Master study programme **Matematical and Computational Modelling in Physics** is designed for students of physics. (The programme for students of mathematics is called **Mathematical Modelling in Physics and Technology**, details concerning this study programme can be found here.)

## Contacts

If you have any **questions concerning the study of mathematical modelling**, please contact Martin Čížek (cizek@mbox.troja.mff.cuni.cz) or Vít Průša (prusv@karlin.mff.cuni.cz).

## Admission

Concerning the general overview of the admission procedure, please see the webpages of the Faculty of Mathematics and Physics. **If you are not yet familiar with advanced subjects such as functional analysis but you are still strongly motivated to study mathematical modelling, please contact us, and we will try to recommend you textbooks and other material for self-study.**

Note also, that if you have a sufficiently strong academic record, you can apply for the exemption from the entrance examination.

The (new) study programme Matematical and Computational Modelling in Physics is expected to start in 2015/2016. See the faculty webpages for the prerequisities concerning the “old” study programme, the prerequisities concerning the new study programme can be found at faculty webpages.

Detailed guidelines should appear on this page soon.

## Curriculum

The expected length of the study is **two years**. You are expected to choose the topic of your thesis during the first year of your study. There are three sets of courses: **compulsory**, **compulsory elective** and **optional**.

Concerning the **compulsory** courses you **must complete all of them**. The strange term **compulsory elective** means that you **must complete some of these courses**. (The term “some” means that you need certain number of credits from these subjects, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you! The choice of the **optional** courses is completely up to you. Ask your thesis supervisor, classmates or the people responsible for the study programme, see contacts, for a hint.

The (new) study programme Computational and Mathematical Methods in Physics is expected to start in 2015/2016. Detailed curriculum should appear on this page soon, in the mean time please see the faculty webpages.

### Compulsory courses

You **must complete all these courses**. The next to the last column indicates the recommended year of study. Besides the courses listed below you must also complete three “virtual” one semester courses called “Diploma Thesis Seminar”. You will get the credit for this course provided that your thesis supervisor confirms that you work on the thesis.

Course | Annotation | Year | Semester |
---|---|---|---|

Introduction to Functional Analysis | normed linear spaces, Banach and Hilbert spaces, linear functionals and operators, Hahn-Banach theorem, uniform boundedness principle, open mapping theorem, duality and reflexive spaces, Riesz-Schauder theory of compact operators, Fredholm theorems, convolutions and basic properties of L_p, distributions and tempered distributions | 1st | winter |

Partial Differential Equations 1 | notion of the weak solution, Lebesgue spaces, Sobolev spaces, linear elliptic equations | 1st | winter |

Analysis of Matrix Calculations 1 | fundamentals of numerical linear algebra, solving of linear algebraic equations, least squares method, eigenvalue problems | 1st | winter |

Finite Element Method 1 | foundations of the theory of finite element method, linear elliptic equations | 1st | winter |

Numerical Solution of Ordinary Differential Equations | one-step and multi-step methods (algorithms, analysis, convergence), discrete and continuous dynamical systems | 1st | winter |

Computer Simulations in Many-particle Physics | principles of the Monte Carlo and molecular dynamics methods | 1st | winter |

Partial Differential Equations 2 | Bochner spaces, parabolic and linear hyperbolic equations of the second order | 1st | summer |

Matrix Iterative Methods | iterative methods, Krylov subspace methods | 2nd | winter |

### Compulsory elective courses

The courses are grouped into five categories — continuum mechanics, molecular dynamics, quantum systems, general relativity and particle physics. Once you have chosen the particular branch of physics you want to specialize in, you should focus on the courses in the corresponding category. You **must complete some of these courses**. (The term “some” means that you need certain number of credits from the given category, see faculty webpages for details.) The choice is yours, but it should reflect the topic of your thesis. Talk to your thesis supervisor if you want to check which courses are suitable for you!

