**Big ideas in applied math**

- Markov chains
- Concentration inequalities
- Low-rank matrices
- The fast Fourier transform
- Galerkin approximation
- Sparse matrices
- Smoothness and degree of approximation
- The Schur complement

**Markov musings (introduction to the mathematical analysis of Markov chains)**

**Low-rank approximation toolbox**

- What is Nyström approximation?
- Nyström approximation, Cholesky factorization, and the Schur complement
- Randomized SVD
- Analysis of the randomized SVD

**Sketching**

**Trace estimation**

- Stochastic trace estimation
- How good can stochastic trace estimates be?
- Don’t use Gaussians in stochastic trace estimation

**Numerical linear algebra: Do’s and don’ts**

- Don’t solve the normal equations
- The better way to convert an SVD into a symmetric eigenvalue problem

**Explainers**

- Rejection sampling
- Five interpretations of kernel quadrature
- Chebyshev polynomials
- Sherman–Morrison for integral equations
- The Vandermonde decomposition
- The elegant geometry of generalized eigenvalue perturbation theory
- Why randomized algorithms?
- Minimal rank completions

**High-dimensional probability**

- The hard way to prove Jensen’s inequality
- Norm of a Gaussian random vector
- The Hanson–Wright inequality
- Gaussian hypercontractivity
- Hanson–Wright and trace estimation with random vectors on the sphere

**Book reviews**