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.

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.

**Research project:** ”Yang-Baxter equation and related algebraic 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.

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

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.

**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

**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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