Deformed mirror symmetry for punctured surfaces

In this thesis we devote ourselves to a phenomenon that comes from physics and is known as homological mirror symmetry. The mirror does not concern reflection of light, but is about a mysterious correspondence between two types of geometry. The one geometry is concerned with so-called symplectic structures, that is, structures to which the principle of the Hamiltonian equations of motion applies. The other geometry deals with so-called complex structures, that is, structures where for instance a surface is parameterized by means of the complex numbers.
Kontsevich put this correspondence into a promising mathematical form in 1994. In his form we are dealing with so-called categories, actually an abstraction of the symplectic or complex structure that is extremely flexible. It is unfortunately difficult to show that the deformation theory of a Fukaya category corresponds exactly to symplectic deformations of the geometric object.
In this thesis, we work out the correspondence of deformation theories for the simple case where the spaces under consideration are so-called punctured surfaces. In the first part of the thesis, we determine the entire deformation theory of the Fukaya category of such a punctured surface. In the second part, we determine all the information we might need to achieve the corresponding deformation on the complex side. In the third part, we actually construct the corresponding deformation and show that it is indeed the correct one. This way, we have successfully deformed homological mirror symmetry in the case of punctured surfaces.