In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

Branching processes, which are the focus of this chapter, arise naturally in the study of stochastic processes on trees and locally tree-like graphs. Similarly to martingales, finding a hidden branching process within a probabilistic model can lead to useful bounds and insights into asymptotic behavior. After a review of the extinction theory of branching processes and of a fruitful random-walk perspective, we give a couple examples of applications in discrete probability. In particular we analyze the height of a binary search tree, a standard data structure in computer science. We also give an introduction to phylogenetics, where a “multitype” variant of the Galton–Watson branching process plays an important role; we use the techniques derived in this chapter to establish a phase transition in the reconstruction of ancestral molecular sequences. We end this chapter with a detailed look into the phase transition of the Erdos–Renyi graph model. The random-walk perspective mentioned above allows one to analyze the “exploration” of a largest connected component, leading to information about the “evolution” of its size as edge density increases.

In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

In mathematics, it simply is not true that “you can’t prove a negative.” Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness, and the universe and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs.

This book is the first to present complete proofs of these theorems for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic – and are elementary, starting only from first principles and definitions.

Very little background knowledge is required, and no specialized mathematical training – all you need is the discipline to follow logical arguments and a pen in your hand.

In this chapter, we turn to martingales, which play a central role in probability theory. We illustrate their use in a number of applications to the analysis of discrete stochastic processes. After some background on stopping times and a brief review of basic martingale properties and results, we develop two major directions. We show how martingales can be used to derive a substantial generalization of our previous concentration inequalities – from the sums of independent random variables we focused on previously to nonlinear functions with Lipschitz properties. In particular, we give several applications of the method of bounded differences to random graphs. We also discuss bandit problems in machine learning. In the second thread, we give an introduction to potential theory and electrical network theory for Markov chains. This toolkit in particular provides bounds on hitting times for random walks on networks, with important implications in the study of recurrence among other applications. We also introduce Wilson’s remarkable method for generating uniform spanning trees.