The book “Smooth manifolds and their applications in the theory of homotopies” by Lev Pontryagin is an outstanding work that has become a fundamental work in the field of differential geometry and topology. Written in the mid-20th century, this work continues to be relevant and in demand among mathematicians and students interested in modern trends in homotopy theory and geometry. In his book, Pontryagin immerses the reader in a world of smooth manifolds - objects that are a generalization of familiar multidimensional spaces. The author masterfully explains how these manifolds can be used to solve complex problems in mathematics and physics. He introduces the reader to the foundations of homotopy theory, showing how smooth manifolds help in the study of the properties of spaces and their interrelationships. Pontryagin not only outlines the theoretical aspects, but also demonstrates the practical application of his ideas, which makes the book especially valuable for those who seek to understand how abstract mathematical concepts can be used in real-world problems. This book will be of interest to both professional mathematicians and senior students who study differential geometry and topology. It is suitable for those who are fond of mathematics and want to deepen their knowledge in these areas. Readers interested in theoretical physics will also find much useful information in it, as many of Pontryagin's ideas find application in physics, especially in relativity theory and quantum mechanics. The topics raised in the book cover a wide range of issues related to geometry and topology. Pontryagin discusses the properties of smooth manifolds, their classification and structure, as well as methods to investigate them using algebraic and topological tools. He raises important questions about the relationship between geometry and analysis, which makes his work not only theoretically significant, but also practical. The reader will be able to see how mathematical ideas intersect with physical concepts, and how they can be used to solve real-world problems. Pontryagin's style is distinguished by clarity and logical presentation, which allows even complex ideas to be perceived with ease. His approach to explaining the material makes the book available to a wide audience, despite the high degree of abstraction of the topics under consideration. Pontryagin is known for his other works, such as Basics of Topology and An Introduction to Differential Geometry, which are also noteworthy and may be useful for a thorough understanding of the topics covered in Smooth Varieties. In conclusion, “Smooth Varieties and Their Applications in Homotopy Theory” is not just a textbook, but a real encyclopedia of knowledge for anyone who wants to understand the world of mathematics more deeply. This book will become an indispensable companion for students, graduate students and researchers seeking to expand their horizons in the field of geometry and topology. Do not miss the chance to touch the heritage of one of the greatest mathematicians of the 20th century and discover the amazing world of smooth diversity!