Ilaria Colazzo
PhD
University of Exeter Department of Mathematics Exeter, UK
Programming Language
GAP
Python
Language
Italian
English

Education
PhD in Mathematics and Informatics
University of Salento and University of Basilicata, Lecce, ItalyJuly 20, 2017

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.

Master's Degree in Mathematics
University of Salento, Lecce, ItalyOctober, 25, 2012

Title: Clifford Algebras (it. Algebre di Clifford)

Supervisor: Prof. Francesco Catino

Grade: 110/110 cum laude

Degree (it. Licenza Scuola Normale Superiore ISUFI Pre-Laurea I livello
University of Salento – Scuola Normale Superiore ISUFIJune 15, 2009

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

Supervisor: Prof. Massimo De Vittorio

Bachelor's Degree in Mathematics
University of Salento, Lecce, ItalyDecember, 18, 2008

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

Research Experience
Postdoctoral Researcher fellow
University of Exeter01/06/202021 – Present

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

Postdoctoral Researcher in Mathematics
Vrije Universiteit Brussel, Brussels, Belgium01/10/2019 – 28/02/2021

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.

Postdoctoral Researcher in Mathematics
University of Salento, Lecce, Italy01/12/2017 – 30/11/2018

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.

Postdoctoral Researcher Visiting (host Prof. E. Jespers)
Vrije Universiteit Brussel, Brussels, Belgium01/09/2018 – 30/11/2018

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.

PhD Student
University of Salento and University of Basilicata, Lecce, Italy03/03/2014–20/07/2017

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.

Visiting PhD Student (host Prof. J. Okninski)
University of Warsaw, Warsaw, Poland15/11/2016–20/12/2016

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

Pubblication

Published papers

Teaching experience
University of Salento, Lecce, Italy
  • 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.
Talks
  • 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.
Summer School
  • 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.
Other Activities
Member of the Scientific Committee
2019 – present
  • 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)
Member of the Organizing Committee
2015 – present
Co-organizer of the Department seminar
2015 – present

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

Representative Ph.D. Student
University of Salento2014-2016
Representative Student

University of Salento2010 – 2012