This paper presents methods using algebraic topology for showing that pseudo-trajectories are close to trajectories of a dynamical system. Our emphasis is the case where the trajectories are unstable in two or more dimensions. We develop the algebraic topology for guaranteeing the existence of such trajectories.

In this paper, the existence of multiple positive solutions for a class of third-order three-point discrete boundary value problem is studied by applying algebraic topology method. Keywords—Positive solutions, Discrete boundary value problem, Third-order, Three-point, Algebraic topology

Project Summary: Some mathematical notions can be described in both algebraic and geometric ways. The geometric descriptions are sometimes richer and can reveal hidden symmetries. K-theory is an example of a theory that has both and algebraic and geometric description. It links algebraic topology with geometry on curved spaces, to the mutual benefit of both disciplines. We believe that the noti...

These notes are based on lectures in algebraic topology taught by Peter May and Henry Chan at the 2016 University of Chicago Math REU. They are loosely chronological, having been reorganized for my benefit and significantly annotated by my personal exposition, plus solutions to in-class/HW exercises, plus content from readings (from May’s Finite Book), books (e.g. May’s Concise Course, Munkres’...

We consider a closed semi-algebraic set X ⊂ R and a C semi-algebraic function f : R → R such that f|X has a finite number of critical points. We relate the topology of X to the topology of the sets {f ∗ α}, where ∗ ∈ {≤,=,≥} and α ∈ R, and the indices of the critical points of f|X and −f|X . We also relate the topology of X to the topology of the links at infinity of the sets {f ∗ α} and the in...

In this paper, an algorithm to determine the topology of an algebraic space curve and to compute a certified G rational parametric approximation of the algebraic space curve is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the sa...

We begin this paper with a discussion of n-dimensional algebraic Vector bundles. Vector bundles of rank n may be identified with locally free Ox-modules of rank n. Several sections are dedicated to reviewing the basic notions of scheme-theory. We show how to construct the projectization of arbitrary rings. This construction corresponds to the construction of the tautological line bundle over an...

We use methods of algebraic topology to study the eigenvalue structure of a complex Osserman algebraic curvature tensor. We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.

An MI-group is an algebraic structure based on a generalization of the concept of a monoid that satisfies the cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In this paper, a topology is defined on an MI-group $G$ under which $G$ is  a topological MI-group. Then we will identify open, discrete and compact MI-subgroups. The connected components of th...

We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and reformulate this question as a question of categoricity of the universal covering space of a complex algebraic variety in a natural countable in nitary language. We provide partial positive results towards the question. Assuming a conjecture of Shafarevich and s...