Relativistický seminář
Ústavu teoretické fyziky


Seminář se koná v úterý ve 13:10 v posluchárně ÚTF MFF UK
v 10. patře katedrové budovy v Tróji, V Holešovičkách 2, Praha 8


Příští semináře:

V nejbližší budoucnosti nejsou plánované žádné semináře.


Minulé semináře:

12. října 2021
Universality of curvature invariants in critical vacuum gravitational collapse
Tomáš Ledvinka, Anton Khirnov
ÚTF
(seminář v angličtině)

We consider several families of asymptotically flat initial data which, similar to the well-known Choptuik's discovery, can be fine-tuned between dispersal into empty space and collapse into a black hole. We discuss the applied methods and then we show that near-critical spacetimes exhibit behavior similar to scalar-field collapse: For different families of initial data, we observe universal "echoes" in the form of irregularly repeating, approximate, scaled copies of the same piece of spacetime.


Záznam:
19. října 2021
General Relativity on AdS_3 and the AKNS hierarchy
Kristiansen Lara
Department of Physics, University of Santiago de Chile
(seminář v angličtině)

It is shown that the Ablowitz-Kaup-Newell-Segur (AKNS) integrable hierarchy can be obtained as the dynamical equations of three-dimensional General Relativity with a negative cosmological constant. This geometrization of AKNS system is possible through the construction of novel boundary conditions for the gravitational field. These are invariant under an asymptotic symmetry group characterized by an infinite set of AKNS commuting conserved charges. Gravitational configurations are studied by means of SL(2,R) conjugacy classes. Conical singularities and black hole solutions are included in the boundary conditions.


Záznam:
26. října 2021
Polarisation of reflected X-ray emission from accretion discs in Active Galactic Nuclei
Jakub Podgorný
Astronomical Institute of the Czech Academy of Sciences, Ondřejov/Spořilov
(seminář v angličtině)

We present the theoretical spectrum and polarisation properties of X-ray emission from accretion discs in active galactic nuclei (AGN) including relativistic effects. The reflected radiation from an accretion disc illuminated by a hot corona above is computed using the Monte Carlo radiative transfer code STOKES. The structure of the disc, modelled as a constant density slab, has been computed with the code for optically thick and hot photo-ionized media TITAN. The resulting simulated spectra locally emitted by the accretion disc in 0.1–100 keV energy band are compared to the results obtained by various, published radiative transfer codes. In addition to these, the STOKES code also provides the polarisation degree and polarisation angle of the outcoming radiation that depend on the geometry of scattering and other key parameters such as the ionisation of the disc. Further, we use the relativistic code KYNSTOKES to account for relativistic effects on radiation near a black hole and to compute the total disc emission as would be detected by a distant observer. In this way it is then possible to predict X-ray spectral and polarisation properties of an accretion disc in AGN and their dependence on various parameters of the system (spin of the black hole, geometry of the corona, ionization of the accretion disc, observer inclination). This is fundamental for the interpretation of AGN spectra and polarisation obtained by the upcoming X-ray polarimetric missions, such as NASA IXPE mission or Chinese-lead eXTP mission.


Záznam:
5. listopadu 2021
Parametrization of black-hole spacetimes and the Blandford-Znajek mechanism for the general slowly rotating black holes
Dr. Roman Konoplya
Research Centre for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava
(seminář v angličtině)

We will briefly review the general parametrization of the axially symmetric black hole spacetime, which is similar in spirit to the parametrized post-Newtonian approximation, but valid in the whole space outside the event horizon, including the near horizon region. Then, we consider the Blandford-Znajek process of electromagnetic extraction of energy from a general axially symmetric asymptotically flat slowly rotating black hole given in the parametrized form. Using the general parametrization of the black-hole spacetime we construct formulas for the flux of the magnetic field and the rate of energy extraction, which are valid not only for the Kerr spacetime, but also for its arbitrary axially symmetric deformations. We show that in the dominant order these quantities depend only on a single deformation parameter, which relates the spin frequency of a black hole with its rotation parameter.


