This text presents an informal overview on how, in accordance with some deeply rooted principles of the philosophy of Alexander Grothendieck concerning the practice of mathematics, recent progress in anabelian arithmetic geometry led to the Inter-universal Teichm¨uller theory (IUT) of Mochizuki Shinichi. The new geometry of monoids furnished by IUT may be understood as the result of a seminal encounter between Grothendieck’s principle of resolving the tension between the discrete and continuous realms, on the one hand, and p-adic Hodge theory and height theory, on the other, and opens a new research frontier that goes beyond the Grothendieck geometry of rings-schemes by providing a unifying framework for Diophantine and anabelian arithmetic geometry

The article gives a brief overview of the results of the recovery signalvector by modules of measurements and by norms of orthoprojectors in finitedimensional Euclidean and unitary spaces, and in infinite-dimensional real space ℓ2. Three unsolved problems are formulated. Possible solutions are given, and theorems that give partial answers to the questions posed

The present paper is based on a lecture given by the author at the conference “Complex Analysis and Related Problems”, which was held from 29 June to July 4 in Kazan. The lecture was devoted to the Glicksberg conjecture and a related but independent assertion that the sums of certain ideals in a uniform algebra are never closed. This text contains several new formulations in comparison with the lecture and the existing literature.

A class of billiards is found, the geometry of which can change with a change in the energy of a ball moving on a «billiard table». Such billiards are called force or evolutionary. They make it possible to implement important integrable Hamiltonian systems (with two degrees of freedom) on the entire phase 4-dimensional space of the system at once. That is, simultaneously on all regular isoenergetic 3-dimensional surfaces. The author and V.V. Vedyushkina proved that force billiards implement the Euler and Lagrange integrable cases in the dynamics of a heavy body in three-dimensional space. It is found that these two well-known systems «billiard equivalent», although they have integrals of different degrees – quadratic (Euler) and linear (Lagrange).

The paper is devoted to an estimation of the index set of finitelky generated structures which have precisely one punctual presentation up to punctual isomorphisms. It is shown that this index set is Π03-complete

We study arithmetic properties of integer sequences counting amount of tilings of circular strips with two pieces consisting of equal cells. We also investigate recurrent sequences related to Pascal’s triangle