Abstracts of lectures

Abstract: Cantor sets appear naturally in smooth, holomorphic, and Hamiltonian dynamics, number theory, spectral theory, and many other areas of mathematics. The class of dynamically defined Cantor sets is characterized by strong self-similarity properties. The standard “one-third” Cantor set is an example of a dynamically defined Cantor set. In this mini-course we will discuss the most important properties and examples of dynamically defined Cantor sets, including their fractal dimension. We will see how those sets appear in the problems related to homoclinic tangencies, continued fraction expansions, and spectra of one-dimensional quasicrystals. Questions related to sums and intersection of Cantor sets will rise naturally in this way. Finally, several open problems and conjectures in the field will be formulated. 

Lecture 1. Dynamically defined Cantor sets.
   Definition and examples of Cantor sets and other fractals. Iterated function systems. Dynamically defined Cantor sets. Examples (tent map, bounded type continued fraction sets, homogeneous and affine Cantor sets). Hausdorff and box counting dimension. Moran’s formula. 
Lecture 2. Sums of Cantor sets. 
   Problems related to sums and intersection of Cantor sets in dynamics, number theory, and spectral theory. Sum of the standard Cantor set with itself. Thickness of a Cantor set. Newhouse Gap Lemma. Palis’ Conjectures on sums of Cantor sets. Some open questions. 
Lecture 3. Cantor sets in Spectral Theory. 
   Quasicrystals. Mathematical models of quasicrystals. Ergodic discrete Schrodinger operators, examples. Fibonacci Hamiltonian and its spectrum. Operators with separable potentials and sums of Cantor sets. Known results and open questions. 

The mini-course is 4 lectures long and created to be an elementary introduction to C_0-semigroup theory for experts in nonlinear dynamical systems.

It will be shown how to make a linear space from a manifold without local coordinates, and how to make linear function from a nonlinear function without linearization. Basic notions of linear functional analysis (linear spaces, linear operators) will be reminded. The definition of a bounded linear operator will be given and the difference with the definition of bounded nonlinear function will be shown. In particular, it will be shown that for a linear operator being bounded is equivalent to being continuous and being uniformly continuous. The definition of C_0-semigroup will be given, and it will be shown that it is a particular case of the definition of a flow in theory of dynamical systems. The connection between C_0-semigroups and evolution partial differential equations will be presented, and also it will be shown how to find approximate solutions to that equations with the methods of functional analysis. To understand the lectures it is enough to have mathematical background on the level of the first year of study at the Bachelor's programme in mathematics.

The development of genetic engineering methods has led to the creation of synthetic genetic circuits that can be introduced into bacteria. In 2000, two works were published: in one, a simple switch of two genes was built, i.e. a system with three stationary states, of which only two are stable; in the second, an oscillator in the form of a ring of three genes was proposed and conditions were found for the formation of a stable limit cycle. Both circuits can be used as basic elements in real genetic networks. In the first lecture, the main reactions between the elements of the genetic network will be considered: genes, promoters, RNA, regulatory proteins; systems of differential equations for a switch and an oscillator will be constructed, and bifurcations of these systems will be demonstrated.

The effectiveness of synthetic genetic circuits for regulating the dynamic behavior of an ensemble of cells depends on the presence of interaction between them, which is not so easy to construct, since the physical characteristics of the intracellular medium where genetic reactions occur, limit the possibilities for intercellular interaction. One of the variants of the exchange of regulatory factors, borrowed from bacteria, will be considered in the second lecture using, by way of example, systems of ordinary differential equations describing the formation of structures and synchronization of oscillations in the cell population.

The third lecture will be devoted to mathematical modeling of one mechanism for generating a variety of dynamic modes in a system of coupled genetic circuits.

The purpose of this series of lectures is to explain some basic homotopy invariants of topological spaces and illustrate their calculations and applications in different areas of mathematics.
We will consider the following topics:
1) The concept of homotopy. Homotopy equivalences and homotopy type. Retracts and deformation retracts.
2) The fundamental group and higher homotopy groups of a topological space.
3) A long exact sequence of homotopy groups of a pair of topological spaces.
4) Serre fibrations. Long exact sequence of homotopy groups for Serre fibrations.
5) Compact open topologies. Homotopies as paths in functional spaces.
6) Cell complexes.
7) Seifert - van Kampen theorem. Calculation of the fundamental groups of cell complexes.
8) The construction of a compact topological space with a given finitely presented fundamental group.
The course is designed for students familiar with the basics of mathematical analysis, general topology and group theory.