Záznam:
9. listopadu 2021
TEOBResumS: an Effective-One-Body waveform model for coalescing black hole binaries
Angelica Albertini
Prague Relativistic Astrophysics group, Astronomical Institute, Czech Academy of Sciences
(seminář v angličtině)

The coalescence of a compact binary system generates gravitational waves. In order to detect the signal, fast and accurate waveform templates are needed. Though numerical relativity simulations provide exact results, they come at a high computational cost and cannot cover the whole parameter space. Semi-analytical models allow instead a fast waveform generation and are thus of fundamental relevance from an experimental point of view. Different families of these models come from the Effective-One-Body (EOB) formalism, which reformulates the two-body problem in GR as the motion of a particle with the reduced mass of the system in an effective metric. We will briefly review the building blocks of the EOB approach and then give details about TEOBResumS, a state-of-the-art EOB model that yields complete waveforms for nonprecessing coalescing black hole binaries on quasi-circular orbits.


Záznam:
16. listopadu 2021
Effect of magnetic advection on spectral characteristics of thin accretion disks
Radek Vavřička
ÚTF / ÚČJF, MFF UK
(seminář v angličtině)

Accretion disks of massive central objects are well established in theory and observation as efficient engines of potential-to-radiant energy conversion in our universe. Concretely in the case of a black hole as the massive central object, accretion disk serves as a transmitter for an otherwise not (meaningfully) radiating object, signalling her existence and parameters (due to inescapably relativistic nature of the disk) for an interested distant observer. Theories of accretion disks exist for a variety of disk classes; we are interested in thin, optically thick disks with an inner edge; "thin disks" for short. The eminent theory in this class being the relativistic Novikov-Thorne theory, operating on a fixed Kerr spacetime background and establishing the thin disk composed of magnetized ionized gas, with viscosity prescribed phenomenologically, stemming from turbulences. The spectrum of incoming radiation carries a signature of black hole & accretion disk parameters, most notably the black hole mass, spin and the disk's accretion rate. Fitting a spectrum predicted by the Novikov-Thorne theory to the observed spectrum ought to determine the parameters, for each observation in many. However, after doing so, a clear discrepancy appeared - when the spin and accretion rate extracted from multiple observation were plotted against each other, a decay of spin as depending on accretion rate appeared, when it should be constant. We sought to explain this discrepancy by modifying the standing Novikov-Thorne theory, by assuming that not all energy viscously generated is immediately radiated away, but instead a portion of it is transferred inward, an advective process mediated by magnetic field.


Záznam (bohužel velmi nekvalitní - omlouvám se):


Prezentace:
23. listopadu 2021
Gravitational lensing by massive objects in a dark matter halo
Michal Karamazov
ÚTF
(seminář v angličtině)

Among many other things, gravitational lensing can be used to survey the matter content of galaxy clusters. However, recent studies showed major discrepancy in halo substructure lensing efficiency between observed clusters and those from cosmological simulations. The role of halo substructures in gravitational lensing thus asks for deeper investigation. We take a bottom-up approach, starting with detailed examination of the simplest model composed of dark matter halo with Navarro—Frenk—White profile and single perturbing point mass. We study critical curves and caustics, discovering unexpected richness of caustic transitions. Next, we analyze geometry of images including implications for weak-lensing quantities. Then we proceed further to more realistic models of the galaxy cluster with multiple point mass perturbations and eventually with truncated elliptical NFW subhalos.


Záznam:
29. listopadu 2021
13:10-16:30
Workshop "postdoc 2022"
(seminář v angličtině)

Série šesti krátkých seminářů kandidátů na pozici postdoca na ÚTF.

13:15 Matthew Terje Aadne
(University of Stavanger)
13:45 Arpita Mitra
(Indian Institute of Science Education & Research Bhopal)
14:15 Carlos Peón-Nieto
(University of Salamanca)
15:00 Jorge G. Valcarcel
(University of Tartu)
15:30 Andriú Conroy
(Dublin City University)
16:00 Luca Ciambelli
(Université Libre de Bruxelles)

Příspěvky budou předneseny přes Zoom:
https://cesnet.zoom.us/j/92851596816?pwd=YnUwZTVYYjA5enBLTk9RcjFpUEJXQT09.

Workshop bude živě promítán v přednáškové místnosti ÚTF.

