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This four‑lecture series introduces the Tutte polynomial of matroids, one of the most influential and extensively studied invariants in modern combinatorics. The first half of the course reviews essential notions from graph theory and the classical Tutte polynomial. The second half develops the foundations of matroid theory, including the greedy algorithm, and culminates with the definition and properties of the Tutte polynomial for matroids.

Only a very basic mathematical background is required; familiarity with set theory and elementary linear algebra is sufficient.
 

Chapter 1: Basics of graph theory;
Chapter 2: Tutte polynomial and applications;
Chapter 3: Basics of Matroid Theory,
Chapter 4: Tutte Polynomial of a Matroid.
I could add one more chapter ("Ribbon graphs and Delta-matroids"), depending on the interest of the students...