The field of tile self-assembly originates in the area of tiling. In 1961, Wang introduced Wang tiles, that is, equally sized 2-dimensional unit squares with labels on each edge. The process starts with a specific predetermined configuration called seed, and stops if no further tile can be attached. Berger proved that this model provides a Turing-universal computation model, based on these tiles.

Inspired by the natural self-assembly processes in chemistry and in biology, in the 1980s, Seeman pioneered the design of nanostructures based on DNA molecules. He showed that DNA is the perfect ingredient for assembling nanostructures. Seeman demonstrated that one can manipulate DNA to build very complex nanometric structures.

In 1994, inspired by these works on DNA nanostructures, Adelman used such tiny DNA objects to perform computations. He designed a nanoscopic DNA-based computing system solving the Traveling Salesman Problem (TSP) on an instance with seven cities. In this problem, one is given a set of cities and their pairwise distances, and one must find a tour with shortest total length that passes through all cities. This is a very famous problem in theoretical computer science that is very difficult to solve even for powerful computers. So, this result was very inspiring and demonstrated the potential of nanometric DNA-based computing techniques. He encoded an instance of this famous problem as DNA molecules, and the molecules assembled in such a way as to provide a solution to the problem. This feature is now labeled as DNA computing.

DNA tile self-assembly, combines the ideas of tiling, of self-assembly, and of DNA computing . This field was pioneered by Winfree in his 1998 PhD thesis. Inspired by the works of Wang, Seeman, Adelman and others and combining their approaches, he introduced the abstract Tile Assembly Model (aTAM), that is a mathematical translation of real-life experiments. The goal is to design tiles that can be built experimentally using DNA molecules. This model uses Wang tiling, with the extra information of a integer strength for each label, and the notion of temperature to help controlling the assemblies. These molecular tiles, when placed in a suitable solution in suitable quantities, self-assemble, and produce specific shapes that represent computations. The power of DNA self-assembly enables to compute anything that (given enough time) is computable by a computer: it is Turing-universal. This has proved the importance of this model and increased he interest of researchers.

In the last twenty-five years, DNA tile self-assembly has seen a huge development. We recall the history of DNA tile self-assembly and we explain the classic aTAM model for tile self-assembly \cite{patitz2014introduction}. Most works in this field deal with tile self-assembly on the 2D plane. The central question that we ask, is, can we design a model for doing tile self-assembly on 3D surfaces, and detect the type of surface that it is performed on? We answer positively by designing a model, called Surface Flexible Tile Assembly model (SFTAM), and characterizing the genus of some polycubes via it. This is joint work with Florent Becker, my PhD supervisor.