Ovidiu Calin Book order

The book delves into the connections between Stochastic Analysis, Geometry, and Partial Differential Equations (PDEs), emphasizing how geometric structures like Riemannian and sub-Riemannian geometries influence diffusion processes. It explores the implications for solving PDEs and applications in mathematical finance, while unifying these disciplines through the bracket-generating condition, also known as Hörmander's condition. The primary objective is to highlight the shared conditions on vector fields that link PDEs, nonholonomic geometry, and stochastic processes.

Stochastic Calculus serves as a foundational tool for various scientific fields involving random fluctuations, such as signal processing, financial markets, and population dynamics. The book emphasizes the necessity of a solid mathematical background, particularly in probability, analysis, and measure theory, to effectively understand and apply these concepts. It acknowledges the complexity of Stochastic Calculus, highlighting the time and effort required to master its theoretical framework and practical applications across diverse disciplines.

This book describes how neural networks operate from the mathematical point of view. As a result, neural networks can be interpreted both as function universal approximators and information processors. The book bridges the gap between ideas and concepts of neural networks, which are used nowadays at an intuitive level, and the precise modern mathematical language, presenting the best practices of the former and enjoying the robustness and elegance of the latter. This book can be used in a graduate course in deep learning, with the first few parts being accessible to senior undergraduates. In addition, the book will be of wide interest to machine learning researchers who are interested in a theoretical understanding of the subject.