MCS 549 - Mathematical Foundations of Data Science
University of Illinois at Chicago
Fall 2026

This course covers the mathematical foundations of modern data science from a theoretical computer science perspective. Topics will include random graphs, random walks, Markov chains, streaming algorithms, learning theory, clustering, singular value decomposition, and random projections.

Basic Information

Syllabus: pdf
Time and Location: M-W-F 9:00-9:50 am, 101 Stevenson Hall (SH)
Instructor: Lev Reyzin, SEO 417
Online Textbook: Avrim Blum, John Hopcroft, and Ravi Kannan, Mathematical Foundations of Data Science
Office Hours: T 11:00-11:50am (online), F 11:00-11:50am (in-person)

Presentations

Problem Sets

problem set 1 due 10/2/26

Lectures and Readings

Lecture 1 (8/24/26)
covered material: intro to the course, preview of the material
reading: chapter 1

Lecture 2 (8/26/26)
covered material: some concentration inequalitiess
reading: chapters 2.1 - 2.2

Lecture 3 (8/28/26)
covered material: geometry in high dimensions, the unit ball, sampling from a high-dimensional sphere
reading: chapters 2.3 - 2.5

Lecture 4 (8/31/26)
covered material: Gaussian annalus theorem, random projections and the Johnson-Lindesnstrauss (JL) lemma
reading: chapters 2.6 - 2.7

Lecture 5 (9/2/26)
coveredmaterial: singular value decomposition (SVD), best-fit subspaces, and optimality of greedy algorithm
reading: chapters 3.1 - 3.4

Lecture 6 (9/4/26)
covered material: computing SVD, power iteration, separating Gaussians
reading: chapters 3.7 - 3.9

Lecture 7 (9/9/26)
covered material: applications of JL
reading: N/A

lecture 8 (9/11/26)
covered material: SVD in recommender systems
reading: N/A

Lecture 9 (9/14/26)
covered material: intro to Markov chains, stationary distribution, Fundamental Theorem of Markov Chains
reading: chapter 4 intro, 4.1

Lecture 10 (9/16/26)
covered material: Markov chain Monte Carlo (MCMC), Metropolis-Hasting, Gibbs Sampling
reading: chapter 4.2

Lecture 11 (9/18/26)
covered material: volume estimation of convex bodies
reading: chapter 4.3

Lecture 12 (9/21/26)
covered material: bounding mixing time with normalized conductance
reading: chapter 4.4

Lecture 13 (9/23/26)
covered material: electrical networks and random walks, harmonic functions
reading: begin chapter 4.5

Lecture 14 (9/25/26)
covered material: probabilistic interpretation of voltage, current, and effective resistence
reading: finish chapter 4.5

Lecture 15 (9/28/26)
covered material: hitting and commute times on unweighted graphs
reading: chapter 4.6

Lecture 16 (9/30/26)
covered material: cover time, random walks on lattices, the web as a Markov Chain
reading: chapters 4.7 and 4.8

Lecture 17 (10/2/26)
covered material: intro to machine learning, PAC learning
reading: chapter 5.1

Lecture 18 (10/5/26)
covered material: Occam's razor theorem
reading: chapter 5.4

Lecture 19 (10/7/26)
covered material: VC dimension
reading: chapter 5.5

Lecture 20 (10/9/26)
covered material: SVM, part I
reading: begin Support-Vector Networks by Cortes and Vapnik

Lecture 21 (10/12/26)
covered material: SVM, part II
reading: finish Support-Vector Networks by Cortes and Vapnik

Lecture 22 (10/14/26)
covered material: kernel functions
reading: chapter 5.3