@incollection{future_1,
  author = {Colazzo, I.},
  title = {Indecomposable solutions to the {Y}ang--{B}axter equation},
  note = {Lecce (Italy), 5-9/06/2023},
  publisher = {{Advances in Group Theory and Applications 2023}},
  year = {2023-06-05},
  keywords = {upcoming}
}

We will mention combinatorial results about the structure of finite indecomposable solutions to the Yang-Baxter. 
In particular, we will focus on the differences between the involutive and non-involutive cases. A crucial difference between these two classes of solutions is that inovlutive ones naturally embed into solutions associated with the structure group. In contrast, it is well known that this is not always the case for non-involutive solutions. Therefore, our first aim will be to connect the decomposability of solutions with the decomposability of their injectivisation. As a consequence, we will obtain some criteria for decomposability.
The talk will also include concrete examples to understand the construction and some open questions. 

@incollection{future_2,
  author = {Colazzo, I.},
  title = {{D}erived-indecomposable solutions and skew braces with a finiteness condition},
  note = {Omaha/Online, 29/05–02/06/2023},
  publisher = {Hopf algebras and Galois module theory 2023},
  year = {2023-05-29},
  keywords = {upcoming}
}

The problem of finding set-theoretic solutions to the Yang-Baxter equation goes back to Drienfeld. A first attempt to tackle this problem is studying indecomposable solutions.

This talk, based on joint work with M. Ferrara and M. Trombetti, will introduce skew braces in which every element has finitely many conjugates, which might be considered the brace-theoretical counterpart of FC-groups. Moreover, I will show that fundamental examples of such skew braces are structure groups of derived-indecomposable solutions (i.e. solutions whose derived is indecomposable).

@incollection{interplay,
  author = {Colazzo, I.},
  title = {The structure monoid of set-theoretic solutions to the {YBE}},
  note = {Keele University (UK), 18/04/2023},
  publisher = {{The Interplay Between Skew Braces and Hopf-Galois Theory}},
  year = {2023-04-18},
  keywords = {invited},
  file = {keele2023.pdf}
}

In this talk, we will introduce the structure monoid of a set-theoretic so-
lution to the Yang-Baxtr equation and explore some combinatorial aspects.
We will show that the structure monoid is useful for investigating bijective
non-degenerate solutions to the Yang-Baxter equation. This talk is based
on joint work with Jespers, Van Antwerpen and Verwimp and an ongoing
project with Kubat, Jespers, and Van Antwerpen.

@incollection{talk_22,
  author = {Colazzo, I.},
  title = {Derived-indecomposable solutions and skew braces whose elements have a finite number of conjugates},
  note = {Oberwolfach (Germany), 26/02-03/03/2023},
  publisher = {Oberwolfach mini-workshop: Skew Braces and the Yang–Baxter Equation},
  year = {2023-02-26},
  keywords = {invited}
}

(joint work with M. Ferrara and M. Trombetti)

Studying indecomposable solutions is a first attempt to tackle the problem of funding all set-theoretic solutions to the Yang-Baxter equation.
This talk will focus on a class of indecomposable solutions called derived-indecomposable, i.e. solutions, as the name suggests, such that their derived solution is indecomposable. I will show that the study of derived-indecomposable solutions naturally leads to the study of skew braces that behave like FC-groups since this captures the features of the structure skew brace (i.e. the structure group regarded as a skew brace) associated with such solutions.
I will also present some structure results for this class of skew braces and state some open problems.

@incollection{talk_7,
  author = {Colazzo, I.},
  title = {{YB}-semitrusses and left non-degenerate solutions to the {Y}ang-{B}axter equation},
  note = {Omaha (US)/Online, 30/05–03/06/2022},
  publisher = {Hopf algebras and Galois module theory},
  year = {2022-06-01},
  keywords = {invited},
  file = {Colazzo_Omaha_22.pdf}
}

The study of large classes of set-theoretic solutions can be reduced to the study of some associative algebraic objects, such as braces, skew braces, and YB-semitrusses. A YB-semitruss is an algebraic associative structure that forms a sub-category of semitrusses, allowing one to determine and analyse left non-degenerate solutions to the Yang-Baxter equation. Based on joint work with E. Jespers, A. Van Antwerpen and C. Verwimp, this talk will describe the algebraic structure of YB-semitrusses and its relation with solutions to the Yang-Baxter equation. Then, we will show that skew braces are examples of YB-semitrusses.Finally, we will use YB-semitrusses as a tool to prove that for a finite left non-degenerate solution being right non-degenerate is equivalent to being bijective.

