Admission to the Final Examination - PHD Programme in Mathematics
The final exam seminar will take place in presence and online through the ZOOM platform.
Thursday 18 January 2024
10:30 Michele Battagliola - Algebraic constructions for multi party protocols with focus on threshold signatures
Abstract: Group actions are fundamental mathematical tools, with a long history of use in cryptography. Indeed, the action of finite groups at the basis of the discrete logarithm problem is behind a very large portion of modern cryptographic systems. With the advent of post-quantum cryptography, however, other group actions, like isogeny or code based ones, received interest from the cryptographic community, attracted by the possibility of translating old discrete logarithm based functionalities. In this talk we show that isomorphism problems which stem from (non-abelian) cryptographic group actions can be viable building blocks for threshold signature schemes. Moreover we show how cryptographic group actions can be used to design other multi party protocols, such as oblivious transfers.
Supervisor: Nadir Murru
10:50 Marzio Mula - Pairings and graphs of elliptic curves
Abstract: The recent attacks on the isogeny-based protocol SIDH have raised the question of whether other isogeny-based protocols, such as CSIDH, are still secure. After introducing CSIDH and its variants, we describe a strategy, based on the construction of suitable pairings, which can be combined with the SIDH attack to break some "weak" variants of CSIDH.
In the last part of the talk, we turn to graphs of elliptic curves that are not the classic isogeny graphs used in SIDH or CSIDH. We call them Hessian graphs, as they arise from the notion of Hessian variety, and we show how their regularity can also be relevant in the light of cryptographic applications.
Supervisors: Nadir Murru, Federico Pintore
11:10 Giulio Binosi - Fueter and Almansi theorems for slice regular functions of several quaternionic variables
Abstract: We broaden some definitions and give new results about the theory of slice functions of several quaternionic variables. We introduce the notions of partial spherical value and derivative for functions of several variables that extend those of one variable, recovering some of their properties and discovering new ones. This leads to a generalization of Fueter’s theorem for slice regular functions of several quaternionic variables. Furthermore, partial spherical derivatives can be use to obtain different Almansi de compositions for slice functions of several variables. The components of each decomposition, defined explicitly through partial spherical derivatives, exhibit desirable properties, such as harmonicity and circularity. As consequences of these decompositions, we give another proof of Fueter’s theorem in H^n , establish the biharmonicity of slice regular functions in every variable and, time permitting, derive some integral formulas for them.
Supervisor: Alessandro Perotti
11:30 Gloria Tabarelli - Edge-colorings and flows in Class 2 graphs
Abstract: We consider edge-colorings and flows problems in Graph Theory that are hard to solve for Class 2 graphs. Most of them are strongly related to some outstanding open conjectures, such as the Cycle Double Cover Conjecture, the Berge-Fulkerson Conjecture, the Petersen Coloring Conjecture and the Tutte’s 5-flow Conjecture. We obtain some new restrictions on the structure of a possible minimum counterexample to the former two conjectures. We prove that the Petersen graph is, in a specific sense, the only graph that could appear in the Petersen Coloring Conjecture, and we provide evidence that led to propose ananalogous of the Tutte’s 5-flow conjecture in higher dimensions.
Supervisors: Giuseppe Mazzuoccolo (Università di Verona), Peter Michael Schuster (Università di Verona)
16:30 Agnese Del Zozzo - Geogebrizzazione di testi matematici come processo di oggettivazione
Abstract: Alan Shoenfeld evidenzia due propositi per la ricerca in didattica della matematica: uno puro sulla comprensione della natura del pensiero, dell’apprendimento e dell’insegnamento della matematica; e uno applicato che riguarda l’implementazione in termini di ingegneria didattica delle conoscenze elaborate nell’ambito puro. Questa tesi abbraccia principalmente il primo proposito e riguarda la definizione e la caratterizzazione del processo di geogebrizzazione di un testo matematico, un processo strutturato di natura semiotica che parte da una porzione di testo matematico stampato e arriva a una combinazione appropriata di risorse della GeoGebra Service Platform (www.geogebra.org) che “spacchetta” la matematica incorporata nel testo. L’outcome di tale processo è un artefatto online utilizzabile per scopi comunicativi, divulgativi o didattici.
L’attività di geogebrizzazione di un testo verrà inquadrata alla luce della Teoria dell'Oggettivazione di Radford e la domanda di ricerca generale affrontata è: quando si geogebrizza un testo matematico, cosa succede al modo di pensare/ragionare in relazione al contenuto matematico del testo stesso? Per rispondere, è stata progettata un’attività sperimentale basata su un estratto del libro Lezioni di geometria analitica e proiettiva di Guido Castelnuovo (1904-1905), svolta da 4 coppie di partecipanti. Per considerare diverse prospettive e livelli di esperienza nello studio, i partecipanti hanno background differenti. I principali risultati raggiunti riguardano:
• Natura del processo: geogebrizzare un testo matematico è un’esperienza di apprendimento;
• Prospettive di implementazione: linee di progettazione per un’attività di geogebrizzazione di un testo matematico generico;
• Possibili finalità per l’outcome finale: geogebrizzare un testo matematico come esempio di pratica di valorizzazione e attualizzazionedi testi storici e i loro contenuti.
Supervisors: Claudio Fontanari, Giorgio Bolondi (Università di Bolzano)
16:50 Giacomo Vianello - Boundary regularity properties for almost-minimizers of the relative perimeter
Abstract: Given a convex set K in the Euclidean n-space, we focus on the boundary behavior of an almost-minimizer E for the relative perimeter in K. We show that, provided n=3, the closure of the internal boundary of E cannot contain vertex-type singularities of the boundary of K. One of the intermediate results, that for instance allows us to consider a larger class of almost-minimizers, is a boundary monotonicity formula valid under some mild, extra assumptions on K.
Supervisor: Gian Paolo Leonardi
Abstract: In this doctoral research, an exploration of gene regulatory networks unfolds, where the focus is on enhancing inference methodologies through the incorporation of prior knowledge. Gene regulatory networks, depicted as directed graphs with causal edges, serve as the backdrop. A comprehensive review of existing inference methods reveals the superior performance of approaches grounded in structural equation models (SEM). Notably, SEM integrates both gene expression data and cis-eQTLs — genetic variations influencing gene expression — as insightful components. This incorporation of cis-eQTLs is instrumental in unraveling the causality and directionality of edges within the gene network.
Building upon this foundation, the investigation delves into refining the inference process. The study extends into the realm of bioinformatic databases housing functional gene networks, where connections emerge based on shared pathways or biological processes. Additionally, databases housing information on transcription factors and their targets contribute valuable insights to further optimize the precision and depth of gene network inference. This research seeks to advance the field by synthesizing diverse data sources, pushing the boundaries of inference accuracy in gene regulatory networks.
Supervisor: Mauro Lauria
