Copyright 2026
Paperback
ISBN 9781032987149
484 Pages 30 B/W Illustrations
September 11, 2025 by Chapman & Hall
Adopting a student-centered approach, this book anticipates and addresses the common challenges that students face when learning abstract concepts like limits, continuity, and inequalities. The text introduces these concepts gradually, giving students a clear pathway to understanding the mathematical tools that underpin much of modern science and technology. In addition to its focus on accessibility, the book maintains a strong emphasis on mathematical rigor. It provides precise, careful definitions and explanations while avoiding common teaching pitfalls, ensuring that students gain a deep understanding of core concepts, and blending algebraic and geometric perspectives to help students see the full picture. The theoretical results presented in the book are consistently applied to practical problems. By providing a clear and supportive introduction to real analysis, the book equips students with the tools they need to confidently engage with both theoretical mathematics and its wide array of practical applications.
Student-Friendly Approach making abstract concepts relatable and engaging
Balanced Focus combining algebraic and geometric perspectives
Comprehensive Coverage: Covers a full range of topics, from real numbers and sequences to metric spaces and approximation theorems, while carefully building upon foundational concepts in a logical progression
Emphasis on Clarity: Provides precise explanations of key mathematical definitions and theorems, avoiding common pitfalls in traditional teaching
Perfect for a One-Semester Course: Tailored for a first course in real analysis
Problems, exercises and solutions
To the Instructor To the Student Logic and Sets Natural Numbers and Induction Real Numbers The Real Number Line Functions Sequences Infinite Series Continuous Functions Integration Differentiation The Fundamental Theorems of Calculus Exponential Functions Circular Functions Complex Numbers Linear Spaces Metric Spaces Approximation Theorems A Solutions to Selected Exercises Bibliography Index
The goal of this book is to provide a mathematical perspective on some key elements of the so-called deep neural networks (DNNs). Much of the interest in deep learning has focused on the implementation of DNN-based algorithms. Our hope is that this compact textbook will offer a complementary point of view that emphasizes the underlying mathematical ideas. We believe that a more foundational perspective will help to answer important questions that have only received empirical answers so far.
The material is based on a one-semester course Introduction to Mathematics of Deep Learning" for senior undergraduate mathematics majors and first year graduate students in mathematics. Our goal is to introduce basic concepts from deep learning in a rigorous mathematical fashion, e.g introduce mathematical definitions of deep neural networks (DNNs), loss functions, the backpropagation algorithm, etc. We attempt to identify for each concept the simplest setting that minimizes technicalities but still contains the key mathematics.
Accessible for students with no prior knowledge of deep learning.
Focuses on the foundational mathematics of deep learning.
Provides quick access to key deep learning techniques.
Includes relevant examples that readers can relate to easily
1 About this book 1
2 Introduction to machine learning: what and why? 2
3 Classification problem 4
4 The fundamentals of artificial neural networks 6
5 Supervised, unsupervised, and semisupervised learning 19
6 The regression problem 24
7 Support vector machine 40
8 Gradient descent method in the training of DNNs 52
9 Backpropagation 67
10 Convolutional neural networks 93
A Review of the chain rule 119
Bibliography 121
Index
In this book, we explore the degeneration of pseudoholomorphic disks bounding a Lagrangian in a symplectic manifold in the large complex structure limit corresponding to a multiple cut. The limit objects, called broken disks, have underlying tropical graphs which in the case of pseudoholomorphic spheres were studied by Brett Parker. In particular, we study the limit of the Fukaya algebra of a Lagrangian submanifold, which is an A
∞
algebra whose higher composition maps involve counts of pseudoholomorphic disks.
The goal of the book is to prove an A
∞
homotopy equivalence between the ordinary Fukaya algebra of a Lagrangian and a tropical version of the Fukaya algebra defined via counts of broken disks with rigid tropical graphs.
The exposition is self-contained and includes details of the transversality scheme. Various computations of disk potentials of Lagrangian submanifolds, such as those in cubic surfaces and flag varieties, are included.
1 Statement of results pp. 1–21
2 Applications to disk counting pp. 23–61
3 Broken manifolds pp. 63–102
4 Broken disks pp. 103–131
5 Stabilizing divisors pp. 133–153
6 Coherent perturbations and regularity pp. 155–190
7 Hofer energy and exponential decay pp. 191–230
8 Gromov compactness pp. 231–275
9 Gluing pp. 277–297
10 Broken Fukaya algebras pp. 299–345
11 Potentials for semi-Fano toric surfaces pp. 347–358
References pp. 359–363
Index pp. 365–367
An accessible, motivated introduction to one of the most dynamic areas of mathematics
Hardcover
ISBN: 9780691268668
Copyright: 2025
Pages: 688
Size: 7 x 10 in
Decades ago, Mumford wrote that algebraic geometry “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics.” The revolution has now fully come to pass and has fundamentally changed how we think about many fields of mathematics. This book provides a thorough foundation in the powerful ideas that now shape the landscape, with an informal yet rigorous exposition that builds intuition for understanding the formidable machinery. It begins with a discussion of categorical thinking and sheaves and then develops the notion of schemes and varieties as examples of “geometric spaces” before discussing their specific aspects. The book goes on to cover topics such as dimension and smoothness, vector bundles and their natural generalizations, and important cohomological tools and their applications. Important optional topics are included in starred sections.
Provides a comprehensive introduction certain to become the standard on the subject
Features a wealth of exercises that enable students to learn by doing
Requires few prerequisites, developing the tools students need to succeed, from category theory and sheaf theory to commutative and homological algebra
Uses an example-driven approach that builds mathematical intuition
Is a self-contained textbook for graduate students and an essential reference for researchers