@incollection{talk_10,
  author = {Colazzo, I.},
  title = {{YB}-semitrusses with associated bijective solutions},
  note = {Swansea University (UK) - Online, 04–06/01/2022},
  publisher = {Braces in Bracelet Bay},
  year = {2022-01-05},
  keywords = {invited},
  file = {BracesinBraceletBay-I.Colazzo.pdf}
}

A YB-semitruss is an algebraic associative structure that forms a sub-category of semitrusses, allowing one to determine and analyse left non-degenerate solutions to the Yang-Baxter equation. Based on joint work with E. Jespers, A. Van Antwerpen and C. Verwimp, this talk will focus on YB-semitrusses with bijective associated solutions. First, we will show that skew braces are examples of YB-semitrusses. Then, we will use YB-semitrusses as a tool to prove that any finite left non-degenerate set-theoretic solution of the Yang-Baxter equation is right non-degenerate if and only if it is bijective, extending a result obtained by Castelli,Catino and Stefanelli.

@incollection{talk_12,
  author = {Colazzo, I.},
  title = {Skew braces and solutions of the {Y}ang-{B}axter equation},
  note = {Online, 24–27/05/2021},
  publisher = {Hopf algebras and Galois module theory},
  year = {2021-05-26},
  keywords = {invited},
  file = {ColazzoOmaha21.pdf}
}

The Yang-Baxter equation, named after the authors of the seminal papers in which the equation arose, is a fundamental tool in several research fields such as statistical mechanics, quantum group theory, and low-dimensional topology. In 1992, Drinfel’d suggested focusing on set-theoretical solutions of the Yang-Baxter equation as a first step to studying solutions of the Yang-Baxter equation. These solutions were initially approached in a combinatorial fashion by Etingof, Soloviev and Schedler and Gateva-Ivanova and Van den Bergh. More recently, Rump introduced braces, an algebraic structure that generalises radical rings, to describe involutive non-degenerate solutions of the Yang-Baxter equation. To study non-involutive solutions, one needs skew braces, a non-commutative analogue of braces. In this talk, we briefly discuss some basic properties of skew braces and how these structures are related to solutions of the Yang-Baxter equation. We also introduce a useful construction technique that allows one to characterise a subclass of skew braces.

(The talk will be mainly based on joint work with F. Catino and P. Stefanelli.)

@incollection{talk_15,
  author = {Colazzo, I.},
  title = {Braces: between regular subgroups and solutions of the {Y}ang-{B}axter equation},
  note = {L'Aquila (Italy)},
  publisher = {1st workshop Algebra for Cryptography},
  year = {2019-10-11},
  keywords = {invited},
  file = {A4C2019Ilaria_Colazzo.pdf}
}

The Quantum Yang-Baxter equation, named after the authors of the seminalpapers in which the equation arose,  Yang and Baxter ,  is a fundamental toolin several research fields such as statistical mechanics, quantum group theory,and low-dimensional topology.  In 1992, Drinfel’d [Dd92] suggested focusing onset-theoretical solutions of the Yang-Baxter equation as a first step to study-ing solutions of the Quantum Yang-Baxter equation.  These solutions were ini-tially approached in a combinatorial fashion by Etingof, Soloviev and Schedler[ESS99]  and  Gateva-Ivanova  and  Van  den  Bergh  [GIVdB98].   More  recently,Rump  [Rum07]  showed  that  involutive  non-degenerate  set-theoretic  solutionscould be constructed using braces,  an algebraic structure generalizing radicalrings.  Furthermore, it turns out that such structure answers the question posedby Liebeck, Praeger, and Saxl [LPS00] of finding regular subgroups of the affinegroup:   indeed,  a  particular  instance  of  braces  called  brace  over  a  field  is  inone-to-one correspondence with all regular subgroups of a specific affine group[CR09]. In this talk we expose the connection between braces and regular subgroups,then we present some constructions [CCS15, CCS16] of braces that lead to describing particular classes of regular subgroups.
  
