Student seminar "Topological Methods in Dynamics"

Coming soon

At the seminar, we proved the fundamental theorem on homomorphisms. For this purpose, we introduced the concept of groups and studied their properties, illustrated by various examples.

At the seminar, we proved main properties of metric spaces and studied continuous maps from one metric space to another. In addition, we considered various metrics and open sets generated by them.

A solid torus is considered. By analogy with a usual torus, the classification of knots on a solid torus is carried out in the homotopy and topological sense. Namely, homotopy classes are listed, conditions for ambient isotopy and equivalence by means of an ambient homeomorphism are established for knots on a solid torus. 

A solid torus is considered. By analogy with a usual torus, the classification of knots on a solid torus is carried out in the homotopy and topological sense. Namely, homotopy classes are listed, conditions for ambient isotopy and equivalence by means of an ambient homeomorphism are established for knots on a solid torus. 

A solid torus is considered. By analogy with a usual torus, the classification of knots on a solid torus is carried out in the homotopy and topological sense. Namely, homotopy classes are listed, conditions for ambient isotopy and equivalence by means of an ambient homeomorphism are established for knots on a solid torus. 

A solid torus is considered. By analogy with a usual torus, the classification of knots on a solid torus is carried out in the homotopy and topological sense. Namely, homotopy classes are listed, conditions for ambient isotopy and equivalence by means of an ambient homeomorphism are established for knots on a solid torus. 

The seminar considers the properties of the homomorphism of the group Aut (T ^ 2) - homeomorphisms of the torus onto itself and the group GL (2, Z) of integer-valued matrices invertible over integers. The main fact is that the group of self-homeomorphisms Aut (T ^ 2), modulo ambient isotopy, is isotopic with group GL (2, Z). That faсt implies a chain of consequences: two homeomorphisms of the torus are ambient isotopic <=> they have the same matrices <=> they are homotopic as maps.

The seminar considers the properties of the homomorphism of the group Aut (T ^ 2) - homeomorphisms of the torus onto itself and the group GL (2, Z) of integer-valued matrices invertible over integers. The main fact is that the group of self-homeomorphisms Aut (T ^ 2), modulo ambient isotopy, is isotopic with group GL (2, Z). That faсt implies a chain of consequences: two homeomorphisms of the torus are ambient isotopic <=> they have the same matrices <=> they are homotopic as maps.

The seminar considers the properties of the homomorphism of the group Aut (T ^ 2) - homeomorphisms of the torus onto itself and the group GL (2, Z) of integer-valued matrices invertible over integers. The main fact is that the group of self-homeomorphisms Aut (T ^ 2), modulo ambient isotopy, is isotopic with group GL (2, Z). That faсt implies a chain of consequences: two homeomorphisms of the torus are ambient isotopic <=> they have the same matrices <=> they are homotopic as maps.

The seminar considers the properties of the homomorphism of the group Aut (T ^ 2) - homeomorphisms of the torus onto itself and the group GL (2, Z) of integer-valued matrices invertible over integers. The main fact is that the group of self-homeomorphisms Aut (T ^ 2), modulo ambient isotopy, is isotopic with group GL (2, Z). That faсt implies a chain of consequences: two homeomorphisms of the torus are ambient isotopic <=> they have the same matrices <=> they are homotopic as maps.

The seminar considers the properties of the homomorphism of the group Aut (T²) - homeomorphisms of the torus onto itself and the group GL (2, Z) of integer-valued matrices invertible over integers. The main fact is that the group of self-homeomorphisms Aut (T²), modulo ambient isotopy, is isotopic with group GL (2, Z). That faсt implies a chain of consequences: two homeomorphisms of the torus are ambient isotopic <=> they have the same matrices <=> they are homotopic as maps.

This time we will deduce several corollaries from the theorem proven at the previous seminar. Particular attention will be paid to the equivalence of knots, laying in a plane, in three dimensional space. Also, the chord theorem will be proven in more strict formulation. Moreover, we will give a counterexample to the case of n being non-integer.

This time we will deduce several corollaries from the theorem proven at the previous seminar. Particular attention will be paid to the equivalence of knots, laying in a plane, in three dimensional space. Also, the chord theorem will be proven in more strict formulation. Moreover, we will give a counterexample to the case of n being non-integer.

We are about to proof the chord theorem, which asserts that for any straight line segment C with its endpoint in path-connected subset X of the plane and for any positive integer n there exists straight-line segment parallel to C with its endpoints in X and of the length 1/n of length of C.

