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Geometric and Topological Inference
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<h2 id="topics">Topics</h2> <p>The book is subdivided into four parts and 11 chapters.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The first part covers basic tools from topology needed in the study,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> including <a href="/facts/Simplicial_complex/fF2282CF">simplicial complexes</a>, <a href="/facts/%C4%8Cech_complex/b3c4jvqh">Čech complexes</a> and <a href="/facts/Vietoris%E2%80%93Rips_complex/Ex4Px4wC">Vietoris–Rips complex</a>, <a href="/facts/Homotopy_equivalence/1nWrPxdj">homotopy equivalence</a> of topological spaces to their <a href="/facts/Nerve_of_a_covering/oeV3qb6f">nerves</a>, <a href="/facts/Filtration_(mathematics)/qcvLvKRI">filtrations</a> of complexes, and the <a href="/facts/Data_structure/yNsNvL1I">data structures</a> needed to represent these concepts efficiently in computer <a href="/facts/Algorithm/fnl5NmRt">algorithms</a>. A second introductory part concerns material of a more geometric nature, including <a href="/facts/Delaunay_triangulation/6VRk7cS9">Delaunay triangulations</a> and <a href="/facts/Voronoi_diagram/AhQY5LD8">Voronoi diagrams</a>, <a href="/facts/Convex_polytope/2pRHcRgC">convex polytopes</a>, <a href="/facts/Convex_hull/D4Id3JDS">convex hulls</a> and <a href="/facts/Convex_hull_algorithms/Wr70A2AI">convex hull algorithms</a>, <a href="/facts/Lower_envelope/GICJ5Bjz">lower envelopes</a>, <a href="/facts/Alpha_shape/LmQYAasU">alpha shapes</a> and alpha complexes, and witness complexes.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>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.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> 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 <a href="/facts/Homology_(mathematics)/VFJmryEW">homology</a> and <a href="/facts/Persistent_homology/Mfmn1sIc">persistent homology</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> <h2 id="audience-and-reception">Audience and reception</h2> <p>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.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a><a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Reviewer Michael Berg evaluates it as an "excellent book" aimed at a hot topic, inference from large data sets,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> and both Berg and Mark Hunacek note that it brings a surprising level of real-world applicability to formerly-pure topics in mathematics.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Berg, Michael (April 2019), "Review of Geometric and Topological Inference", MAA Reviews, Mathematical Association of America <a href="https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0" target="_blank">https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Rodrigues, Kévin Allan Sales, "Review of Geometric and Topological Inference", zbMATH, Zbl 1457.62006 <a href="/wiki/ZbMATH" target="_blank">/wiki/ZbMATH</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Adams, Henry Hugh, "Review of Geometric and Topological Inference", MathSciNet, MR 3837127 <a href="/wiki/MathSciNet" target="_blank">/wiki/MathSciNet</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Hunacek, Mark (February 2021), "Review of Geometric and Topological Inference", The Mathematical Gazette, 105 (562): 184–185, doi:10.1017/mag.2021.37, S2CID 233859967 <a href="/wiki/The_Mathematical_Gazette" target="_blank">/wiki/The_Mathematical_Gazette</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Adams, Henry Hugh, "Review of Geometric and Topological Inference", MathSciNet, MR 3837127 <a href="/wiki/MathSciNet" target="_blank">/wiki/MathSciNet</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Berg, Michael (April 2019), "Review of Geometric and Topological Inference", MAA Reviews, Mathematical Association of America <a href="https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0" target="_blank">https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Hunacek, Mark (February 2021), "Review of Geometric and Topological Inference", The Mathematical Gazette, 105 (562): 184–185, doi:10.1017/mag.2021.37, S2CID 233859967 <a href="/wiki/The_Mathematical_Gazette" target="_blank">/wiki/The_Mathematical_Gazette</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Berg, Michael (April 2019), "Review of Geometric and Topological Inference", MAA Reviews, Mathematical Association of America <a href="https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0" target="_blank">https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Adams, Henry Hugh, "Review of Geometric and Topological Inference", MathSciNet, MR 3837127 <a href="/wiki/MathSciNet" target="_blank">/wiki/MathSciNet</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Hunacek, Mark (February 2021), "Review of Geometric and Topological Inference", The Mathematical Gazette, 105 (562): 184–185, doi:10.1017/mag.2021.37, S2CID 233859967 <a href="/wiki/The_Mathematical_Gazette" target="_blank">/wiki/The_Mathematical_Gazette</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Hunacek, Mark (February 2021), "Review of Geometric and Topological Inference", The Mathematical Gazette, 105 (562): 184–185, doi:10.1017/mag.2021.37, S2CID 233859967 <a href="/wiki/The_Mathematical_Gazette" target="_blank">/wiki/The_Mathematical_Gazette</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Rodrigues, Kévin Allan Sales, "Review of Geometric and Topological Inference", zbMATH, Zbl 1457.62006 <a href="/wiki/ZbMATH" target="_blank">/wiki/ZbMATH</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Berg, Michael (April 2019), "Review of Geometric and Topological Inference", MAA Reviews, Mathematical Association of America <a href="https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0" target="_blank">https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Berg, Michael (April 2019), "Review of Geometric and Topological Inference", MAA Reviews, Mathematical Association of America <a href="https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0" target="_blank">https://www.maa.org/press/maa-reviews/geometric-and-topological-inference-0</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Hunacek, Mark (February 2021), "Review of Geometric and Topological Inference", The Mathematical Gazette, 105 (562): 184–185, doi:10.1017/mag.2021.37, S2CID 233859967 <a href="/wiki/The_Mathematical_Gazette" target="_blank">/wiki/The_Mathematical_Gazette</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>