References
[CCS15] F. Catino, I. Colazzo, and P. Stefanelli. On regular subgroups of the affine group. Bull. Aust. Math. Soc., 91(1):76–85, 2015.
[CCS16] F. Catino,  I. Colazzo,  and P. Stefanelli.  Regular subgroups of the affine groupand asymmetric product of radical braces. J. Algebra, 455:164–182, 2016.
[CR09] F. Catino and R. Rizzo.  Regular subgroups of the affine group and radical circle algebras. Bull. Aust. Math. Soc., 79(1):103–107, 2009.
[Dd92] V.  G.  Drinfel′d.   On  some  unsolved  problems  in  quantum  group  theory.   InQuantum groups (Leningrad, 1990),  volume  1510  of Lecture Notes in Math., pages 1–8. Springer, Berlin, 1992.
[ESS99] P. Etingof, T. Schedler, and A. Soloviev. Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Math. J., 100(2): 169–209, 1999.
[GIVdB98] T.  Gateva-Ivanova  and  M.  Van  den  Bergh.   Semigroups  of I-type. J. Algebra, x206(1):97–112, 1998.
[LPS00] M.  W.  Liebeck,  C.  E.  Praeger,  and  J.  Saxl.   Transitive  subgroups  of  primitive permutation groups. J. Algebra, 234(2): 291–361, 2000.  Special issue in honor of Helmut Wielandt.
[Rum07] W.  Rump.   Braces,  radical  rings,  and  the  quantum  Yang-Baxter  equation. J.Algebra, 307(1):153–170, 2007.

@incollection{talk_21,
  author = {Colazzo, I.},
  title = {The matched product of the solutions of the {Y}ang-{B}axter equation},
  note = {Malta, 11–15/03/2018},
  publisher = {Noncommutative and non-associative structures, braces and applications},
  year = {2018-03-13},
  keywords = {invited},
  file = {Malta2018-IC.pdf}
}

The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld in [4]. In particular, the class of non-degenerate solutions has received considerable attention [5, 6, 7, 8, 3]. Although interesting and remarkable results on classifying non-degenerate solutions have been presented, there are still many open related problems. One of these problems is how to construct new families of solutions. Initial contribution to this aspect has been given by Etingof, Schedler, and Soloviev, who present a method to obtain a new solution via retraction. Recently, Vendramin in [9] and Bachiller, Cedó, Jespers, Okniński in [1] provide methods to obtain families of solutions starting from others, which are known.
In this talk we introduce a novel construction technique, called matched product of solutions [2], which allows one to obtain new solutions from two generic (not necessarily non-degenerate) solutions. In addition, we prove that the matched product of two left non-degenerate involutive solutions is still left non-degenerate and involutive.

References
[1] D. Bachiller, F. Cedó, E. Jespers, J. Okniński: A family of irretractable square-free solutions of the Yang-Baxter equation, Forum Math., 29 (2017), 1291-1306.
[2] F. Catino, I. Colazzo, P. Stefanelli: The matched product of set-theoretic solutions of the Yang-Baxter equation, in preparation.
[3] F. Cedó, E. Jespers, J. Okniński: Braces and the Yang-Baxter equation, Comm. Math. Phys., 327 (2014), 101-116.
[4] V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math. 1510, Springer, Berlin, (1992), 1-8.
[5] P. Etingof, T. Schedler, A. Soloviev: Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209.
[6] T. Gateva-Ivanova, M. Van den Bergh: Semigroups of I-type, J. Algebra, 206 (1998), 97-112.
[7] J.-H. Lu, M. Yan, Y.-C. Zhu: On the set-theoretical Yang-Baxter equation, Duke Math. J., 104 (1) (2000) 1-18.
[8] W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153-170.
[9] L. Vendramin: Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova, J. Pure Appl. Algebra, 220 (2016) 2064-2076.

