**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.

**Published papers**

**I. Colazzo**, E. Jespers, A. Van Antwerpen, C. Verwimp, Left non-degenerate set-theoretic solutions of the Yang-Baxter equation and semitrusses, J. Algebra (2022)**I. Colazzo**, Braces: between regular subgroups and solutions of the Yang-Baxter equation, Algebra for Cryptography, Collectio Ciphrarum, (Extended abstract), ISBN 9791259943286 (2021).- F. Catino,
**I. Colazzo**, P. Stefanelli, Set-theoretic solutions to the Yang-Baxter equation, Forum Math., 33 (2021), n. 3 **I. Colazzo**, A. Van Antwerpen,*The algebraic structure of left semi-trusses*, J. Pure Appl. Algebra, 225 (2021) n.2, 106467.**I. Colazzo**, E. Jespers, L. Kubat,*Set-Theoretic Solutions of the Pentagon Equation*, Commun. Math. Phys., 380 (2020).- F. Catino,
**I. Colazzo**, P. Stefanelli,*The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order*, Mediterr. J. Math., 17 (2020), n. 2. - F. Catino,
**I. Colazzo**, P. Stefanelli,*The matched product of solutions to the Yang-Baxter equation*, J. Pure Appl. Algebra 224 (2020) n. 3, 1173–1194. - F. Catino,
**I. Colazzo**, P. Stefanelli,*Skew left braces with non-trivial annihilator*, J. Algebra Appl. 18 (2019) n.2, 1950033, 23 pp. - F. Catino,
**I. Colazzo**, P. Stefanelli,*Semi-braces and the Yang–Baxter equation*, J. Algebra, 483, (2017), 163–187. - F. Catino,
**I. Colazzo**, P. Stefanelli,*Regular subgroups of the affine group and asymmetric product of radical braces*, J. Algebra, 455, (2016), 164–182. - F. Catino,
**I. Colazzo**, P. Stefanelli,*On regular subgroups of the affine group*, Bull. Aust. Math. Soc., 91 (2015), 76–85.

**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*.

*Braces; between regular subgroups and solutions of the Yang-Baxter equation*, Workshop Algebra for Cryptography, L'Aquila (Italy) October 10-11, 2019 (Invited talk).*The matched product of shelves,*Conference*Advances in Group Theory and Applications 2019*, Lecce (Italy), June 25-28, 2017.*Regular subgroups and left semi-braces*, ALGB Seminar, Vrije Universiteit Brussel (Belgium), October 31, 2018 (Invited talk).*The matched product of the solutions of the Yang-Baxter equation*, Meeting*Noncommutative and non-associative structures, braces and applications*, 11-15 March, 2018 (Invited talk).*Skew braces with non-trivial annihilator,*International Conference*Advances in Group Theory and Applications 2017*, Lecce (Italy), September 5-8, 2017.*Semi-braces and the Yang-Baxter equation*, International Conference*Groups, Rings and the Yang- Baxter equation*: Spa (Belgium), June 18-24, 2017.*The algebraic structure of semi-brace*, International Conference*Young Researchers Algebra Conference 2017*, Napoli (Italy), May 23–24, 2017.*Regular subgroups of the affine group*, Seminar Algebra, University of Warsaw (Poland), November 24, 2016 (Invited talk).*Radical braces and the Yang–Baxter equation*, International*Conference on Rings and Polynomials*: Graz (Austria), July 3-8, 2016.*Regular subgroups of an affine group*,*Meeting Group Theory in Florence - A meeting in honour of Guido Zappa*: Florence (Italy), June 16-17, 2016.*The Asymmetric Product of radical braces*, International Conference*Advances in Group Theory and Applications 2015*, Porto Cesareo (Lecce, Italy), June 16-19, 2015.

*Advances in Group Theory and Applications 2016 - The School*, Vietri sul Mare (Salerno, Italy), June 6-10, 2016, organized by F. Catino, M. De Falco, F. de Giovanni, C. Musella, courses held by C. Casolo, A ́. del R ́ıo, A. Facchini, A. J. Macintyre.*Bicocca Ph.D. School on Representation Theory 2014*, Università di Milano- Bicocca (Italy), June 16-18, 2014, organized by M. Avitabile, F. Dalla Volta, L. Di Martino, A. Previtali, P. Spiga and Th. Weigel, courses held by M. Isaacs and G. Navarro.*Advances in Group Theory and Applications 2014 - The School*, Porto Cesareo (Lecce, Italy), June 3-6, 2014, organized by F. Catino and F. de Giovanni, courses held by E. Aljadeff, A. Ballester- Bolinches, F. de Giovanni and H. Laue.

- 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

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