Betti number

Abstract: This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets — persistent homology — and a novel representation of this algebraic characterization — barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a four-dimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object?” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer three-dimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations in a strictly lower dimension. The inference of global structure occurs on much finer scales as well, with regards to converting discrete data into continuous images. Dot-matrix printers, scrolling LED tickers, televisions, and computer displays all communicate images via arrays of discrete points which are integrated into coherent, global objects. This also is a skill we have practiced from childhood. No adult does a dot-to-dot puzzle with anything approaching anticipation. 1.1. Topological data analysis. Problems of data analysis share many features with these two fundamental integration tasks: (1) how does one infer high dimensional structure from low dimensional representations; and (2) how does one assemble discrete points into global structure. The principal themes of this survey of the work of Carlsson, de Silva, Edelsbrunner, Harer, Zomorodian, and others are the following: (1) It is beneficial to replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. This converts the data set into global topological objects. (2) It is beneficial to view these topological complexes through the lens of algebraic topology — specifically, via a novel theory of persistent homology adapted to parameterized families. (3) It is beneficial to encode the persistent homology of a data set in the form of a parameterized version of a Betti number: a barcode. The author gratefully acknowledges the support of DARPA # HR0011-07-1-0002. The work reviewed in this article is funded by the DARPA program TDA: Topological Data Analysis.

Abstract: Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

Abstract: (Note: There are no symplectic forms on X unless b and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the Seiberg-Witten invariants of a manifold X, as in the theorem, which can be split by an embedded 3-sphere as X−∪X+ where neither X− nor X+ have negative definite intersection forms. Thus, no such manifold admits a symplectic form. That is,

Abstract: PROOF Approximate fi, * *, fm by real polynomials F1, * , Fm of the same degrees whose coefficients are algebraically independent Now consider the variety Vc in the complex Cartesian space C'defined by the equations F1 =0, * * *, Fm =0 It follows from van der Waerden [9, ?41 ] that the number of points in Vc is equal to (deg fi) (deg f2) (deg fm) Since each point of V0 lies close to some real point of Vc; this proves Lemma 1