''Geometric and Topological Inference'' is a monograph in computational geometry, computational topology,

geometry processing Geometry processing, or mesh processing, is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, reconstruction, analysis, manipulation, simulat ...

, and

topological data analysis In applied mathematics, topological based data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challengin ...

, on the problem of inferring properties of an unknown space from a finite

point cloud Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

of noisy samples from the space. It was written by

Jean-Daniel Boissonnat Jean-Daniel Boissonnat (born 18 May 1953) is a French computer scientist, who works as a director of research at the French Institute for Research in Computer Science and Automation (INRIA). He is an invited professor of computational geometry at ...

, Frédéric Chazal, and

Mariette Yvinec Mariette Yvinec is a French researcher in computational geometry at the French Institute for Research in Computer Science and Automation (INRIA) in Sophia Antipolis. She is one of the developers of CGAL, a software library of computational geomet ...

, and published in 2018 by the

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...

in their Cambridge Texts in Applied Mathematics book series. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.

Topics

The book is subdivided into four parts and 11 chapters. The first part covers basic tools from topology needed in the study, including

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial ...

es,

Čech complex In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distributi ...

es and

Vietoris–Rips complex In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space ...

homotopy equivalence In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defo ...

of topological spaces to their

nerves A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system. A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...

, filtrations of complexes, and the

data structure In computer science, a data structure is a data organization, management, and storage format that is usually chosen for Efficiency, efficient Data access, access to data. More precisely, a data structure is a collection of data values, the rel ...

s needed to represent these concepts efficiently in computer

algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...

s. A second introductory part concerns material of a more geometric nature, including

Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle ...

s and

Voronoi diagram In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. In the simplest case, these objects are just finitely many points in the plane (called seeds, sites, or generators). For each seed ...

s, convex polytopes, convex hulls and

convex hull algorithms Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science. In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of poin ...

, lower envelopes, alpha shapes and alpha complexes, and witness complexes. With these preliminaries out of the way, the remaining two sections show how to use these tools for topological inference. The third section is on recovering the unknown space itself (or a topologically equivalent space, described using a complex) from sufficiently well-behaved samples. The fourth part shows how, with weaker assumptions about the samples, it is still possible to recover useful information about the space, such as its homology and persistent homology.

Audience and reception

Although the book is primarily aimed at specialists in these topics, it can also be used to introduce the area to non-specialists, and provides exercises suitable for an advanced course. Reviewer Michael Berg evaluates it as an "excellent book" aimed at a hot topic, inference from large data sets, and both Berg and Mark Hunacek note that it brings a surprising level of real-world applicability to formerly-pure topics in mathematics.

References

{{reflist, refs= {{citation, title=Review of ''Geometric and Topological Inference'', first=Henry Hugh, last=Adams, work=

MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal '' Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

, mr=3837127 {{citation, title=Review of ''Geometric and Topological Inference'', url=https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0, date=April 2019, first=Michael, last=Berg, work=MAA Reviews, publisher= Mathematical Association of America {{citation, last=Hunacek, first=Mark, date=February 2021, doi=10.1017/mag.2021.37, issue=562, journal=

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, pages=184–185, title=Review of ''Geometric and Topological Inference'', volume=105, s2cid=233859967 {{citation, title=Review of ''Geometric and Topological Inference'', first=Kévin Allan Sales, last=Rodrigues, work=

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

, zbl=1457.62006 Mathematics books Computational geometry Computational topology Geometry processing 2018 non-fiction books Cambridge University Press books