Clasificación de formas cuadráticas con un enfoque en el lema morse y el lema de separación

Abstract

Quadratic forms are fundamental mathematical objects that have intrigued mathematicians for centuries. These second-degree polynomial functions have found their application in a wide range of disciplines, from physics and engineering to economics and computer science. Their study has not only revealed properties inherent to algebraic structures, but has also allowed practical problems to be tackled with elegance and precision. In this monograph, we explore in depth quadratic forms and their classification through two powerful tools: Morse's Lemma and the Separation Lemma. These lemmas, although seemingly abstract, provide insightful insight into the nature of quadratic forms and how they behave in different contexts. The monograph begins with a comprehensive review of quadratic forms, from their basic definitions to their more advanced properties. Then, we dive into the fascinating world of Morse's Lemma, a result that establishes a deep connection between topology and the theory of quadratic forms. We will explore how this lemma allows us to understand the structure of manifolds defined by quadratic forms and how we can use it to effectively classify them. We then delve into the Separation Lemma, another crucial result that sheds light on the relationship between quadratic forms and sets in Euclidean space. This lemma provides us with tools to distinguish between different quadratic forms and understand their behavior in terms of convex sets. Throughout this monograph, we not only focus on the abstract theory, but also explore concrete applications and illustrative examples that demonstrate the relevance and versatility of these concepts.