties new departments were established engaged in solid state physics, chemical physics, nuclear and subnuclear physics. The scientific community of the School responds vividly to the modern trends in physics (for example low temperature physics, quantum optics, physics of molecular and biological structures, cosmic physics, surface physics). The experimental activities are supported by applied physics research, by well equipped laboratories (cryogenic laboratory, optical methods laboratory, surface physics laboratory, solid state laboratory, high vacuum and plasma laboratory, low energy accelerator laboratory) and by the School's development and production base (glass and optics workshop, Association Vacuum/MFF UK Praha). Applied research has been an integral part of the School of Physics' output and covers a wide spectrum of problems  from modelling the influence of pollution on the quality of the environment to the construction of the hygrometer or radon concentration meter.
Modern physics is developing broad collaborations on joint projects with other Czech universities as well as with the Academy of Sciences Institutes. Twosided agreements are the main form of collaboration with colleagues from universities and research institutions around the world. The top quality collaboration represents the active participation in the Europewide projects such as the LHC accelerator at CERN.
