This course is a mathematical introduction to artificial intelligence and large language models. We will study vectors, matrices, tensors, embeddings, gradient descent, automatic differentiation, attention mechanisms, and generative models. The course will emphasize hands-on mathematical examples, especially simple low-dimensional versions of ideas that appear in modern AI systems.
Term: Summer 2026 (June 25, 2026 - August 26, 2026)
Class meetings: Monday, Wednesday, Friday, 2:10–3:15 PM
X-hour: Thursday, 1:20–2:10 PM
| Component | Points |
|---|---|
| Homework | 10 |
| Class Activities | 10 |
| Quiz | 20 |
| Final Project | 20 |
| Midterm | 30 |
| Final Exam | 10 |
| Total | 100 |
1) Homework and class activities are graded for completion (you don't need to get a complete correct answer to get the points), with no feedback.
1) Quiz problems are selected from homework problems of that week, with possibly a minor modification.
Overview of AI, neural networks, LLMs, tensors, training, and generative models. Review of vectors, matrices, and functions.
Vectors, matrices, matrix multiplication, linear classifiers, and neural network layers.
Norms, distances, dot products, cosine similarity, projections, high-dimensional geometry, word embeddings, matrix factorization, and encoder-decoder models.
Tensors, tensor products, tensor networks, contractions, batching, and tensor shapes.
Partial derivatives, gradients, chain rule, loss functions, gradient descent, and training.
Dual numbers, computational graphs, automatic differentiation, and backpropagation.
Midterm: The midterm will take place during this part of the course.
Vector fields on a high-dimensional sphere. Vector fields represented by kernels.
Many-particle dynamics on the sphere, next-token prediction, and the attention mechanism.
Project presentations. Time permitting: probability distributions, conditional probability, maximum likelihood, diffusion models, and final project discussions.
By the end of the course, students should be able to:
Students with disabilities who may need disability-related academic adjustments and services for this course are encouraged to see me privately as early in the term as possible. Students requiring disability-related academic adjustments and services must consult the Student Accessibility Services office (Carson Hall, Suite 125, 646-9900). Once SAS has authorized services, students must show the originally signed SAS Services and Consent Form and/or a letter on SAS letterhead to me. As a first step, if students have questions about whether they qualify to receive academic adjustments and services, they should contact the SAS office. All inquiries and discussions will remain confidential.