| Category |
Main topic |
Subtopics |
Some useful references |
| Math 1 |
Real Analysis |
- The basics of mathematical logic
- The language of set theory
- Sequences, subsequences, liminf/limsup, limits
- Functions, continuity, intermediate value theorem
- Differentiation, chain rule
- Series, integration, fundamental theorem of calculus, integration by parts
- Univariate Taylor series
- Overview of metric spaces, metric and examples
- Basic definitions: limit/interior points, closed/open/bounded/connected/compact sets
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| Prob 1 |
Basic Probability Theory |
- Probability axioms and basic definitions
- Conditional probability and independence
- Bayes' theorem
- Combinatorics and counting techniques
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| Math 2 |
Linear Algebra |
- Vector spaces, linear independence, span, basis
- Linear maps, matrices, rank, nullity
- Eigenvalues and eigenvectors
- Determinant and trace
- Matrix decompositions (Cholesky, EVD, SVD)
- Counting parameters in the different factorizations
- Matrix phylogeny
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| Prob 2 |
Random Variables and Distributions |
- Important discrete random variables
- Important continuous random variables
- Joint, marginal, and conditional distributions
- Important multivariate random variables
- Transformations and functions of random variables
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| Math 3 |
Vector Calculus |
- Partial differentiation and gradients
- Gradients of vector-valued functions
- Gradients of matrices
- Some useful identities
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| Prob 3 |
Expectation and Probabilistic Inequalities |
- Expectation, variance, and higher moments
- Markov's and Chebyshev's Inequalities
- Holder’s and Jensen’s Inequalities
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| Math 4 |
Continuous Optimization |
- Unconstrained optimization, gradient descent
- Constrained optimization, Lagrange multipliers
- Overview of convex optimization
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| Prob 4 |
Stochastic Convergences |
- Almost sure convergence
- Convergence in probability
- Convergence in distribution
- Convergence in mean
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