A categorical view of algebraic theories

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Carles Casacuberta[en] Classically, algebraic structures such as groups, rings, and many others were jointly studied with the language of universal algebra. It was later found that certain tools from category theory, called monads, are especially suitable to encode the whole amount of information contained in algebraic theories. In this work we discuss monads, and, in particular, some monads that are relevant in functional programming in Computer Science. We give a proof of the equivalence between the category of algebraic theories (formalized as Lawvere theories) and the category of finitary monads on the category of sets. We also prove that there is an equivalence between the category of algebras over a monad and the category of models of the associated Lawvere theory. Finally, we apply this equivalence of categories to give a new proof of the fact that all localizations on the category of abelian groups can be uniquely lifted to $R$-modules for every ring $R$

Wavelet pooling for convolutional neural networks

Treballs Finals de Grau d'Enginyeria Informàtica, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Petia Radeva,[en] Wavelets are mathematical functions that are currently used in many computer vision problems, such as image denoising or image compression. In this work, first we will study all the basic theory about wavelets, in order to understand them and build a basic knowledge that allows us to develop another application. For such purpose, we propose two pooling methods based on wavelets: one based on simple wavelet basis and one that combines two basis working in parallel. We will test them and show that they can be used at the same level of performance as max and average pooling