@incollection{future_3,
  author = {Colazzo, I.},
  title = {Combinatorial solutions to the {Y}ang-{B}axter equation and skew braces},
  note = {Keele University (UK)},
  publisher = {Discrete Maths seminar series},
  year = {2023-04-19},
  keywords = {seminar}
}

The Yang-Baxter equation is a fundamental equation in theoretical physics and pure mathematics, with many applications in different fields of mathematics. The importance of this equation led Drinfeld to propose the following problem: studying combinatorial solutions. A key tool for exploring this family of solutions is the concept of a skew brace, introduced by Rump and Guarnieri and Vendramin. In this talk, we will focus on combinatorial solutions to the YBE and skew braces, providing a variety of examples and discussing their interplay.

@incollection{UCLouvain,
  author = {Colazzo, I.},
  title = {Combinatorial solutions to the {Y}ang-{B}axter equation and applications},
  note = {Louvain-La-Neuve (Belgium)},
  publisher = {Séminaire de topologie algébrique},
  year = {2023-01-17},
  keywords = {seminar}
}

The first part of the talk will connect the topological problem of distinguishing mathematical knots with combinatorial solutions to the Yang-Baxter equation. The second one will be devoted to a new algebraic structure (skew braces) that provides the perfect tool to study and classify such solutions.

@incollection{talk_6,
  author = {Colazzo, I.},
  title = {Retractability problem for quasi-linear cycle sets},
  note = {University of Warsaw (Poland)},
  publisher = {Seminar Algebra},
  year = {2022-10-20},
  keywords = {seminar},
  file = {IC_warsaw_seminar_compressed.pdf}
}

Involutive set-theoretic solutions to the Yang-Baxter equation have been widely studied in the last two decades, and various connections with algebraic structures have been introduced and explored. Among these structures, in 2005, Rump introduced cycle sets which are in one-to-one correspondence with left non-degenerate involutive solutions. One of the questions yet to be solved in the context of involutive set-theoretic solutions to the Yang-Baxter equation is the following: When is such a solution retractable?

In this talk, we will focus on cycle sets with an underlying compatible abelian group structure called quasi-linear cycle sets, introduced by Rump. We will state basic properties of this structure and discuss some conjectures about the retraction problem for the class of solutions associated with a quasi-linear cycle set.

@incollection{talk_5,
  author = {Colazzo, I.},
  title = {Set-theoretic solutions to the {Y}ang-{B}axter equation and skew braces},
  note = {University of Leeds (UK)},
  publisher = {Algebra Seminar},
  year = {2022-06-12},
  keywords = {seminar}
}

The Yang-Baxter equation is a fundamental equation in theoretical physics and pure mathematics, with many applications in different fields of mathematics. The importance of this equation led Drinfeld to propose the following problem: studying set-theoretical solutions. A key tool for exploring this family of solutions is the concept of a skew brace, introduced by Rump and Guarnieri and Vendramin. In this talk, we will review the basic theory of this algebraic structure and this family of solutions, describe their connection, discuss some problems, and give some applications.

@incollection{talk_8,
  author = {Colazzo, I.},
  title = {Set-theoretic solution of the {P}entagon {E}quation: the involutive case},
  note = {Vrije Universiteit Brussel (Belgium)},
  publisher = {QA Seminar},
  year = {2022-05-18},
  keywords = {seminar}
}

The pentagon equation appears in various contexts:  For example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation.
This talk, based on joint work with E.Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation. Furthermore, we will look at bijective solutions, focusing on the involutive case. In the latter case, we provide a complete description of all involutive solutions and discuss when two involutive solutions are isomorphic.

