Van Kampen theorem for persistent fundamental group

Download
2019
Batan, Mehmet Ali
Persistent homotopy is one of the newest algebraic topology methods in order to understand and capture topological features of discrete objects or point data clouds (the set of points with metric defined on it). On the other hand, in algebraic topology, the Van Kampen Theorem is a great tool to determine fundamental group of complicated spaces in terms of simpler subspaces whose fundamental groups are already known. In this thesis, we show that Van Kampen Theorem is still valid for the persistent fundamental group. Finally, we show that interleavings, a way to compare persistences, among subspaces imply interleavings among total spaces.

Suggestions

Knotting of algebraic curves in CP2
Finashin, Sergey (2002-01-01)
For any k⩾3, I construct infinitely many pairwise smoothly non-isotopic smooth surfaces homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve and having abelian fundamental group ⧹ . This gives an answer to Problem 4.110 in the Kirby list (Kirby, Problems in low-dimensional topology, in: W. Kazez (Ed.), Geometric Topology, AMS/IP Stud. Adv. Math. vol 2.2, Amer. Math. Soc., Providence, 1997).
On equivelar triangulations of surfaces
Adıgüzel, Ebru; Pamuk, Semra; Department of Mathematics (2018)
Persistent homology is an algebraic method for understanding topological features of discrete objects or data (finite set of points with metric defined on it). In algebraic topology, the Mayer Vietoris sequence is a powerful tool which allows one to study the homology groups of a given space in terms of simpler homology groups of its subspaces. In this thesis, we study to what extent does persistent homology benefit from Mayer Vietoris sequence.
Mayer Vietoris sequence for persistent homology
Yılmaz, Yağmur; Önder, Mustafa Turgut; Department of Mathematics (2018)
Persistent homology is an algebraic method for understanding topological features of discrete objects or data (finite set of points with metric defined on it). In algebraic topology, the Mayer Vietoris sequence is a powerful tool which allows one to study the homology groups of a given space in terms of simpler homology groups of its subspaces. In this thesis, we study to what extent does persistent homology benefit from Mayer Vietoris sequence.
Energy preserving methods for lattice equations
Erdem, Özge; Karasözen, Bülent (2010-11-27)
Integral preserving methods, like the averaged vector field, discrete gradient and trapezoidal methods are to Poisson systems. Numerical experiments on the Volterra equations and integrable discretization of the nonlinear Schrodinger equation are presented.
On the Poisson sum formula for the analysis of wave radiation and scattering from large finite arrays
Aydın Çivi, Hatice Özlem; Chou, HT (1999-05-01)
Poisson sum formulas have been previously presented and utilized in the literature [1]-[8] for converting a finite element-by-element array field summation into an alternative representation that exhibits improved convergence properties with a view toward more efficiently analyzing wave radiation/scattering from electrically large finite periodic arrays. However, different authors [1]-[6] appear to use two different versions of the Poisson sum formula; one of these explicitly shows the end-point discontinui...
Citation Formats
M. A. Batan, “Van Kampen theorem for persistent fundamental group,” Thesis (M.S.) -- Graduate School of Applied Mathematics. Mathematics., Middle East Technical University, 2019.