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Nicolau C. Saldanha: Domino tilings in dimensions 2 and 3

Tid: Må 2018-11-19 kl 11.00 - 12.00

Plats: Room 14, building 5, Kräftriket, Department of Mathematics, Stockholm University 

Medverkande: Nicolau C. Saldanha (PUC-Rio)

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Abstract:
The problem of counting and classifying domino tilings (or dimer partitions) has applications in physics and chemistry and uses tools of several areas of mathematics. Far more is known about the case of dimension 2 than about higher dimension.

This talk focuses on local moves, particularly the flip, the only non-trivial move which involves only two dominoes. We give a sketch of the proof that, for connected and simply connected planar regions, the space of tilings is connected via flips. The corresponding result does not hold in dimension 3, not even if the region is a box. There exists an integer valued number associated to each tiling, called the twist, which assumes several values and which is invariant under flips. There are also examples of tilings for which no flip is possible. We present a detailed description of the example of the 4x4x4 box, a few theorems and several conjectures.

Includes joint work with J. Freire, C. Klivans, P. Milet, C. Tomei and others. Includes results due to W. Thurston and many others.