Title: Left Semi-Braces and the Yang-Baxter equation

Supervisor: Prof. Francesco Catino

Final Judgement: cum laude

Description: My thesis focuses on studying set-theoretical solutions of the Yang-Baxter equation via a new algebraic structure called semi-braces. In 1992, Drinfel'd introduced set-theoretical solutions, posed the question of finding all of these solutions and suggested to narrow solutions that are also involutive. In 2007, Rump presented braces (a ring-like structure) to answer Drinfel'd question in the involutive case using group-theoretical and ring-theoretical tools.In my thesis, I introduced semi-braces that cover, as a particular case, braces and allows obtaining solutions that are left non-degenerate. In the first chapter, I gave the notion of semi-braces. Then, in the next chapter, I presented the asymmetric product of semi-braces, a new technique for obtaining several examples of semi-braces. In the third chapter, I introduced the matched product of solutions a novel construction technique of semi-braces that allow analyzing the solution associated with a semi-brace. Finally, in the last chapter, I focused on the application of semi-brace in two fields: the Yang-Baxter equation and regular subgroups. In the first part of this chapter, I proved that every semi-brace leads to a solution. Whereas, in the second part, we advance the connection between semi-brace and regular subgroups of the holomorph. I use a particular semi-braces, the braces over a field, to answer the problem of finding regular subgroups of the affine group posed by Liebeck, Praeger, and Saxl.

Title: Clifford Algebras (it. Algebre di Clifford)

Supervisor: Prof. Francesco Catino

Grade: 110/110 cum laude

Disssertation Title: Development of geometry manipulation tools for atomistic simulations"

Supervisor: Prof. Massimo De Vittorio

Title: On the Galois group of a polynomial of degree five (it.  Sul gruppo di Galois di un polinomio di quinto grado)

Supervisor: Prof. Maria Maddalena Miccoli

Grade: 110/110 cum laude

I am working on the EPSRC project: Hopf-Galois Theory and Skew Braces (PI: N. Byott).

Scientific advisor: Prof. Eric Jespers

Studied the algebraic structure of a semi-truss. Deepen the connection between set-theoretic solutions associated to a semi-truss and set-theoretic solutions associated to semi-braces. Classified the involutive solutions of the Pentagon equation.

Research project: ”Yang-Baxter equation and related algebric structures”.

Scientific advisor: Prof. Francesco Catino

Introduced a novel construction technique for set-theoretical solutions of the Yang-Baxter equation that to obtain new classes of involutive and idempotent solutions. This technique led to characterizing completely the solutions associated with a finite semi-brace.

Established collaborations with E. Jespers, A. Van Antwerpen and C. Verwimp to obtain a description of solutions of the Yang-Baxter equation associated with almost semi-braces.

Presented new construction techniques (i.e., the Hochschild product and the asymmetric product) for regular subgroups of an affine group using their relationship with braces over a field. Advanced the state-of-the-art of the algebraic structures related to set-theoretical solutions of the Yang- Baxter equation by introducing semi-braces.

Collaborated with J. Okninski to study solutions of the Yang-Baxter equation which are not necessarily involutive.

• Instructor
  Duties included preparing lectures, fielding of student inquiries, and office hours.
  • Mathematics preliminary course for bachelor’s students in Mathematics and in Physics - A.Y. 2017–18
• Teaching assistant
  Duties included preparing lectures for tutorial classes for recitation classes and office hours.
  • Group Theory (Course Professor F. Catino) - A.Y. 2017–18
  • Group Theory (Course Professor F. Catino) - A.Y. 2016–17
  • Group Theory (Course Professor F. Catino) - A.Y. 2015–16
  • Algebra II (Course Professor M. M. Miccoli) - A.Y. 2015–16
• Tutor
  Duties included preparing lectures for tutorial classes, fielding of student inquiries, and office hours.
  • Algebra I and II (Course Professors F. Catino and M. M. Miccoli) - A.Y. 2018–19
  • Algebra I (Course Professor F. Catino) - A.Y. 2016–17
• Algebra, Exam committee member (it Cultore della materia) - July 2016 – present
  Duties included grading final exams.

• International Conference Young Researchers Algebra Conference 2019 held in Naples (Italy) on September 16–18, 2019 (with M. Brescia, M. Ferrara, P. Stefanelli, and M. Trombetti)

Algebra Reading Seminar, University of Salento, Department of Mathematics and Physics