Lecture notes and exercises

MA 764
Spring 2021
Max Kutler

Written solutions to starred exercises may be submitted at any time but no later than April 30. Exercises will be graded and returned to you within two weeks.



Lecture 1: Independent sets and examples (Exercises)

Lecture 2: Circuits and cryptomorphism (Exercises)

Lecture 3: Circuits continued (Exercises)

Lecture 4: Rank, closure, flats (Exercises)

Lecture 5: Flats of graphic matroids, uniform matroids (Exercises)

Lecture 6: Simple matroids (Exercises)

Lecture 7: Geometric representation (Exercises)

Lecture 8: Examples (Exercises)

Lecture 9: Strong basis exchange, duality (Exercises)

Lecture 10: Duality (Exercises)

Lecture 11: More duality (Exercises)

Lecture 12: Duality for representable matroids (Exercises)

Lecture 13: The lattice of flats

Lecture 14: More about flats (Exercises)

Lecture 15: Minors (Exercises)

Lecture 16: Excluded minor theorems

Lecture 17: Direct sum, truncation (Exercises)

Lecture 18: Connectivity (Exercises)

Lecture 19: More connectivity (Exercises)

Lecture 20: The characteristic polynomial (Exercises)

Lecture 21: Properties of the characteristic polynomial (Exercises)

Lecture 22: The chromatic polynomial of a graph (Exercises)

Lecture 23: Log-concavity

Lecture 24: Möbius inversion (Exercises)

Lecture 25: The finite field method

Lecture 26: Whitney numbers and Weisner's theorem (Exercises)

Lecture 27: Coefficients of the reduced characteristic polynomial (Exercises)

Lecture 28: Initial descending flags (Exercises)

Lecture 29: The Orlik-Solomon algebra (Exercises)

Lecture 30: Cohomology of the complement (Exercises)

Lecture 31: The Poincaré polynomial (Exercises)

Lecture 32: Towards a compactification of the complement (Exercises)

Lecture 33: The wonderful compactification

Lecture 34: The boundary of the wonderful compactification

Lecture 35: The cohomology of the wonderful compactification (Exercises)

Lecture 36: The Chow ring of a matroid (Exercises)

Lecture 37: The Chow ring and flags of flats (Exercises)

Lecture 38: The top-degree part of the Chow ring

Lecture 39: The Kähler package

Lecture 40: Log-concavity via the Hodge-Riemann relations

Lecture 41: The Bergman fan

Lecture 42: Minkowski weights, the degree map, the permutohedral variety, etc.