Cohomological field theories and global spectral curves

In my thesis I consider interplay between several different structures in mathematical physics. These structures are used to solve a large class of problems in enumerative algebraic geometry and combinatorics in a universal way. The problems can range from counting certain one-dimensional drawings on two-dimensional surfaces to counting maps of certain type from a two-dimensional surface to some higher-dimensional space. The structures that we study in this thesis allow to encode the solutions to this type of enumerative and combinatorial problems in some general compact form. In one approach the solutions to the enumerative problems are encoded in a complex algebraic curve with certain functions on it. From this initial small set of data one can reconstruct the full solution with the help of a recursive procedure that is absolutely universal and does not depend on a particular problem. In another approach the solutions to the enumerative problems are encoded as certain integrals over some complicated spaces that parametrize different complex structures on two-dimensional surfaces. This reveals that the solutions to the enumerative problems reflect the geometric properties of the space of complex structures, also in a universal way. These two approaches turn out to be related in many different ways. In this thesis their relation is studied in the framework of an advanced differential geometric structure called Frobenius manifold