Sets

Collections of objects

• Logic

  The formal language of mathematics

• Sets

  Collections of elements

• Functions

  Mappings between sets

Real Analysis

Real-valued numbers and functions

• Real Numbers

  Scalar quantities

• Sequences

  Ordered collections of numbers

• Limits

  Behaviour as points are approached

• Series

  Sums of numbers

• Continuity

  Whether a function is unbroken

• Derivatives

  A function's rate of change

• Integrals

  The space under a function

Linear Algebra

Representations of linear systems

• Vector Spaces

  Spaces supporting addition and scalar multiplication of elements

• Inner Product Spaces

  Vector spaces with additional structure

• Matrices

  Multi-dimensional data structures

• Determinants

  Fundamental property of square matrices

• Eigenvalues and Eigenvectors

  Fundamental decomposition of square matrices

• Matrix Decompositions

  Decompositions of square and non-square matrices

• Linear Transformations

  Mappings between vector spaces

Probability

Occurrences of uncertain events

• Events

  Sets of experiment outcomes

• Probability Spaces

  Model for assigning probabilities to events

• Conditional Probability

  Using events that have occurred

• Random Variables

  Mapping from outcomes to a measurable space

• Distributions

  Common functions for probability models

• Moments

  Information about a distribution's shape

• Limit Theorems

  Behaviour after many outcomes have been observed

• Concentration Bounds

  Inequalities describing deviation of random variables

• Probabilistic Convergence

  Behaviour after randomness is observed for a while

• Stochastic Processes

  Models for random systems

Frequentist Statistics

Drawing conclusions from data

• Samples

  Selecting observations from a population

• Sufficiency

  Information in statistics

• Estimators

  Calculating parameters from data

• Point Estimation

  Finding parameter values from data

• Interval Estimation

  Finding plausible ranges of parameter values from data

• Hypothesis Testing

  Deciding whether data supports an idea

Bayesian Statistics

Modelling parameters as random variables

Optimisation

Maximising and minimising functions

• Convexity

  Fundamental property of a function's shape

• Unconstrained Optimisation

  Finding maximum and minimum values

• Constrained Optimisation

  Reducing set of valid optimal values