We present a new and complete algorithm for computing the topology of an algebraic surface S given by a squarefree polynomial in ◗[X,Y, Z]. Our algorithm involves only subresultant computations and entirely relies on rational manipulation, which makes it direct to implement. We extend the work in [15], on the topology of non-reduced algebraic space curves, and apply it to the polar curve or app...

We provide a systematic reconstruction of Abramsky's category SProc of synchronous processes Abr93]: SProc is isomorphic to a span category on a category of traces. The signiicance of the work is twofold: It shows that the original presentation of SProc in mixed formulations is unnecessary | a simple categorical description exists. Furthermore, the techniques employed in the reconstruction sugg...

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...