''Geometric and Topological Inference'' is a

monograph A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...

in computational geometry,

computational topology Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory. A primary concern of algorithmic topology, as its ...

geometry processing Geometry processing is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, 3D reconstruction, reconstruction, analysis, manipulation, simulation and ...

, and

topological data analysis In applied mathematics, topological 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 challenging. TDA ...

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

point cloud A point cloud is a discrete set of data Point (geometry), points in space. The points may represent a 3D shape or object. Each point Position (geometry), position has its set of Cartesian coordinates (X, Y, Z). Points may contain data other than ...

of noisy samples from the space. It was written by Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec, and published in 2018 by the

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

in their Cambridge Texts in Applied Mathematics book series. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university A university () is an educational institution, institution of tertiary edu ...

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 structured Set (mathematics), set composed of Point (geometry), points, line segments, triangles, and their ''n''-dimensional counterparts, called Simplex, simplices, such that all the faces and intersec ...

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, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

of topological spaces to their

nerves A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the electrochemical nerve impulses called ...

, filtrations of complexes, and the

data structure In computer science, a data structure is a data organization 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 relationships amo ...

s needed to represent these concepts efficiently in computer

algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...

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

Delaunay triangulation In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points; that is, each circumcircle has its gen ...

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. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

convex hull In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...

s and

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

lower envelope In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value at every point is the minimum of the values of the functions in the given set. The concept of ...

s, 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 In topological data analysis, persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to ...

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=

{{citation, last=Hunacek, first=Mark, date=February 2021, doi=10.1017/mag.2021.37, issue=562, journal=

The Mathematical Gazette ''The Mathematical Gazette'' is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, 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 Infrastru ...

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