7. prosince 2021
Nonlinear evolution of oscillations in thin accretion disks
Dr. Jiří Horák
Prague Relativistic Astrophysics group, Astronomical Institute of the Czech Academy of Sciences
(seminář v angličtině)

Due to general relativistic effects, accretion disks surrounding black holes can host self-trapped oscillations close to their inner edges. This talk is devoted to their nonlinear evolution. We will discuss change of the trapping region and oscillation frequency of the g-modes when the amplitude of the oscillations is significant. When non-linearities in the hydrodynamic equations are taken into account, the g-modes are coupled to other oscillation modes. We will also discuss their impact on the original oscillations.


Záznam:
14. prosince 2021
Schwarzschild-Bach black holes in the Eddington-Finkelstein-like coordinates
Šimon Knoška
ÚTF
(seminář v angličtině)

We analyse spherically symmetric solutions in the quadratic gravity which represents the GR natural modification including quadratic corrections in its action. We extend previous works employing the Eddington–Finkelstein-like coordinates. This allows to find the explicit solution in the form of power series, analogously to the previous approach which used the conformal-to-Kundt metric form, however, our results provide more direct transition to the classic Schwarzschild-like coordinates and more naturally elucidate physical and geometrical properties of the resulting solution such as basic aspects of geodesic motion, the quadratic gravity mass contribution, or presence of the cosmological horizon.


Záznam:
21. prosince 2021
Action-angle coordinates for black-hole geodesics
Vojtěch Witzany, PhD
Relativity group @ University College Dublin
(seminář v angličtině)

Imagine there is a way to solve any set of equations of motion by transforming them into trivial ones. That is what action-angle (AA) coordinates do for Hamiltonian mechanics. Of course, the issue is that by doing so, you just move the problem from solving the equations of motion to finding the transformation. Nevertheless, even a perturbative transformation to AA coordinates represents an efficient and systematic way of storing and representing the solution. Additionally, AA coordinates are an extremely useful starting point for perturbation theory of the original equations of motion. In this talk I show how transformations to and from action-angle coordinates can be found for black-hole geodesics and how that simplifies gravitational-wave inspirals both in the extreme and comparable mass ratio regimes.


Záznam:
4. ledna 2022
Gravitational multipole moments for asymptotically de Sitter spacetimes
Sk Jahanur Hoque, PhD
ÚTF
(seminář v angličtině)

We discuss a prescription to compute the gravitational multipole moments of compact objects for asymptotically de Sitter spacetimes. Our prescription builds upon a recent definition of the gravitational multipole moments in terms of Noether charges associated to specific vector fields, within the residual harmonic gauge, dubbed multipole symmetries. We derive the multipole symmetries for spacetimes which are asymptotically de Sitter. We apply our prescription to the Kerr-de Sitter black hole and compute its multipole structure. Our result recovers the Geroch-Hansen moments of the Kerr black hole in the limit of vanishing cosmological constant.


Záznam:
11. ledna 2022
Energy conditions, physicality, and reverse-engineered solutions
Sebastian Schuster, PhD
ÚTF
(seminář v angličtině)

Many familiar metrics of general relativity have been achieved by integrating the Einstein equations for a particular source stress-energy tensor. In other cases, physical considerations lead to a motivation for studying metrics of a particular form; here, then, the question is to derive the stress-energy tensor from a given metric. While mathematically simpler—as differentiation is simpler than integration—this opens a long list of questions which stress-energy tensors should be considered physical. An early heuristic for such an evaluation were so-called energy conditions. In particular for many science-fiction-inspired metrics, attempts to produce these without violating the (point-wise) energy conditions has become a staple of the search for new reverse-engineered solutions. Here, I will present a quite general proof for the violation of the null energy condition in "warp field" space-times, and describe a new engineered solution, "tractor beams". These can serve as a cautionary tale of what to expect of energy condition violations, and I will try to ask—and partially answer: "Quo vadis, energy conditions?"


Záznam:
8. února 2022
The effect of peculiar velocities on the null deceleration parameter and other tales
Dr. Jessica Santiago
Department of Astrophysics, Astronomy and Mechanics, Aristotle University of Thessaloniki
(seminář v angličtině)

I will introduce and discuss the concepts of bulk flows and peculiar velocities, the importance of which has progressively been shown to be more significant in modern cosmology. Alongside this, I will also discuss recent theoretical developments of a new way of expanding luminosity distances in terms of null geodesics, allowing us to obtain null analogues of the well-known expansion and deceleration parameters. I will wrap up by connecting both topics and analyzing the effect of peculiar velocities in the null deceleration parameter, as well as open a discussion on the impacts of such results on the existence of dark energy.