Our Earth is in continuous motion and change, and that may often lead to natural disasters. Every now and then, radio and television give reports on floods, tropical cyclones, earthquakes, tsunamis, catastrophic heat, drought, asteroid falls and other events involving human casualties and numerous destruction.       In this course, we will focus on natural disasters occurring in the aquatic environment (oceans, seas, rivers and lakes). Thus, only in December 2018, occurred the events that strike in their size. So, on December 11, a landslide descended into the Bureya River (the Far East of Russia), leading to the river rise in places up to 90 m. The river stop would have caused disastrous consequences at the Bureyskaya hydroelectric power station, therefore, it was necessary to carry out blasting operations to clear the riverbed. Fortunately, this event occurred in uninhabited places and did not lead to human deaths. On December 22, there was a catastrophic eruption of the Anak Krakatau volcano in Indonesia, which led to the large wave formation and human casualties. The maximum water rise on the island closest to the volcano was 85 m. On December 3, 2019, the port of Korsakov on Sakhalin was flooded. Similar examples of floods can be given in our region as well. In 1896 in Nizhny Novgorod (near the Fair city district) and in 1908 in Moscow (Red Square city area) it was possible to move only by boat. All these events indicate the importance of catastrophic event prediction, which can be done by using computer models based on reliable theoretical models of the phenomenon.

The lectures will give an overview of various disasters in the aquatic environment occurring in different parts of the world. We will also discuss several basic idealized models allowing us to assess the natural disaster characteristics. Several test tasks aiming at the quick water disaster forecast will be offered to solve. In conclusion, the ways how the prognostic models of the phenomena work nowadays will be demonstrated.

In the first lecture, I'll present all without equations, (accessible also for high-school students),

in the second lecture will explain in simple terms basics of circle maps and corresponding properties of ODEs, maybe if there will be time  will introduce how one describes collective synchronization for many oscillators.

In these lectures we aim to provide an overview on how variational methods can be applied to the study of the dynamics of Hamiltonian systems. In particular, we shall describe the main ideas behind what is nowadays called Aubry-Mather theory, which allows one to construct a plethora of compact invariant subsets for the dynamics of the system. Besides being very significant from a dynamical systems point of view, these objects also appear in other different contexts: such as in the study of the Hamilton-Jacobi equation, in billiard dynamics, etc. We shall describe some of these applications.

Mathematical billiards arise in many domains of mathematics, mechanics and physics. Their investigation is on cross-road of several branches of mathematics, including dynamical systems, Riemannian and symplectic geometry, algebraic geometry. In these lectures we will see and example: how right triangular billiards arise in mechanics.We will discuss several long-standing problems on billiards, including Birkhoff's Conjecture on integrable billiards and Ivrii's Conjecture on periodic orbits, and recent advances in their investigation.

This talk will present a generalization of billards called projective billards. In such billards, the law of reflexion is not defined by the usual orthogonal symetries with respect to the tangent lines of the billard tables. Instead, the curves defining the billard tables are endowed with a field of lines, giving place to another reflexion law at each point of the borders. Playing on these tables, we can therefore investigate the same questions as for the usual billards, and for example try to answer Ivrii's conjecture: are there billard tables on which one can find a two-dimensionnal set of periodic orbits ? Even more, is it possible to classify such tables ? I will present a result for triangular periodic orbits, and on the pretext of giving ideas of the proof I will try to display some basics of analytic geometry.

O-Minimality serves as a very successfull framework for tame geometry beyound the algebraic category. Important concepts of analysis can be realized in o-minimal structures. This talk provides a gentle introduction to o-minimal structures.

Mini-course will be an introduction to the basics of dynamical system theory. 
Will be start with some examples of dynamical systems, their bifurcations 
and their occurrence, and will then pass to introducing tools and proving first theorems 
(circle diffeomorphisms, rotation numbers, invariant measure, Poincaré map, Lyapunov 
exponent, distortion of a map, Denjoy theorem).

Talk will be devoted to the random dynamical systems on the real line. 
For a dynamical system on a compact set, the existence of an invariant 
measure is a classical theorem of Krylov and Bogoluybov, that generalizes 
verbatim to the existence of a stationary measure for a random dynamics. 
At the same time, invariant and stationary measures are basic tools in 
the theory of the dynamical systems. 
However, these results cannot be applied in the non-compact case, 
the simplest of which is the dynamics on the real line. 
A theorem by Deroin, Kleptsyn, Navas and Parwani states that 
under some reasonable assumptions for a _symmetric_ dynamical system 
on R there is no probability stationary measure, but there is a (Radon) stationary 
measure of an infinite total mass. 
We generalize this theorem to the non-symmetric dynamical systems on R, 
obtaining a classification of possible types of their behaviour. 
This is also related to earlier work by Guivarc’h, Brofferio, Homburg, and others.