Notes

These 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.

Topology 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. It includes discussions of basic definitions, connectedness, the separability and countability axioms, compactness, metric spaces and theorems about them, and topological groups.

These notes are partially 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.

These 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, \(L_p\) spaces, relationships between types of convergence, and large-number results such as the Weak Law of Large Numbers, the Strong Law of Large Numbers, the Central Limit Theorem, and so on. In particular, it contains an extensive description (with a diagram) of the various ways that one can translate between different sorts of convergence.

These notes serve as an introduction to elementary graph theory. They begin with the basic definition of a graph as a collection of vertices with edges between them, move on to introducing important types of graphs. The rest of the notes consist of introductions to various graph-theoretic topics, including Eulerian and Hamiltonian cycles, planar graphs and their relation to platonic solids, introductory Ramsey theory, and the Lindström-Gessel-Viennot Lemma and applicatoins.

Combinatorics is the math of counting things. These notes begin by introducing the foundation of combinatorics, the binomial coefficient, and then move on to discuss topics such as recurrence relations, generating functions, applications to complexity theory, the Catalan, Bell, and Stirling numbers, and connections between combinatorics and extremal geometry. Finally, it concludes with a brief look at algebraic combinatorics with group actions and Burnside's Lemma.

Chip-firing is a discrete form of diffusion, where a certain quantity of chips are placed at various points in a graph and stabilize by spreading out from areas of high concentration. The resulting field mixes graph theory, combinatorics, and group theory, translating between each field as varying perspectives offer fitting solutions. These notes introduce the basic of chip-firing, the correspondence between critical configurations and superstable configurations, Dhar's burning algorithm, and sandpile groups.

These notes are an introduction to set theory and the set-theoretic construction of basic mathematical objects. They begin with the Zermelo-Frankael axioms and the axioms of choice, moving on to using these axioms to construct objects such as tuples, functions, relations, and so on. The next three sections are respectively associated with the construction of the real numbers, the ordinal numbers, and the cardinal numbers. The notes conclude with a discussion of the continuum hypothesis, cofinality, inacessible cardinals, and applications of set theory.

These notes discuss commutative algebra, an extension of ring theory. Commutative algebra studies the structure of various types of rings (such as Neotherian rings, local rings, discrete valuation rings, etc.) and their properties. These topics provide useful algebraic tools for other fields, particularly algebraic geometry. As an example, the basics of varieties (including the Nullstellensatz and the Zariski topology) are presented from the perspective of commutative algebra.

These notes introduce category theory, an approach to the foundations of math which offers a unifying language for many different areas of the field. It begins with basic definitions before moving to the abstract notion of limits and colimits, of which various constructions (products, kernels, images, pullbacks, etc.) are specific examples. Then, it moves on to the discussion of relationships between categories with functors, natural transformations, adjoints, and Yoneda's Lemma. Finally, it concludes with an application of these ideas to algebraic topology.

These notes cover algebraic geometry, and are sourced from a combination of Hartshorne, Brian Conrad's notes, and Foundations of Algebraic Geometry by Ravi Vakil. They discuss varieties and elementary facts about them 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.