Introduction to the Analysis of Boolean Functions

Boolean functions are at the heart of computer science, cryptography, and complexity theory, yet analyzing their behavior requires a specialized mathematical toolkit. This course introduces you to the powerful intersection of discrete mathematics, probability, and analysis. You will learn how to decompose complex Boolean operations and understand their structural properties through a rigorous, step-by-step approach. By studying these mathematical frameworks, you will gain a deeper appreciation of theoretical computer science and combinatorics. What you'll learn: - Understand the foundational definitions of Boolean functions and discrete probability spaces. - Apply Fourier analysis techniques to represent and simplify Boolean functions. - Analyze key properties such as influence, noise sensitivity, and hypercontractivity. - Explore the role of Boolean analysis in modern computer science, including social choice theory and circuit complexity. - Practice solving combinatorial problems using algebraic and analytic proofs. The course begins with essential terminology, basic algebraic concepts, and foundational definitions of Boolean spaces. From there, you will progress through the core principles of Fourier expansion on the hypercube, culminating in the study of noise stability and advanced structural theorems. This text-only course is designed for beginners in discrete mathematics and theoretical computer science, requiring no advanced prerequisites beyond basic linear algebra and probability. Start your journey into discrete analysis today.