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En aquest treball s'introdueixen les principals nocions de teoria de matroides, matroides transversals i extensions d'un sol element. El nostre primer objectiu és estudiar quines extensions d'una matroide transversal són també transversals. En particular, centrem el nostre estudi en la família de matroides uniformes. El conjunt d'extensions d'una matroide és un reticle sota l'anomenat weak order. Intentem respondre a la pregunta de si el conjunt d'extensions transversals d'una matroide uniforme és també un reticle o no. També dissenyem un algoritme per catalogar i contar matroides transversals fins a una certa mida del ground set. Per fer-ho, les construim a partir de la més petita, a base de iterativament extendre presentacions minimals, fent servir les eines que hem desenvolupat anteriorment. En vista que l'estratègia no troba realment totes les matroides transversals, també estudiem com són les matroides que no es poden construir amb aquesta estratègia i quines propietats satisfan. En este trabajo introducimos las nociones básicas de teoría de matroides, matroides transversales y extensiones de un solo elemento. Nuestro primer objetivo es estudiar las extensiones de matroides transversales que también son transversales. En particular, centramos nuestro estudio en la familia de las matroides uniformes. El conjunto de extensiones de una matroide es un retículo bajo el llamado weak order. Intentamos responder a la pregunta de si el conjunto de extensiones transversales de un matroide uniforme es también un retículo o no. También diseñamos un algoritmo que cataloga y cuenta matroides transversales hasta un tamaño fijo del ground set. Para ello, las construimos desde cero extendiendo iterativamente presentaciones minimales utilizando las herramientas que hemos visto anteriormente. Dado que la estrategia no cuenta realmente todas las matroides transversales, estudiamos también qué matroides no son alcanzables utilizando esta estrategia y qué propiedades satisfacen. In this work we introduce the basic notions of matroid theory, transversal matroids and single-element extensions. Our first objective is to study which extensions of a given transversal matroid are also transversal. In particular, we focus our study on the family of uniform matroids. The set of single-element extensions of a matroid is a lattice under the so-called weak order. We try to answer the question of whether the set of transversal extensions of a uniform matroid is also a lattice or not. We also design an algorithm that catalogs and counts transversal matroids up to a fixed size of the ground set. To do so, we build them from scratch by iteratively extending minimal presentations using the tools that we have previously seen. In view that the strategy does not actually count all transversal matroids, we also study which matroids are not reachable using this strategy and what properties do they satisfy.

The main objective of this bachelor's thesis will be to present the concept of quasirandomness in various mathematical contexts while proving all the pertinent results. We will introduce the results of Fan Chung and Ronald Graham on quasirandom graphs and quasirandom sets, and the results of Timothy Gowers on quasirandom groups. To further display the connection between randomness and structuredness, we will also present some other influential results on the subject, i. e. Roth's Theorem on arithmetical progression on subsets of Zn, Erdős' Theorem on sum-free sets of integers, and the Alon-Boppana Bound for the spectral expansion of a graph.

Matroids arise from the abstract notion of dependency. Matroids can be studied from different points of view. From linear algebra we know matrices, which can be seen as matroids, however matroids generalise the concept of dependency. In matroids there also is combinatorics, graph theory and geometry. This project wants to be an introduction to matroids throughout two its families: positroids and lattice path matroids. Starting with basic definitions and examples the project reaches deeper and more interesting relations between these two families.

Inspired by prior results by Stanley and Leroux showing what information can be recovered from an isomorphism of incidence algebras, we investigate the very same idea applied to decomposition spaces. We review the work of Stanley and Leroux and provide sufficient background on the homotopy theory of groupoids to be able to define decomposition spaces, equivalences of them as linear functors and solve the isomorphism problem for both the groupoid-level coalgebra and for the numerical incidence algebra.

The main objective of this thesis is to present and prove Plünnecke's Inequality, a theorem that gives bounds for sumsets in commutative groups. An introduction to the theory of set addition is presented. Three different proofs of Plünnecke's Inequality are presented, two of them relying strongly on graph theory, the third being more elementary. Some other tools and techniques are introduced to obtain generalizations of Plünnecke's Inequality. The most important are the Plünnecke-Ruzsa Inequality, that gives bounds to general sum-and-difference sets in commutative groups, and some generalizations to the non-commutative case, related to Tao's Theorem. Some other generalizations involve the sum of different sets. The results are used to prove an important structural result.