Whenever I read a book, I try to create notes for myself on the most important parts. When I think that a set of notes I've made that I think are polished enough to be helpful in some capacity, they are added to The Tree, making a sort of personal encyclopedia of mathematics. My hope is that by the time that I graduate, the Tree will be roughly complete. Indeed, about half of these sections have been completed, they just haven't been polished yet. Following are links to the notes that have been published:
Analytic Number TheoryThese notes, made during Math 155 taught by Lynnelle Ye at Stanford, discuss various techniques and problems from analytic number theory. Analytic number theory generally concerns the asymptotic behavior of various sequences related to the behavior of the integers. This includes a dicsussion of the Riemann zeta-function, the Riemann Hypothesis, and proofs of the Prime Number Theorem, Dirichlet's Theorem on Primes in Arithmetic Progressions, and Waring's Problem.
Elementay Number TheoryThese notes are based on Math 152, a class taught by Kannan Soundararajan at Stanford. They are an introduction to elementary number theory, including the proofs of unique factorization and the basic properties of the integers, an introduction to arithmetical functions, quadratic reciprocity, Hensel's Lemma, quadratic forms, and some elements of computational number theory. Finally, there is a section on perfect numbers vs. Mersenne primes and Diophantine approximations.
Probability TheoryThese notes, made during Math 230A taught by Sourav Chatterjee at Stanford. They discuss elementary measure theory, Lebesgue integration, random variables, and various probabilistic tools such as inequalities, Lp spaces, relationships between types of convergence, and large-number results such as the WLLN, SLLN, and CLT.
Graph TheoryGraph theory starts with a simple premise: dots and lines between them. Yet as often happens in math, the simplest assumptions give rise to complex and intrincate structures. These notes introduce graph theory and offer some applications to different areas of math.
CombinatoricsCombinatorics is the math of counting things. These notes begin with the basics - the binomial coefficient - and work through recurrence relations, generating functions, and even some basic algorithms. Finally, it concludes with a brief look at algebraic combinatorics with Burnside's Lemma.
Category TheoryCategory theory offers a unifying language for many different areas of math. It helps formalize the notion of "portals" between different subfields of math, with its notion of functors, and helps place the meta-structure of math in sharp relief.
Set TheorySet theory is the formal foundation of all math. It codifies everything from the definition of the natural numbers to a precise notion of infinity. It also contains numerous useful tools for other areas of math - in particular, Zorn's Lemma will be used to prove any vector space has a basis.
Chip-FiringRoughly speaking, chip-firing can be understood as a discrete form of diffusion, where a certain quantity of chips are placed in a graph and stabilize by spreading out from areas of high concentration. As such, the game of chip-firing is combinatorial in nature, but it is usually analyzed with tools from algebra.
TopologyTopology is a foundational language used in many different fields of math, from algebraic topology to differential geometry to algebraic geometry to functional analysis. This summary of the first half of Munkres' text on the subject can serves as a reference for terms and theorems about topological/metric spaces in their full generality.
Commutative AlgebraCommutative algebra is an extension of ring theory, studying the structure of various types of rings (Noetherian, local, etc.) and their properties. Commutative algebra provides numerous algebraic tools which are essentially for algebraic geometry, and indeed the basics of varieties are discussed here.
Algebraic Geometry (In Progress)These notes, made while reading Hartshorne with additions from Brian Conrad's notes and FOAG by Ravi Vakil, cover algebraic geometry. They discuss elementary methods of varieties before moving into a description of schemes and morphisms between them. In the future, a discussion of Chapter III of Hartshorne, curves and their intersections, arithmetic geometry, and computational algebraic geometry will be added.
Disclaimer: I am not an expert in any of these fields: my notes are just meant to help you get a glimpse at the notes' subject. Also, even widely-used textbooks have mistakes, so verify each theorem and argument for yourself!