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This book collects selected contributions presented at the INdAM Workshop "Geometric Challenges in Isogeometric Analysis", held in Rome, Italy on January 27-31, 2020. It gives an overview of the forefront research on splines and their efficient use in isogeometric methods for the discretization of differential problems over complex and trimmed geometries. A variety of research topics in this context are covered, including (i) high-quality spline surfaces on complex and trimmed geometries, (ii) construction and analysis of smooth spline spaces on unstructured meshes, (iii) numerical aspects and benchmarking of isogeometric discretizations on unstructured meshes, meshing strategies and software. Given its scope, the book will be of interest to both researchers and graduate students working in the areas of approximation theory, geometric design and numerical simulation. Chapter 10 is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.

Spline theory --- Data processing. --- Spline functions --- Approximation theory --- Interpolation --- Geometría combinatòria --- Anàlisi combinatòria --- Geometries finites --- Geometria discreta --- Matroides --- Mathematics --- Mathematics. --- Computer science --- Algorithms. --- Computational Mathematics and Numerical Analysis. --- Applications of Mathematics. --- Mathematical Applications in Computer Science. --- Algorism --- Algebra --- Arithmetic --- Computer mathematics --- Electronic data processing --- Math --- Science --- Foundations --- Geometria combinatòria.

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The book, based on the INdAM Workshop "Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology" provides a bridge between different communities of mathematicians who utilize splines in their work. Splines are mathematical objects which allow researchers in geometric modeling and approximation theory to tackle a wide variety of questions. Splines are interesting for both applied mathematicians, and also for those working in purely theoretical mathematical settings. This book contains contributions by researchers from different mathematical communities: on the applied side, those working in numerical analysis and approximation theory, and on the theoretical side, those working in GKM theory, equivariant cohomology and homological algebra.

Approximation theory. --- Numerical analysis. --- Algebra. --- Geometry. --- Topology. --- Approximations and Expansions. --- Numerical Analysis.