We will consider the theorem which asserts that a closed homeomorphic image of a line divides the plane into two parts. Also, we will prove that this theorem can be deduced from the Jordan curve theorem and vice versa.

Some corollaries of the Annulus theorem, proven in the previous seminar. Primarily, we are interested in the theorem about existence of ambient isotopy for any two knots in the sphere. The statement establishing classification of links in the sphere and in the plane will be considered and used as well. Finally, we will sharpen the main theorem. It turns out that the ambient isotopy can be fixed outside any preassigned neighbourhood of the closed annular region between the knots (assuming they are disjoint). Analogous facts will be proven for the plane as well.

The talk is devoted to the proof of the Annulus theorem, which asserts that the closure of the domain bounded by two disjoint knots in the plane is homeomorphic to the standard annulus, while one of the knots lies in the domain bounded by another. The proof will be based on the Schönflies theorem and the fact that for any two knots there are two arcs with its endpoint on these knots which interiors are disjoint and do not intersect the knots

We will consider the Jordan curve theorem, which asserts that every knot in a plane divides it into two domains one of which is bounded, and Schönflies theorem, which states that this bounded component is a disk. The proofs will not be provided due to its complicity. Instead, some corollaries will be proved. For example, Alexander’s lemma. Last but not the least will be to proof that any two knots in the plane or in the sphere are equivalent. Moreover we will consider classical objects such as trefoil knot and the Vaad’s lakes.

In this talk the basic notions of the knot theory such that a knot, a link, equivalence, map equivalence, oriented equivalence and ambient isotopy will be considered. Also, the statement about relation between these notions of equivalence will be formulated and proved.

In this lecture the basic notions of general and algebraic topology will be considered. The link between some problems of the knot theory and different branches of mathematics, especially qualitative theory of dynamical systems, will be analyzed.

We will continue to consider orientation preserving homeomorphism of circle with irrational rotation number. It will be proven that if map is transitive then it conjugates rigid rotation with the same rotation number and if map is not transitive then it semiconjugates the such rigid rotation. We will consider Denjoy example. Also we will prove theorem about continuous dependence of rotation number on homeomorphism of circle.

We will consider orientation preserving homeomorphism of circle with irrational rotation number. We will prove that for such map omega-limit set of every point of circle does not depend on point and is whole circle or Cantor set. Also it will be proven that if map is transitive then it conjugates rigid rotation with the same rotation number and if map is not transitive then it semiconjugates the such rigid rotation.

Будет дана классификация орбит в случае гомеоморфизма с рациональным числом вращения. Будет доказана теорема Пуанкаре о полусопряжённости каскадов, задаваемых гомеоморфизмом с иррациональным числом вращения и чистым поворотом.

Будет введено число вращения для сохраняющих ориентацию гомеоморфизмов окружности и выведены его свойства. Будет дана классификация орбит в случае гомеоморфизма с рациональным числом вращения. Будет доказана теорема Пуанкаре о полусопряжённости каскадов, задаваемых гомеоморфизмом с иррациональным числом вращения с чистым поворотом.

Отображения окружности принципиально отличаются от отображений прямой разнообразием предельных множеств. Одним из важнейших инвариантов гомеоморфизма окружности является число вращения Пуанкаре. В рамках семинара это понятие будет введено и будут доказаны его основные свойства.

Подведена мотивация изучения дискретных динамических систем (каскадов). Изучены базовые понятия обратимых отображений на прямой, проиллюстрированы отличия потоков от каскадов. Введено понятие диаграммы Ламерея -- геометрической интерпретации поведения динамической системы. Введено понятие и получен критерий структурной устойчивости диффеоморфизма прямой.

Введено, восходящее к А.А. Андронову понятие грубого векторного поля. Для векторных полей в ограниченной части  прямой установлен критерий грубости гладкого векторного поля, состоящий в конечности и гиперболичности состояний равновесия. Для доказательства критерия применена классическая теорема Сарда. Результаты обобщены на векторные поля на окружности.

Изложен подход к решению автономных дифференциальных уравнений, приводящий к к общему понятию динамической системы.  Введено понятие интегральных кривых, траекторий, фазового и объемлющего пространства, состояний равновесия, их гиперболичности. Дан геометрический подход к решению дифференциальных уравнений первого порядка.