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Sets
Collections of objects
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The formal language of mathematics
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Collections of elements
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Mappings between sets
Linear Algebra
Representations of linear systems
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Spaces supporting addition and scalar multiplication of elements
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Vector spaces with additional structure
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Mappings between vector spaces
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Multi-dimensional data structures
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Fundamental property of square matrices
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Fundamental decomposition of square matrices
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Decompositions of square and non-square matrices
Real Analysis
Real-valued numbers and functions
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Scalar quantities
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Ordered collections of numbers
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Behaviour as points are approached
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Sums of numbers
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Whether a function is unbroken
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A function's rate of change
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The space under a function
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Calculus of fields, curves and surfaces
Topology
Nearness, limits and continuity in general spaces
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Sets with a notion of distance
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Spaces without missing limits
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Nearness without distance
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Finiteness for infinite sets
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Spaces that are all of a piece
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Functions as points of a space
Differential Equations
Equations relating functions to their derivatives
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Equations relating a function to its derivatives
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Equations in a first derivative
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Equations in a second derivative
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Several unknowns that evolve together
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Solving equations through series and integrals
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Equations in several variables
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Equations with conditions at both ends
Optimisation
Maximising and minimising functions
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Fundamental property of a function's shape
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Finding maximum and minimum values
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Reducing set of valid optimal values
Probability
Occurrences of uncertain events
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Sets of experiment outcomes
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Model for assigning probabilities to events
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Using events that have occurred
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Mapping from outcomes to a measurable space
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Information about a distribution's shape
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Common functions for probability models
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Inequalities describing deviation of random variables
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Behaviour after randomness is observed for a while
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Behaviour after many outcomes have been observed
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Models for random systems
Frequentist Statistics
Drawing conclusions from data
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Selecting observations from a population
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Information in statistics
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Calculating parameters from data
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Finding parameter values from data
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Finding plausible ranges of parameter values from data
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Deciding whether data supports an idea
Bayesian Statistics
Modelling parameters as random variables
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Updating beliefs about parameters with data
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Representing beliefs before data are observed
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Estimates, intervals and decisions from the posterior
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Approximating posteriors that cannot be found in closed form
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Comparing, combining and checking models
Causal Inference
Effects of interventions from data
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Effects as comparisons of outcomes under different treatments
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Encoding causal assumptions as directed graphs
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When causal effects can be computed from observed data
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Estimating effects by adjusting for observed covariates
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Identifying effects through variation that affects only the treatment
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Identification from the structure of observational data
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Effects through intermediate variables and over time
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Effects that vary across units and robustness to unmeasured confounding
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Learning causal graphs from data