We begin this note, by asking the following question: for which angles can you rotate the graph of a differentiable function about the origin and still obtain the graph of a function? Using only methods from calculus we are able to provide an answer. In particular, we observe a distinction in the answer among odd and even degree polynomials.
This question fits nicely in the curriculum of a first course of calculus and many more question stem from this one, making it a nice project to continue.
The Stable Derived Category of a Ring was introduce for Noetherian rings, by Henning Krause. In my thesis I focus in applying Model Categories to obtain this category as it's Homotopy Category for any ring R. The model structure is constructed over the category of Chain Complexes. The fact that we can construct a model category over this homotopy category helps us to understand this category at the model category level, which also allow us to perform some computations. In particular we consider the ring of polynomials on two variables modulo the quadratic forms. This ring it's not quasi-Frobenius nor Gorenstein; these last two classes of rings are well documented cases.
In this work, we studied a recursive formula for k-Schur fucntions. In my master thesis I implemented a combinatorial proof of a recursive formula for Schur functions; this combinatorial approach served us as the initial step. We generalized this recursive formula, along with its proof, to the more general k-Schur functions.