#### Continuum mechanics

Course | Annotation | Year | Semester |
---|---|---|---|

Continuum Mechanics | introduction to continuum mechanics, kinematics, balance equations, linearized elasticity, Navier–Stokes fluid | 1st/2nd | winter |

Mixture Theory | mechanics and thermodynamics of mixtures | 2nd | winter |

Computer Solutions of Problems in Continuum Mechanics | basic commercial/open source software for numerical computation (Matlab, Femlab, FEniCS) and its application to solution of partial differential equations, basic numerical libraries (Blas, Lapack, Petsc), finite element libraries (Feat, Featflow), libraries for parallel computation (MPI, OpenMP) | 1st | summer |

Thermodynamics and Mechanics of non-Newtonian Fluids | non-Newtonian phenomena, constitutive relations for non-Newtonian fluids, thermodynamics of non-Newtonian fluids | 1st/2nd | summer |

Thermodynamics and Mechanics of Solids | mechanics and thermodynamics of nonlinear solids, constitutive relations for non-linear solids | 1st/2nd | summer |

Nonlinear Functional Analysis | nonlinear operators in Banach and Hilbert spaces, monotone, pseudomonotone and potential operators | 2nd | summer |

Numerical Solution of Evolutionary Equations | theoretical and practical aspects of the numerical solution of evolutionary differential equations | 2nd | summer |

Solution of Nonlinear Algebraic Equations | theoretical and practical aspects of the numerical solution of nonlinear equations and their systems | 2nd | winter |

#### Molecular dynamics

Course | Annotation | Year | Semester |
---|---|---|---|

Methods of Mathematical Statistics | principles of estimation and hypothesis testing | 2nd | summer |

Advanced Simulations in Many-particle Physics | advanced methods of Monte Carlo and molecular dynamics and their applications to various problems (critical phenomena, complex molecules, non-equilibrium phenomena, transport coefficients, kinetic MC, growth processes, optimalization problems, quantum MC, ab initio simulations) | 2nd | summer |

Introduction to Plasma and Computational Physics I | basics of computational physics, introduction to plasma physics | 2nd | winter |

Introduction to Plasma and Computational Physics II | elementary processes in plasma, transport effects in plasma, advanced techniques of particle modelling, particle and fluid modelling in plasma physics and in plasma chemistry | 2nd | summer |

Computer Modelling of Biomolecules | drug design, searching and visualization of biomolecular structures | 2nd | summer |

Thermodynamics and Statistical Physics II | thermodynamic limit, Gibbs paradox, identical particles, quantum statistical ensembles, the classical limit, fluctuation theory, equivalence of statistical ensembles, ideal Bose and Fermi gas, interacting systems (virial expansion, critical phenomena, mean field approximation, the scaling hypothesis), transport phenomena, Boltzmann kinetic equation | 2nd | summer |

#### Quantum systems

Course | Annotation | Year | Semester |
---|---|---|---|

Quantum Theory of Molecules | Born-Oppenheimer and adiabatic approximation, Hückel method, Hartree, Hartree-Fock and Roothaan equations, semiempirical and ab initio methods, correlation energy, symmetry of molecules, intermolecular interactions, polarizability, vibrations of molecules, chemical reactivity | 2nd | summer |

Quantum Scattering Theory | theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, formal collision theory and methods for solving scattering problems, single-particle approach in atomic physics | 2nd | winter |

Theory of Collisions of Atoms and Molecules | advanced theory of atomic processes with applications in non-relativistic astrophysics and chemical physics, introduction to many-particle atomic and molecular theory, computational methods for collision processes and applications to collisions of electrons with atoms and molecules | 2nd | summer |

Group Theory and its Applications in Physics | group theory and representation theory for both finite and continuous (Lie) groups | 2nd | winter |

Quantum Mechanics II | identical particles and many-particle systems, angular momentum in quantum theory, mean field and theory of atoms and molecules, symmetry and conservation laws, time-dependent perturbation theory, density matrix and open systems | 2nd | summer |

#### Relativistic physics

Course | Annotation | Year | Semester |
---|---|---|---|

General Theory of Relativity | general relativity and its applications in astrophysics and cosmology | 2nd | summer |

