Magnitude homology, as introduced by Hepworth and Willerton, is a bigraded homology theory of metric spaces that categorifies Leinster's notion of magnitude. The definition of magnitude was motivated, and naturally arose, from considerations on Euler characteristic in enriched category theory, and it has found many interesting connections with other areas of mathematics; such as differential geometry, graph theory, category theory, entropy, and even biological diversity. Magnitude homology satisfies all reasonable axioms of homology theories: Mayer-Vietoris and Kunneth theorems are prominent examples. The goal of the thesis is to study the magnitude homology of metric spaces in general, and its relation with reachability homology using the magnitude-path spectral sequence. 

Hochschild cohomology plays a central role in the study of algebraic structures, representation theory and investigations of associative algebras, capturing both algebraic and homotopical properties. It has been extensively studied in contexts ranging from representation theory to algebraic geometry and topology. To each simplicial complex X, one can consider the so-called incidence algebra I(X) of the face poset of X. Then, a classical theorem of Gerstenhaber and Schack shows that Hochschild cohomology of I(X) is isomorphic to the homology of X. Goal of the thesis is to study the diagrammatic Hochschild homology of algebras and prove the theorem of Gerstenhaber and Schack using path homology of graphs. 

Sam and Snowden have recently introduced the notion of (quasi)-Grobner categories, that is categories satisfying "good" combinatorial conditions. The advantage of the framework developed by Sam and Snowden is that one can study representation theory of categories and talk about Noetherianity of functors in a way that generalizes the classical notions. The goal of the thesis is to explore the theory of(quasi)-Grobner categories and explore its applications; such as a criterion for a “rationality” result for Hilbert series of representations, connecting it to the theory of formal languages. 

The goal of the thesis is to study homological stability properties of mapping class groups. We will be generally interested in homological invariants of families of groups, and in their “stable” homology. In fact, homological stability has shown that, for certain families such as the mapping class groups, it is easier to compute infinitely many homology groups at once - rather than computing single homology groups of the family. The proof of homological stability uses a spectral sequences argument, known as the Quillen argument, and in the case of mapping class groups of 3-manifolds this is due to Hatcher and Wahl. 

At the crossroads between geometry, algebra, model theory, and the computer science, the Tarski-Seidenberg theorem deals with subsets of the Euclidean space, showing that a projection of such a set on a hyperplace is still of this form. This project aims at exploring this fascinating result and its implications in logic, algebraic geometry, or computational complexity, just to name a few of the possibilities.

Does there exist an explicit criterion to determine whether a given Diophantine equation admits integer solutions? This problem, formalized in the context of computability theory, was shown to have a negative solution. The aim of this project is to unravel the proof of this deep result and explore similar decidability problems in number theory.

The goal of this project is to consider (quasicoherent?  perverse?) sheaves in classical algebraic topology, and investigate how they can be recast in the definable context, where spaces of sections are endowed with the structure of (phantom) Polish spaces

Schauder bases are a fundamental notion in Banach Space theory. In this project their fundamental properties will be studied, and higher-order generalizations indexed by walks on ordinals will be constructed using methods from analysis and set theory.

TBA

TBA

• Diamond and all things Suslin

• Szemeredi past and present

• An Introduction to Sheaf Theory and the Foundations of Condensed Mathematics, UNIBO, Bachelor Thesis

• Caratteristiche Cardinali del Continuo e Teorema di Bell, UNIBO, Bachelor Thesis

• Introduzione alla Teoria Dei Modelli e sue Applicazioni ai Campi Algebricamente Chiusi, UNIBO, Bachelor Thesis

• Introduzione alla Teoria Descrittiva degli Insiemi Effettiva, UNIBO, Bachelor Thesis

• Rango di Morley, Forking e Indipendenza, UNIBO, Master Thesis

• Sistemi deduttivi in teoria della dimostrazione e didattica, UNIBO, Master Thesis

• Homological Algebra for pro-Lie and Locally Compact Polish groups, UniTrento, Master Thesis