Topology: l2 methods in topology, low-dimensional manifolds, differential topology (Morse theory and surgery), classification of manifolds, hyperbolic geometries
Geometric group theory: random groups, coarse embeddings into Hilbert spaces, Kazhdan's Property (T)
Differential geometry: foliations, mapping class groups, exotic differential structures on manifolds
Quantum relativistic theory of information: what happens with the wave function of particle when it's falling into a black hole? How the curvature of space influence the quantum mechanics in this space?
Special and general relativity: Thomas-Wigner precession, flow of time in the curved space, black holes, geometry of the universe
Cubulating random groups in the square model (preprint) We prove that a random group in the square model at densities < 3/10 with overwhelming probability acts freely and cocomplactly on CAT(0) cube complex. First we construct a space with walls on which a random group acts metrically properly and then using the Sageev contruction we obtain desired CAT(0) complex. Finally we apply the Agol's Virtual Haken theorem to conclude that such random groups are residually finite.
The square model for random groups accepted to Colloqium Mathematicum We introduce a new random group model called the square model: we quotient a free group on n generators by a random set of relations, each of which is a reduced word of length four. We prove, as in the Gromov density model, that for densities > 1/2 a random group in the square model is trivial with overwhelming probability and for densities <1/2 a random group is with overwhelming probability hyperbolic. Moreover we show that for densities 1/4 < d < 1/3 a random group in the square model does not have Property (T). Inspired by the results for the triangular model we prove that for densities < 1/4 in the square model, a random group is free with overwhelming probability. We also introduce abstract diagrams with fixed edges and prove a generalization of the isoperimetric inequality. .
Half-page derivation of the Thomas precession(with Andrzej Dragan) - American Journal of Physics 8/2013 Composition of two non-collinear
Lorentz boosts, results in a Lorentz transformation that is not a pure boost but a composition of a boost and a spatial rotation, known as the Wigner rotation. As a consequence, a body moving on a curvilinear trajectory undergoes a rotational precession, that was first discovered by Thomas. In the vast majority of textbooks this phenomenon is either omitted or described with very
sophisticated mathematical tools, such as gyrogroups, associative-commutative groupoids, etc. Here we present a half-page derivation of the Thomas precession formula using only basic vector operations. Our approach is not only simple and clear, but also builds a better physical intuition of this relativistic effect.
Coarse embeddings of the discrete metric spaces into a Hilbert spaces (only in Polish) - my Bachelor thesis In this paper I have fixed some errors in the preprint of the book "Large scale geometry" written by Nowak and Yu. This work partially answers the question: "when coarse embeddability of the discrete metric space X into some Hilbert space implies that X has Property A?".
Problems I am working on:
Determining sharp density thresholds for Property (T) and Haagerup Property in random groups It can be easily proven that there exist sharp density thresholds for Property (T) and Haagerup Property in the Gromov random group model and the square model. However precise values of this thresholds are unknown (we know only some estimates). I would like to determine exact values of that thresholds.
Find a geometric criterion for residual finiteness There is an open question: Are the hyperbolic groups residually finite? We know from papers of Gromov and Ollivier that random groups for densities < 1/2 are with overwhelming probability infinite and hyperbolic. Hence I will try to prove that ranodm groups are residually finite. I will try to conclude the residuall finiteness from the fact that a group acts on a space with some additional geometric structure.
l2-Homologies of random complexes Consider a flag complex, such that it's 1-skeleton is a random graph in the G(n,p) model. Khale determined for which values of p such a flag complex has non trivial homolgies with coefficients in Q. I would like to achieve similar results for l2-homologies.
First l2-Betti numbers of random groups I would like to prove that random groups in the Gromov density model have vanishing first l2-Betti number w.o.p. for all densities. Moreover, I want to prove that for densities > 1/4 random groups in the square model have have vanishing first l2-Betti number w.o.p..
Links to mathematical pages with content connected with my research: