Instructor: Nadia Lafrenière

Course on canvas.dartmouth.edu.⇗

Course Schedule

Date Content References
     
January 4 What is geometric combinatorics? Download What is geometric combinatorics? [MR07]
January 6 Class is cancelled  
January 9 Posets: General definitions Download Posets: General definitions [EC1, §3.1, 3.3]
January 11 Lattices Download Lattices [AOC, §5.3]
January 13 Möbius function & inversion Download Möbius function & inversion [EC1, §3.5, 3.6, 3.7, 3.8], [AOC, §5.4, 5.5]
January 16 Class is postponed, Martin Luther King day  
January 18 Hyperplane arrangements: General definitions Download Hyperplane arrangements: General definitions [Sta07, Lecture 1]
January 19 Intersection poset, counting regions Download Intersection poset, counting regions (x-hour) [Sta07, Lecture 2]
January 20 Graphical arrangements, characteristic polynomial, Zaslavsky's theorem for counting regions Download Graphical arrangements, characteristic polynomial, Zaslavsky's theorem for counting regions [Sta07, Lecture 2]
January 23 Polytopes: Notions, f-vector Download Polytopes: Notions, f-vector [Zie07]
January 25 Lattice polytopes, Pick's theorem and idea for Ehrhart theory Download Lattice polytopes, Pick's theorem and idea for Ehrhart theory [CCD, §2]
January 27 Ehrhart theorem Download Ehrhart theorem [CCD, §2,3]
January 30 Triangulations of a convex polygon and flips Download Triangulations of a convex polygon and flips [Wil14]
February 1 New direction: Cluster algebras from triangulations of polygons, total positivitiy and Laurent polynomials Download New direction: Cluster algebras from triangulations of polygons, total positivitiy and Laurent polynomials [FR07, §4] Links to an external site., [ST09 Links to an external site.]
February 3 Intro to matroids. Geometric lattices, graphical and oriented matroids (Lectures notes are the beautiful notes of Federico Ardila, page 1-6). [AM07 Links to an external site.]
February 6 Matroid invariants: Characteristic polynomial, independence and broken circuit complexes, Tutte polynomial Download Matroid invariants: Characteristic polynomial, independence and broken circuit complexes, Tutte polynomial (pages 1-5 of the notes) [AM23]
February 8 Combinatorial sequences from matroids (pages 6-10 of the notes) Download Combinatorial sequences from matroids (pages 6-10 of the notes) [AM23], [Eur23]
February 10 Tilings: Escher tilings, the problem, tiling with multiple tiles; periodic and aperiodic tilings; Counting tilings Download Tilings: Escher tilings, the problem, tiling with multiple tiles; periodic and aperiodic tilings; Counting tilings [AS10]
February 13 Convex tilings: Regular polygons and the fifteen pentagons Download Convex tilings: Regular polygons and the fifteen pentagons
February 15 Creating tilings (Beauquier – Nivat's theorem, Conway's criterion) Download Creating tilings (Beauquier – Nivat's theorem, Conway's criterion) [BN91, Fat20]
February 17 Topological methods: Borsuk-Ulam Theorem and Tucker's lemma Download Topological methods: Borsuk-Ulam Theorem and Tucker's lemma [Matousek, §2], [Lon13]
February 20 Brouwer fixed point theorem and Sperner's lemma Download Brouwer fixed point theorem and Sperner's lemma [Matousek, §2], [Lon13, §1]
February 22 Winning at Hex, the ham sandwich and the stolen necklace Download Winning at Hex, the ham sandwich and the stolen necklace [Gal79], [GW85], [Matousek, §3]
February 24 Kneser conjecture Download Kneser conjecture [Matousek, §3.3], [Lon13, §2.1]
February 27 Class is cancelled, Feedback due for buddy's project  
March 1 Student presentations  
March 3 Student presentations  
March 6 Student presentations  
March 8 Written part of project due