Záznam:
15. února 2022
Fun with slowly rotating black holes: generalized Lense-Thirring spacetimes
Dr. David Kubizňák
ÚTF, back again
(seminář v angličtině)

The Lense-Thirring spacetime describes a 4-dimensional slowly rotating approximate solution of vacuum Einstein equations valid to a linear order in rotation parameter. It is fully characterized by a single metric function of the corresponding static (Schwarzschild) solution. We shall discuss a generalization of the Lense-Thirring spacetimes to the case that is not necessarily fully characterized by a single (static) metric function. This generalization lets us study slowly rotating spacetimes in various higher curvature gravities as well as in the presence of non-trivial matter such as non-linear electrodynamics. In particular, we construct slowly multiply-spinning solutions in Lovelock gravity and notably show that in four dimensions Einstein gravity is the only non-trivial theory amongst all up to quartic curvature gravities that admits a Lense-Thirring solution characterized by a single metric function. The "magic square" upgraded version of our ansatz can be cast in the Painlevé-Gullstrand form (and thence is manifestly regular on the horizon) and admits a tower of exact rank-2 and higher rank Killing tensors that rapidly grows with the number of dimensions.


Záznam:
22. února 2022
No seminar
enjoy the 22022022 day!
(seminář v angličtině)
1. března 2022
Carter constant, photon region and black hole shadow in a general axially symmetric stationary spacetime in plasma
Dr. Barbora Bezděková
ÚTF
(seminář v angličtině)

Problem of light rays propagating around compact relativistic objects (e.g. a black hole) is usually solved with a vacuum background. However, in real astrophysical applications a medium (plasma) surrounding a vicinity of the black hole has to be considered. Studies describing ray trajectories around relativistic objects in plasma typically focus on particular examples and discuss concrete spacetime and plasma effects. This problem can be effectively solved as the separation of the Hamilton-Jacobi equation. To be able to find a solution of such system, a Carter constant has to exist. In our work, we state general conditions for both the axially symmetric stationary metric and plasma density which guarantee that the Carter constant exists. General expressions for a photon region and black hole shadow for light rays penetrating around an axially symmetric stationary object in plasma are further derived. Moreover, the obtained general relations are applied to find the boundary of photon region and shadow for particular examples, such as Kerr metric or Melvin metric.


Záznam:
8. března 2022
Gravity of static thin discs: on Laplace transform of Bessel-function products
Petr Kotlařík
ÚTF
(seminář v angličtině)

Thin discs are important structures in astrophysics. Whether treated in Newton's theory or in general relativity, in the static and axially symmetric case they generate potential determined by the Laplace equation. Though clear in principle, its solution is only known explicitly for several surface-density profiles. We show how to derive close-form potentials for the densities composed of powers of the radius. The method, due to Conway (2000), rests on the Laplace transform of products of Bessel functions. Finite as well as infinite discs will be included, possibly empty at their centre (i.e. annular).


Záznam:
15. března 2022
Asymptotic characterization of spacetimes with positive cosmological constant in all dimensions
Dr. Carlos Peón-Nieto
ÚTF
(seminář v angličtině)

We review the formalisms which allow to pose an asymptotic Cauchy problem in all dimensions for Lambda positive vacuum spacetimes: the Friedrich conformal field equations in four spacetime dimensions and the Fefferman-Graham formalism in higher dimensions. We employ the initial data to asymptotically characterize Kerr-de Sitter and related spacetimes.


Záznam:
22. března 2022
How does an Unruh-DeWitt detector respond near an extremal black hole?
Dr. Aindriú Conroy
ÚTF
(seminář v angličtině)

