1. Foundations, Logic, And Discrete Structures Mathematics before numbers: a framework needed to define what truths exist, to organize, and to collect sets of existing things. Mathematical Logic/Set Theory: Boolean logic (AND/OR/NOT) and ZFC set theory, upon which mathematics is built. Sets are expressions that define a collection of things; mathematics is built from sets and their relations. Proof Theory: The art of proving mathematical statements. Mathematicians use direct proofs, induction (proving that the first domino falls, and that whenever one domino falls, the next also falls), and contradictions (proving that something is true by showing that its negation leads to a contradiction). Combinatorics/Graph Theory: Counting theory and graph theory are branches of mathematics concerning discrete, countable entities. Combinatorics develops counting techniques, whereas graphs are diagrams of vertices connected by edges. 2. Pure Architecture: Algebra And Topology This branch of mathematics concerns itself with abstract structures, transformations, and symmetries. Abstract Algebra: Higher-level algebra built from axioms, with groups (structures with operations that follow particular rules), rings, and fields (systems where algebraic operations are consistently definable). Linear algebra concerns itself with vector spaces and linear transformations, allowing problems with infinitely many variables to be solved at once; it underpins machine learning and computer graphics. Topology: Also known as rubber sheet geometry, topology examines the properties of space that are preserved under stretching. Certain shapes can be transformed into others via stretching (a coffee mug and a donut, in topological definitions, are equal). 3. Number Theory And The Arithmetic Cosmos Number theory concerns itself with the properties of integers, particularly prime numbers (numbers divisible only by themselves and 1). Elementary number theory: The fundamental theorem of arithmetic, concerned with the properties of prime numbers and their distribution. Analytic number theory: A branch concerned with real and complex analysis, examining number theory questions with calculus. Cryptographic applications : Modern cryptography is largely based on number theory, particularly on the distribution of prime numbers; RSA encryption, for instance, relies on the difficulty of obtaining the prime factors of large integers. 4. Calculus, Analysis, And Differential Equations Calculus concerns itself with infinitesimal changes and rates of change; it examines how systems change when there is an underlying, non-discrete continuity. Real/complex analysis: Rigorous foundations for calculus, extending it to complex numbers. Differential equations: Equations with derivatives in them, which describe rates of change. They form the backbone of modern science. The Intersecting Machinery Of Change To understand how calculus describes systems changing from an initial state to a final state, one must understand the link between differentiation and integration. 5. Applied Mathematics: Data, Probability, And The Physical Universe Mathematics begins when physical phenomena and processes are converted into statements about quantities. Probability: A powerful branch of mathematics concerning itself with randomness and stochastic processes (random events that follow particular rules). At the lowest level, it examines coin tosses and dice rolls; at higher levels, it describes financial instruments, such as stock indices. Statistics: The science of analyzing data and making quantitative inferences from them. Numerical analysis: Computational algorithms used to solve problems that are too complex to be solved analytically (with a pen and paper). Many mathematical models, such as those that describe the weather, can only be approximated using supercomputers. 6. The Foundations Of Computing And Complexity Theory Computer science concerns itself with automata (computers) and the problems that they can theoretically solve. Automata: Mathematical models of computers, programmed to examine certain mathematical questions. Complexity theory: Closely related to automata, complexity theory defines problems according to how much time and memory a computer would need to solve them. P vs. NP concerns whether a problem that can be quickly solved can also be quickly reversed.
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