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• Pair-Free Tilings in Two and Three Dimensions (in progress)

  We investigate domino tilings in two dimensions and determine geometric conditions that force tilings to contain certain substructures. In particular, for a special class of regions in the square lattice, we derive an inequality involving the number of flippable pairs and "windmills" contained in a tiling. Finally, we extend the result to the three-dimensional cubic lattice.

miscellaneous

• On an Elementary Criterion for Monotonicity

  In this note, we discuss a well-known Chern-Lashof-type criterion providing sufficient conditions for a domain to possess cross-sections homeomorphic to balls along at least one direction. We demonstrate how this can be used to prove that all polygons with five or fewer sides are monotone, answering in the affirmative a question posed by an anonymous user on a mathematics forum.