UNCE - seminar junioru univerzitniho centra Math MAC

UNCE

University Center for Mathematical Modeling, Applied Analysis and Computational Mathematics

8:30 Opening

8:30 Vlasák Václav - Haar meager sets and compact sets

8:50 Cúth Marek - Large separated sets of unit vectors in Banach spaces

9:10 Pokorný Dušan - Removable sets for convex functions

9:30 Coffee Break

9:40 Kalousová Klára - Water generation and transport through the high-pressure ice layer of Titan

10:00 Souček Ondřej - Powering prolonged hydrothermal activity inside Enceladus

10:20 Tůma Karel - Thermomechanical phase-field model for martensitic transformation

10:40 Coffee Break

10:50 Pavelka Michal - Entropy production maximization vs. gradient dynamics

11:10 Vlasák Miloslav - Flux reconstructions and dependence of efficiency estimate on polynomial degree

11:30 Vejnar Benjamin - Fixed points of group actions

11:50 Conclusion

Abstracts

Vlasák Václav: Haar meager sets and compact sets. The notion of Haar meager sets is topological counterpart

of measure theoretic notion of Haar null sets. Collections of those sets are considered as collections of small sets.

So, there arise a natural question, whether other sets, which are generally considered as small are also Haar meager.

I would like to speak about the problem, whether compact subsets of nonlocally compact Polish groups are Haar

meager.

Cúth Marek: Large separated sets of unit vectors in Banach spaces. I will talk about the problem of finding

unit vectors (xi) in a Banach space such that kxi−xjk > 1 for every distinct i, j and about related problems. Finally,

I will mention joint results with Benjamin Vejnar and Ondrej Kurka which concern the case when the Banach space

in consideration is nonseparable and of the form C(K), that is the Banach space of continuos funcitons.

Pokorný Dušan: Removable sets for convex functions. It is well known and easy to see that a function F

defined on a convex set K is convex if and only if it is locally convex (i.e. every point has a convex neighborhood

on which F is convex). Similar to other locally determined classes of functions (analytic functions being a prominent

example) it is natural to ask which conditions must be satisfied for a set H in order to imply an existence of a

continuous (or locally Lipschitz) non-convex function, which is locally convex on the complement of H. I will present

some partial answers to this question, which are mostly based on a joint work with M. Rmoutil.

Kalousová Klára: Water generation and transport through the high-pressure ice layer of Titan.

Ganymede and Titan, the largest icy moons in the solar system, are very similar in mass and radius but their

radial mass distribution is quite different. Both moons harbor a deep liquid water ocean which is separated from the

silicate interior by a layer of high-pressure (HP) ice. This ice layer seems to prevent a direct contact of water with

silicates that is essential from the astrobiological point of view. In our recent study, we found that melting at the

interface with silicates and the subsequent water transport through the HP ice layer might have allowed exchange of

material between silicates and ocean in the case of Ganymede. We now concentrate on Titan’s HP ice layer which is

expected to be a few hundred kilometers thinner than that of Ganymede.