This book, "Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem" (Second Edition), is a comprehensive treatment of the Atiyah-Singer Index Theorem using heat equation methods. The book covers the necessary analysis to define and compute the index of elliptic operators, including pseudo-differential operators and their properties. It introduces the heat equation and its role in computing the index, and discusses variational formulas and Lefschetz fixed point theorems. The book also delves into characteristic classes, invariance theory, and the proof of the Atiyah-Singer Index Theorem, including its extension to manifolds with boundary. Additionally, it explores spectral geometry and asymptotic formulas, providing a detailed guide to the literature on index theory and spectral geometry. The second edition includes substantial new material and significant revisions, making it a valuable resource for advanced mathematicians and researchers in the field.This book, "Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem" (Second Edition), is a comprehensive treatment of the Atiyah-Singer Index Theorem using heat equation methods. The book covers the necessary analysis to define and compute the index of elliptic operators, including pseudo-differential operators and their properties. It introduces the heat equation and its role in computing the index, and discusses variational formulas and Lefschetz fixed point theorems. The book also delves into characteristic classes, invariance theory, and the proof of the Atiyah-Singer Index Theorem, including its extension to manifolds with boundary. Additionally, it explores spectral geometry and asymptotic formulas, providing a detailed guide to the literature on index theory and spectral geometry. The second edition includes substantial new material and significant revisions, making it a valuable resource for advanced mathematicians and researchers in the field.