A sample curriculum for a full year or semester course is as follows (20 lectures based on Bi195 at Caltech):
1. Basics of linear algebra [Ch 2]:
Introduction to the course; vectors; linear operators; basis sets;
2 - Operator algebra [Ch 2]
Addition and multiplication of linear operators; inverse operator; determinant; change of basis; scalar product and metric;
3 - Eigenvalues [Ch 2]
Eigenvalues and eigenvectors, diagonalizing a matrix; special matrix properties
4 - Linear systems [Ch 3]
Superposition; translation invariance; delta function; convolution; sinusoids as eigenfunctions
6 - Applications of linear systems [Ch 4]
Observing molecular dynamics; filtering and signal processing; crystal analysis; point spread function in microscopy; deconvolution;
7 - Basics of probability [Ch 6]
Motivation (e.g. Luria-Delbrück fluctuation test); events and probabilities; Bayes' rule; random variables; combinatorics; binomial distribution; mean and standard deviation
8 - Probability distributions [Ch 6]
Poisson distribution; Gaussian distribution; moments of a distribution; Central limit theorem; examples and applications
9 - Continuous random variables [Ch 6]
Example distributions: uniform/exponential/gaussian; sums and products of random variables; multi-variate distributions;
10 - Inference and model testing [Ch 7]
Inference; maximum likelihood; parameter estimation; hypothesis testing; mathematical origin of common statistical tests; linear regression; bootstrapping;
13 - Random time series [Ch 8]
Random processes; stationarity; Markov property; correlation function; power spectrum; Chapman-Kolmogorov equation;
14 - Information theory [Ch 8]
Entropy; mutual information; capacity; redundancy; channel coding theorem; data processing inequality; Shannon-Hartley theorem; applications
15 - Applications of probability methods [Ch 9]
Fluctuation correlation analysis; elements of population genetics; communication in the nervous system;
16 - Basics of dynamical systems [Ch 11]
Motivation; dynamical system concepts; 1D dynamics (fish farm); fix points; basin of attraction; linearization; bifurcation;
17 - 2D linear systems [Ch 11]
2D systems; linear systems; eigenvalue analysis; classification of fix points;
18 - 2D nonlinear systems [Ch 11]
Bifurcations in 2D systems; Lotka-Volterra models; centers and limit cycles; relaxation oscillator;
19 - Dynamics in 3D and higher [Ch 12]
Asymptotic behavior in 2D; Poincare-Bendixson theorem; Chaos
20 - Applications of nonlinear dynamics [Ch 13]
Turing model of biological pattern formation; circadian rhythms; repressilator model; others.
Mathematics in Biology