**UCLA Course:** Math 285N, Fall 2022.

**Instructor:** Igor Pak

(see email instructions on the bottom of the page).

**Class schedule:** MWF 3:00-3:50 pm, MS 6201.

**Office Hours:** M 4:00-4:50

**Grading:** The grade will be based on attendance and class participation.

**Domino tilings**- W.P. Thurston, Conway's tiling groups (1990); the original article by Thurston describing his approach.
- J.C. Fournier, Tiling pictures of the plane with dominoes (1997); concise description of the algorithm and complexity applications.
- T. Chaboud, Domino tiling in planar graphs with regular and bipartite dual (1996); a small but insightful well written generalization.

**Combinatorial group theory**- J.H. Conway, J.C. Lagarias, Tiling with polyominoes and combinatorial group theory (1990); the original article.
- W.P. Thurston, Groups, Tings and Finite State Automata (1989); unpublished lecture notes describing the connection and giving more context.

**Tilings with two bars**- C. Kenyon, R. Kenyon, Tiling a polygon with rectangles (1992); the original article.
- E. Remila, Tiling a polygon with two kinds of rectangles (2005), a generalization.
- R. Muchnik, I. Pak, On tilings by ribbon tetrominoes (1999); here Lemma 2.1 (the "induction lemma") is given with a proof which is omitted/sketched in the C-L paper.

**Flip graph**- N.C. Saldanha, C. Tomei, M.A. Casarin Jr. and D. Romualdo, Spaces of domino tilings (1995), original article.
- H. Parlier, S. Zappa, Distances in domino flip graphs (2016), a friendly exposition.

**Height functions and extension theorems**- I. Pak, A. Sheffer and M. Tassy, Fast domino tileability (2016).
- A.V. Akopyan, A.S. Tarasov, A constructive proof of Kirszbraun's theorem (2008), a clean proof of (discrete) Kirszbraun theorem
- U. Brehm,
Extensions of distance reducing mappings to
piecewise congruent mappings on L
^{m}(1981), original article. - (continuous) Kirszbraun theorem,
*Wikipedia*. - S.S. Tasmuratov, The bending of a polygon into a polyhedron with a given boundary (1974), the oldest and most similar geometric result to that by Tassy.

**Ribbon tilings and rim hook bijection**- Murnaghan-Nakayama rule,
*Wikipedia*. - S.V. Fomin and D.W. Stanton, Rim hook lattices,
*St. Peterburg Math Journal*, 1997. - I. Pak, Ribbon tile invariants,
*Trans. AMS*(2000). - C. Moore and I. Pak, Ribbon tile invariants from signed area,
*JCTA*(2002). - S. Sheffield, Ribbon tilings
and multidimensional height functions,
*Trans. AMS*(2002).

- Murnaghan-Nakayama rule,
**Matchings via identity testing**- Polynomal identity testing,
*Wikipedia* - Schwartz-Zippel lemma,
*Wikipedia* - M. Goemans, these lecture notes
- A. Sinclair, these lecture notes
- V. Kabanets and R. Impagliazzo, Derandomizing polynomial identity tests means proving circuit lower bounds (conference version)

- Polynomal identity testing,
**Number of domino tilings**- L. Lovasz and M.D. Plummer,
*Matching Theory*, Chapter 8. - R. Kenyon, An introduction to the dimer model
- A. Kaufer, these notes
- N. Robertson, P.D. Seymour, R. Thomas, Permanents, Pfaffian orientations, and even directed circuits
- L. Pachter, Combinatorial approaches and conjectures for 2-divisibility problems concerning domino tilings of polyominoes

- L. Lovasz and M.D. Plummer,
**Tilings of rectangles**- N.G. de Bruijn and D.A. Klarner, A finite basis theorem for packing boxes with bricks (1975), original article (first correct proof in full generality)
- M. Reid, Klarner Systems and Tiling Boxes with Polyominoes (2004), a simplified proof with interesting applications
- D.A. Klarner, Packing a rectangle with
congruent
*N*-ominoes (1969); a number of interesting examples. - M. Reid, Tiling rectangles and half strips with congruent polyominoes (1997); more examples and discussion of the odd order conjecture.
- J. Yang, Rectangular tileability and complementary tileability are undecidable (2012), a complementary hardness result.

**Geometric tilings**- C. Freiling, D. Rinne, Tiling a square with similar rectangles (1994); necessary and sufficient conditions for tiling squares with rectangles similar to a given.
- M. Laczkovich, G. Szekeres, Tiling of the square with similar rectangles (1995); an independent proof.
- C. Freiling, M. Laczkovich, D. Rinne, Rectangling a rectangle (1997); a generalization to rectangular regions.
- M. Prasolov, M. Skopenkov, Tiling by rectangles and alternating current (2010); an alternative proof.

**Counting tilings**- D. Merlini, R. Sprugnoli, M.C. Verri, Strip tiling and regular grammars (2000), rational tilings
- S. Garrabrant, I. Pak, Counting with irrational tiles (2014), irrational tilings.

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*Last updated 11/30/2022*.