There is no well-defined notion of a particle in quantum field theory in curved spacetime due to the lack of global symmetries. The standard procedure in quantum field theory is to treat fields rather than particles as the fundamental object of interest. Nevertheless, in a seminal 1976 paper by W. G. Unruh, an operational meaning was given to the particle concept by examining the absorption and emission of field quanta by a two-level atom. This is the so-called Unruh-DeWitt detector and, in this operational sense, we say a particle is what a particle detector detects! Computing the response of an Unruh-DeWitt detector in black hole spacetimes is technically challenging – a fact which has largely limited the literature to stationary detectors. In this talk, we consider the response of an Unruh-DeWitt detector near an extremal charged black hole, modelling the near-horizon region of this extremal spacetime by the Bertotti-Robinson spacetime. The advantage of employing the Bertotti-Robinson limit is that the two-point functions for a massless scalar field are obtainable in closed form for a number of quantum states of interest, allowing us to explore the full gamut of the parameter space for a broader range of stationary and non-stationary trajectories. Particular attention will be paid to the thermalization of the detector, the anti-Unruh effect, and the anti-Hawking effect. If time allows, I will then move on to discussing some unpublished results on the open question of how to define surface gravity and temperature in dynamical spacetimes by shifting focus to a detector in a cosmological spacetime with an evolving horizon.


Záznam:
29. března 2022
Uniqueness of extremal isolated horizons and their identification with horizons of type D black holes
Dávid Matejov
Matematický ústav AV ČR
(seminář v angličtině)

We investigate axisymmetric extremal isolated horizons (EIHs). They are described by an induced metric characterized by a single function. This function is subject to particular constraints following from Ricci identities in the Einstein-Maxwell theory. At first, we restrict our attention to asymptotically flat spacetimes. In the extremal case the solution is unique and depends on 5 independent parameters with direct geometrical or physical interpretation. The solution is possible to identify with extremal horizons in exact electrovacuum Plebański-Demiański spacetimes, that are all (double aligned) black holes of algebraic type D without a cosmological constant. Several interesting subclasses include accelerating extremely charged Reissner-Nordström black hole (C-metric) or extremal accelerating Kerr-Newman. As a natural extension, I will generalize the previous results to asymptotically (anti-) de Sitter spacetimes with non-zero cosmological constant and identify the EIHs geometry with the corresponding horizon geometry of a black hole in a special case of the famous Kerr-Newman-NUT-(A)dS metric.


Záznam:
5. dubna 2022
A look at the geometry of the 5-dimensional charged rotating black hole
doc. Igor Khavkine
Oddělení algebry, geometrie a matematické fyziky, Matematický ústav AV ČR
(seminář v angličtině)

The Einstein-Maxwell equations for a charged rotating black hole in 5 dimensions with two equal-magnitude angular momenta has been reduced to a system of non-linear ODEs and studied numerically by Kuntz et al. We study some geometric aspects of these solutions, confirming that they have regular horizon and null infinity. In addition, we rule out the possibility that the Weyl tensor of these spacetimes has special algebraic type in the bulk, while verifying the "geometric horizon" conjecture by showing that the Weyl tensor reduces to Type II on the horizon and to Type D on the bifurcation sphere.


Záznam:
12. dubna 2022
Obscuring tori in active galactic nuclei: observations and simulations
Prof. Elena Bannikova
Institute of Radio Astronomy, National Academy of Sciences of Ukraine, V.N. Karazin Kharkiv National University
(seminář v angličtině)

The recent paradigm of AGNs is to explain the differences between type I and II by the different orientation of an optically thick torus relative to the observer. The dynamics of the dusty clouds in such a torus was unclear due to the relative small scale of these objects. The unprecedented resolution of ALMA has recently allowed us to improve significantly the information about dynamics of clouds in the obscuring tori for some AGNs. On this ground, we present our N-body simulations of the toroidal structure of clouds moving around a supermassive black hole (SMBH). We also apply ray-tracing algorithm to take into account effects of obscuration of the central engine by the clouds. As the final result, we obtain the velocity and temperature maps which allow us to interpret the VLTI and ALMA observations. We also estimate the SMBH masses in the nearest Sy2 galaxies and provide a new explanation of the counter-rotation of the torus in NGC 1068.


Záznam:
19. dubna 2022
Interplay between spectral energy distribution and spectral line profiles in the context of gappy accretion discs
Marcel Štolc
Skupina relativistické astrofyziky, Astronomický ústav AV ČR
(seminář v angličtině)

Both spectral energy distribution of the continuum and the spectral lines offer a way to measure and model parameters of the inner and outer region of the black hole accretion disc. We introduce a perturbative term in the orbital plane of the accretion disc and assume different scenarios pointing towards a gappy accretion disc setup. Studying the radiative effects of the perturber, we discuss the application of the methods used to constrain the parameters of the system.


