1. Calculus of one variable
1.1 The exponential function and Euler’s identity
1.2 Gaussian functions and integrals
1.3 Fourier series and transforms
1.4 Sigmoid functions and their properties
1.5 The gamma function and Stirling’s formula
1.6 Exercises and solutions
2. Calculus of several variables
2.1 Differentiable maps f : ℝⁿ → ℝᵐ
2.2 The Hessian matrix and Newton’s method
2.3 Higher dimensional sigmoid functions and classifications
2.4 Backpropagation: chain rule and automatic differentiation
2.5 Exercises and solutions
3. Matrix Algebra
3.1 Systems of linear equations
3.2 Vector spaces
3.3 Bases and dimension
3.4 Inner products and norms
3.5 The determinant of a square matrix
3.6 Eigenvalues, eigenvectors, and eigenspaces
3.7 Special types of matrices and useful factorizations
3.8 Quaternions and rotation matrices
3.9 Convolutions of matrices
3.10 Exercises and solutions
4. Probability
4.1 Some basic probability
4.2 Random variables and distributions
4.3 The Borel-Cantelli lemma
4.4 Weak and strong laws of large numbers
4.5 Exercises and solutions
5. Graphs, shifts, and stochastic matrices
5.1 Graphs
5.2 Entropy and decision trees
5.3 Coding information with symbols
5.4 Markov shifts and subshifts of finite type
5.5 The Perron-Frobenius Theorem and applications
5.6 Exercises and solutions
6. Neural networks
6.1 Mathematical structure of a neural network
6.2 Backpropagation: training a neural network
6.3 Convolution layers
6.4 Exercises and solutions
Biography
Jane Hawkins is a Professor Emerita at the University of North Carolina at Chapel Hill who has held faculty positions at Stony Brook University, Cal Tech, and Duke University, with over 50 research papers published in dynamical systems, ergodic theory, differentiable and complex dynamics, Markov shifts, and HIV and Ebola virus dynamics, and is the author of two books, Ergodic Dynamics and The Mathematics of Cellular Automata. An inaugural American Mathematical Society (AMS) Fellow, she chaired the AMS Committee on Science Policy for two years, testified before Congressional committees on the importance of science and mathematics in the U.S., and served as a Program Director in the Division of Mathematical Sciences at the National Science Foundation. Her teaching spans multivariable calculus, linear algebra, differential equations, probability theory, and dynamical systems, often using computational tools, and she has supervised 20 Ph.D. and master's students, many contributing to mathematical breakthroughs through computer-generated insights. She has delivered over 160 research talks across four continents and numerous public lectures on fractals, virus classification, and HIV dynamics, while also teaching undergraduate courses on dynamics, cellular automata, and probability in society, having received her Ph.D. from the University of Warwick in England as a Marshall Scholar.