Geometrical Methods of Theoretical Physics I | elements of topology, differentiable manifolds, tangent bundles, vector and tensor fields, affine connection, parallel transfer and geodesic curves, torsion and curvature, Riemann and pseudo-Riemann manifolds, Riemann connection, Gauss theory of surfaces, Gauss formula, Lie derivative, Killing vectors, exterior calculus, integration on manifolds, integrable densities | 2nd | winter |

Relativistic Physics I | tensor analysis, curvature and Einstein gravitational law, Schwarzschild solution of Einstein equations, black holes and gravitational collapse, astrophysics of black holes, general relativity in other branches of physics, linearised theory of gravitation, gravitational waves | 2nd | winter |

Foundations of Numerical Study of Spacetimes | basic numerical methods for solving Einstein equations of general relativity | 2nd | winter |

Geometrical Methods of Theoretical Physics II | Riemann geometry in terms of forms, Hodge theory, Lie groups and algebras, fibre bundles, geometry of gauge fields, two-component spinors | 2nd | summer |

Introduction to Analysis on Manifolds | foundations of differential geometry | 2nd | winter |

#### Particle physics

Course | Annotation | Year | Semester |
---|---|---|---|

Elementary Particle Physics | basic properties of elementary particles, models (SU(3), the eightfold way, the quark model, interactions (strong, electromagnetic, weak) and their unification | 2nd | winter |

Fundamentals of Electroweak Theory | phenomenological V-A theory of the weak interactions, idea of unification of weak and electromagnetic interactions, nonabelian gauge fields and the Higgs mechanism, Glashow-Weinberg-Salam standard model of electroweak interactions | 2nd | summer |

Quarks, Partons and Quantum Chromodynamics | quark model of hadrons, parton model and the deep inelastic scattering of leptons off nucleons, synthesis of mentioned models in the framework of the field theory | 2nd | summer |

Particles and Fields I | gauge theories | 2nd | winter |

Selected Topics on Quantum Field Theory I | path integral in quantum mechanics, functional methods and Green functions, Wick rotation and the partition sum, Berezin integral | 2nd | winter |

Software and data processing in particle physics I | overview of software used in particle physics | 1st | summer |

Software and data processing in particle physics II | simulations of particle collisions and thier passage through detector, statistics in evaluation of data acquired in modern detectors | 2nd | winter |

Neural Nets in Particle Physics | neural nets, training, error backpropagation, the Hessian matrix, regularization in neural network, Bayesian neural networks dual use of neural networks – approximations and decision |
2nd | winter |

#### Other

Course | Annotation | Year | Semester |
---|---|---|---|

Functional Analysis 1 | topological spaces, weak topology, vector integration, spectral theory | 2nd | winter |

Ordinary Differential Equations 2 | dynamical systems, Poincaré-Bendixson theory, Carathéodory theory, optimal control, Pontryagin maximum principle, bifurcations, stable, unstable and central manifolds | 2nd | winter |

Partial Differential Equations 3 | linear and nonlinear evolution equations, semigroup theory, asymptotic behaviour of the solutions to differential equations, pptimal control of evolution equations | 2nd | winter |

Theory of Nanoscale Systems I | independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods | 2nd | winter |

Modern Computational Physics I | genetic algorithms, wavelet transform | 2nd | winter |

Modern Computational Physics II | neural networks, advanced algoritms for molecular dynamics | 2nd | summer |

Statistical Methods in High Energy Physics | basic statistical methods frequently used in analysis of experimental data in high energy physics | 2nd | summer |

Theory of Nanoscale Systems I | independent fermion and boson models, Hartree-Fock theory for fermions and bosons, Brueckner-Hartree-Fock theory, density functional theory, quantum dots in a magnetic field, Monte Carlo methods | 2nd | winter |

## State exam

The (new) study programme Computational and Mathematical Methods in Physics is expected to start in 2015/2016. Detailed list of topics for the oral part of the state exam should appear on this page soon, in the mean time please see the faculty webpages.