Lectures of Educational Conference “Mathematical Spring 2019"

In these lectures, the following questions of one-dimensional dynamics will be considered: (i) the combinatorial classification (ii) topological classification and (iii) the ergodic properties

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.

Many essential properties of dynamical system relate to the existence of invariant foliations and lamination in its phase space. Lectures of this course will give information about some topological invariants of foliations on orientable closed surfaces except for spheres and the relationship between foliations with geodesic lamination in the metric with zero curvature on the torus and with negative constant curvature on a surface of negative orbifold Euler characteristic.

The modern qualitative theory of dynamical systems is thoroughly intertwined with the fairly young science of topology. Many strange constructions of topology are found sooner or later in dynamics of discrete or continuous dynamical systems. In the lectures we will show how to realize a topological oject as an invariant set of a dynamical system.

The lecture will about what dynamical chaos is and what of type can it be. Close attention will be payed to Strange attractors, especially to attracting sets with random behavior of trajectories.

In the lecture the a notion of index of a closed curve with respect to a continuous vector field on a plane will be introduced some properties of the index will be established. The index theory will be applied for prooving the existence of at least one (possibly complex) root of an arbitrary polynomial of degree n.

The proof of a famous set theory theorem called Banach-Tarski paradox will be given at the lecture .

The history of soliton (initially wave of translation) has nearly 200 years. Assuming initially to be exceptionally specific type of waves, after the “revolution” of the 1960s – 1970s, when the method of exact solution of nonlinear partial differential equations was invented, solitons turned out to be extremely common, typical solutions for many key equations in physics. In the lecture you will learn how such waves may look, how unusual their dynamics are, how the Inverse Scattering Technique works, and which new actual problems in physics require the further development of the theory of solitons.

Constructive proof contains not only arguments for the existence of an object, but also an algorithm for its construction. One way to record such a proof, known as a natural conclusion, is of interest because the proof itself is an algorithm. This coincidence, called the Curry-Howard Isomorphism, establishes a connection between the proofs and the lambda calculus. This connection is the basis of modern functional programming languages. As part of the lectures you will be shown the difference between the common classical logic and a constructive one. You will also understand the relationship between the natural conclusion and the lambda calculus and see how they are used in formalized mathematics.

Let M be closed 3-manifold with a finite fundamental group. According to Novikov’s theorem every smooth foliation (M, F) of the codimension one on such M admits a Reeb component. We present a method of a reconstruction of a Reeb component of (M, F) allows to get a smooth foliation on M with a countable set of attractors.

The holonomy group of a pseudo-Riemannian manifold gives reach information about the geometry of the manifold. A classical and important result is the classification of the connected holonomy groups of Riemannian manifolds. In the lecture will be represented recent results about classification for the holonomy groups of Lorentzian manifolds and some results about holonomy groups of pseudo-Riemannian manifolds.

One-parameter strongly continuous semigroup of linear operators is a flow in an infinite-dimensional space of functions, in the sense of the theory of dynamic systems. The functional analysis paradigm enables us to rewrite the partial differential equation as an ordinary differential equation but written for an unknown function that takes values in the infinite-dimensional space of functions. Moreover, the equation we get is very simple — a linear first-order equation with separable variables, a solution to which is an exponent. Finding an exponent here means finding a solution to the original partial differential equation. It became apparent that an analogue to the theorem on the Second Remarkable Limit is applicable to exponentials of operators and this statement is called the Chernoff theorem. Our course will explain how one can find an approximate exponent value and solve partial differential equations using Chernoff theorem. The beginning of the course will be elementary, but in the end the author will cover the latest results in this area, including those that he has achieved in recent years.

Almost contact metric manifolds with N-connection form the class of Riemannian-Cartan manifolds (or metric-affine manifolds in another terminology), these are Riemannian manifolds with a connection having a torsion. Metric-affine manifolds lie in the ground of Einstein-Cartan gravity theory. Most of the works on Riemann-Cartan manifolds are written by physicists and reflect the specific of physics. Among all classes of Riemann-Cartan manifolds, only quarter-symmetric and half-symmetric metric spaces have been studied in details. An N-connection on an almost contact metric manifold is introduced be the speaker. It is planned to give an introduction to the geometry of Riemann-Cartan manifolds and to discuss the classification problem for almost contact metric manifolds with N-connection.