Foundations of Arithmetic Circuit Complexity

How do we measure the difficulty of computing algebraic expressions and polynomials? Arithmetic circuit complexity provides the mathematical framework to understand the limits of algebraic computation. In this text-based course, you will transition from basic algebraic concepts to analyzing sophisticated computational models. You will read clear mathematical proofs, analyze circuit depth and size, and understand the core open questions in theoretical computer science. What you'll learn: Understand the fundamental definitions of arithmetic circuits, formulas, and algebraic branching programs; Analyze the classes VP and VNP, exploring the algebraic analog of the P vs NP question; Explore polynomial identity testing (PIT) and its role in randomized algorithm design; Study classic lower bound proofs, including depth-reduction techniques and homogeneous circuits; Examine modern developments in matrix multiplication complexity and algebraic computation. The course starts with essential terminology and foundational definitions before guiding you through structural results, complexity classes, and modern lower-bound techniques. You will practice through structured written exercises and mathematical proofs designed to solidify your theoretical intuition. This course is designed for students and researchers in computer science or mathematics who want a clear introduction to algebraic complexity without needing prior advanced knowledge. Begin your journey into the mathematical heart of computation today.