Záznam:
26. dubna 2022
EMRI from canonical perturbation theory
Lukáš Polcar
Skupina relativistické astrofyziky, Astronomický ústav AV ČR
(seminář v angličtině)

By employing the methods of canonical perturbation theory we can find an approximate analytical solution to a Hamiltonian system. We applied this approach to the geodesic equation in the spacetime describing a black hole superposed with a ring-like source. The solutions to the geodesic equation are then used to compute gravitational wave fluxes using the quadrupole formalism. This allows us to compute the extreme mass ratio inspiral (EMRI) in our spacetime by adiabatically evolving the constants of motion which parametrize the analytical solution of the geodesic equation.


Záznam:
3. května 2022
Higher-dimensional Robinson-Trautman spacetimes coupled to conformally invariant electrodynamics
David Kokoška
Oddělení algebry, geometrie a matematické fyziky, Matematický ústav AV ČR
(seminář v angličtině)

We present the complete family of higher-dimensional Robinson-Trautman (RT) spacetimes sourced by (aligned) conformally invariant non-linear electrodynamics with an arbitrary cosmological constant. We also briefly compare the obtained results with the known ones in the case of RT spacetimes with linear Maxwell field in higher dimensions. Subsequently, some properties of a static subclass are also investigated. It contains dyonic black holes with various horizon geometries and different asymptotics. Finally, we discuss the special case of stealth solutions.


Záznam:
10. května 2022
Holographic reconstruction of asymptotically flat spacetimes
Finnian Gray
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
(seminář v angličtině)

We present a "holographic" reconstruction of bulk spacetime geometry using correlation functions of a massless field living at the "future boundary" of the spacetime, namely future null infinity I+. It is holographic in the sense that there exists a one-to-one correspondence between correlation functions of a massless field in four-dimensional spacetime M and those of another massless field living in three-dimensional null boundary I+. The idea is to first reconstruct the bulk metric by "inverting" the bulk correlation functions and re-express the latter in terms of boundary correlators via the correspondence. This effectively allows asymptotic observers close to I+ to reconstruct the deep interior of the spacetime using only correlation functions localized near I+.


Záznam:
17. května 2022
Self-intersecting marginally outer-trapped surfaces in black holes
Kam To Billy Chan
Department of Physics and Physical Oceanography, Memorial University, St. John’s, Newfoundland and Labrador, Canada
(seminář v angličtině)

The advent of gravitational wave detectors had facilitated a constant stream of black hole merger observations. Despite this, black hole mergers are not fully understood. The details of the two apparent horizons becoming one is unclear due to the non-linear nature of the merger process. Recent numerical work has shown that there is an appearance of self-intersecting marginally outer-trapped surfaces (MOTS) during the black hole merger [Pook-Kolb et. al. arXiv:1903.05626]. Following papers have found similarly behaving MOTS in a simpler and static scenario, that of a Schwarzschild black hole, where a seemingly infinite number of self-intersecting MOTS were found [Booth et. al., arXiv:2005.05350]. This talk introduces new phenomena that occur in the presence of an inner horizon. For Reissner-Nordstrom and Gauss-Bonnet black holes, we find that the maximum number of self-intersections becomes finite with the MOTS parameter space deeply dependent on the interior structure of the black hole and in particular the stability of the inner horizon [Hennigar et. al., arXiv:2111.09373].


Záznam:
10. června 2022
PÁTEK OD 11:00
The strong cosmic censorship conjecture: The fall and the rise
Dr. Sumanta Chakraborty
School of Physical Sciences, Indian Association for the Cultivation of Science (IACS), Kolkata
(seminář v angličtině)
Deterministic nature of general relativity abhors existence of Cauchy horizons through the strong cosmic censorship conjecture. I will first discuss the recent claims, invalidating the conjecture, and then I will describe how it is again restored to its former glory.
Zoom link

Předchozí ročníky:


Jiří Bičák                                                                                                  Oldřich Semerák


© 24. května 2022; Oldřich Semerák <semerak@mbox.troja.mff.cuni.cz>
© 25. června 2022; vygenerováno pomocí aplikace seminar, verze 2.04 (2003-09-02); správce <Pavel.Krtous@utf.mff.cuni.cz>