Bistable elliptic equations with fractional diffusion

This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.

q-Hook length formulas for colored labeled forests

The major index has been deeply studied from the early 1900s and recently has been generalized in different directions, such as the case of labeled forests and colored permutations. In this thesis we define new types of labelings for forests in which the labels are colored integers. We extend the definition of the flag-major index for these labelings and we present an analogue of well known major index hook length formulas. Finally, this study (which has just apparently a simple combinatoric nature) allows us to show a notion of duality for two particular families of groups obtained from the product G(r,n)×G(r,m).