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Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…

This is a survey on the local structure about a fixed point of discrete finite-dimensional holomorphic dynamical systems, discussing in particular the existence of local topological conjugacies to normal forms, and the structure of local…

We present an alternative approach to the vector version of Krasnosel'skii compression-expansion fixed point theorem due to Precup, which is based on the fixed point index. It allows us to obtain new general versions of this fixed point…

We introduce a category of noncommutative bundles. To establish geometry in this category we construct suitable noncommutative differential calculi on these bundles and study their basic properties. Furthermore we define the notion of a…

These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.

A proof of Sharkovsky's Theorem is given. It is shown how this proof naturally generalizes to looking at maps on graphs and to Sharkovsky-type theorems for these maps. The paper is written at an elementary level and is meant as an…

Index theorem is formulated in noncommutative geometry with finite degrees of freedom by using Ginsparg-Wilson relation. It is extended to the case where the gauge symmetry is spontaneously broken. Dynamical analysis about topological…

Establishing the existence of periodic orbits is one of the crucial and most intricate topics in the study of dynamical systems, and over the years, many methods have been developed to this end. On the other hand, finding closed orbits in…

We present new criteria on the existence of fixed points that combine some monotonicity assumptions with the classical fixed point index theory. As an illustrative application, we use our theoretical results to prove the existence of…

In geometric analysis, an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned, sometimes using the heat kernels of two higher-order…

We prove the Conley conjecture for a closed symplectically aspherical symplectic manifold: a Hamiltonian diffeomorphism of a such a manifold has infinitely many periodic points. More precisely, we show that a Hamiltonian diffeomorphism with…

We review the Deligne-Simpson problem, a combinatorial structure of middle convolutions and their relation to a Kac-Moody root system discoverd by Crawley-Boevey. We show with examples that middle convolutions transform the Fuchsian systems…

Consider briefly the equations of fluid dynamics-they describe the enormous wealth of detail in all the interacting physical elements of a fluid flow-whereas in applications we want to deal with a description of just that which is…

Topological bases of behaviour of trajectories for autonomous differential systems of the second order on sphere are stated. Stereographic atlas of trajectories is constructed. Differential connections between trajectories of…

We approach the analysis of dynamical and geometrical properties of nonholonomic mechanical systems from the discussion of a more general class of auxiliary constrained Hamiltonian systems. The latter is constructed in a manner that it…

Continuing the study of Hamiltonian pseudo-rotations of projective spaces, we focus on the conjecture that the fixed-point data set (the actions and the linearized flows at one-periodic orbits) of a pseudo-rotation exactly matches that data…

Many real-world scientific processes are governed by complex nonlinear dynamic systems that can be represented by differential equations. Recently, there has been increased interest in learning, or discovering, the forms of the equations…

Several approaches to cognition and intelligence research rely on statistics-based models testing, namely factor analysis. In the present work we exploit the emerging dynamical systems perspective putting the focus on the role of the…

This note contains some observations on abelian convexity theorems. Convexity along an orbit is established in a very general setting using Kempf-Ness functions. This is applied to give short proofs of the Atiyah-Guillemin-Sternberg theorem…

A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of Hamiltonian vector…