@incollection{talk_9,
  author = {Colazzo, I.},
  title = {Set-theoretic solutions of the {Y}ang-{B}axter equation and related associative structure},
  note = {University of Padua (Italy)},
  publisher = {Algebra Seminar},
  year = {2022-04-22},
  keywords = {seminar}
}

The study of large classes of set-theoretic solutions can be reduced to the study of some associative algebraic objects, such as braces, skew braces, and semi-trusses. This talk will discuss the basic properties of skew braces and how these structures are related to non-degenerate bijective set-theoretic solutions to the Yang-Baxter equation. We will also discuss a possible extension of this theory to left non-degenerate solutions, not necessarily right non-degenerate. These new tools will allow us to prove that for a finite left non-degenerate solution being right non-degenerate is equivalent to being bijective.

@incollection{talk_11,
  author = {Colazzo, I.},
  title = {Bijective set-theoretic solutions of the {P}entagon {E}quation},
  note = {University of Milan – Bicocca, Milan (Italy)},
  publisher = {Al@Bicocca take-away},
  year = {2021-11-12},
  keywords = {seminar},
  file = {Al_Bicocca_Ilaria_Colazzo.pdf}
}

The pentagon equation appears in various contexts: for example, any finite-dimensional Hopf algebra is characterised by an invertible solution of the Pentagon Equation, or an arrow is a fusion operator for a fixed braided monoidal category if it satisfies the Pentagon Equation. This talk, based on joint work with E. Jespers and Ł. Kubat, will introduce the basic properties of set-theoretic solutions of the Pentagon Equation, present a complete description of all involutive solutions, and discuss when two involutive solutions are isomorphic.

@incollection{talk_13,
  author = {Colazzo, I.},
  title = {Involutive solutions of the {P}entagon {E}quation},
  note = {Vrije Universiteit Brussel (Belgium)},
  publisher = {ALGB Seminar},
  year = {2020-05-28},
  keywords = {seminar},
  file = {ALGB_Colazzo_pe.pdf}
}

@incollection{talk_14,
  author = {Colazzo, I.},
  title = {The algebraic structure of brace-like semi-trusses},
  note = {Vrije Universiteit Brussel (Belgium)},
  publisher = {ALGB Seminar},
  year = {2020-02-11},
  keywords = {seminar}
}

To study set-theoretic solutions of the Yang-Baxter equation, several authors introduced algebraic structures. Rump, and Cedó, Jespers and Okniński introduced braces, Guarnieri and Vendramin introduced skew braces and Catino, Colazzo and Stefanelli, and Jespers and Van Antwerpen introduced semi-braces. All these objects are subclasses of (semi-)trusses, an algebraic structure introduced by Brzeziński to study the distributive law of (semi-)braces. In this talk, we focus on a subclass of left semi-trusses, the class of brace-like semi-trusses. First, we prove that in the finite case, the additive structure is a completely regular semigroup. Secondly, we apply this result on a specific instance of a left semi-truss called an almost left semi-brace, introduced by Miccoli to study its algebraic structure. In particular, we show that one can associate a left semi-brace to any almost left semi-brace. Furthermore, we show that the set-theoretic solutions of the Yang-Baxter equation originating from almost left semi-braces arise from this correspondence.
This talk is based on joint work with Arne Van Antwerpen (arXiv:1908.11744)

@incollection{talk_17,
  author = {Colazzo, I.},
  title = {Regular subgroups and left semi-braces},
  note = {Vrije Universiteit Brussel (Belgium)},
  publisher = {ALGB Seminar},
  year = {2018-10-13},
  keywords = {seminar}
}

The Yang-Baxter equation is one of the fundamental equations in mathematical-physics. The problem of finding and classifying the set-theoretical solutions of the Yang-Baxter equation has originally been posed by Drinfeld. Rump introduced braces, as a generalization of radical rings, in order to construct set-theoretical involutive solutions of the Yang-Baxter equation. Braces over a field have an unexpected connection with regular subgroups of an affine group. Finding all regular subgroups of an affine group is an open problem formalized by Liebeck, Praeger and Saxl in 2010. Last year Guarnieri and Vendramin obtained a generalization of braces, namely the skew braces, and Catino, Colazzo and Stefanelli obtained the semi-braces a further generalization. In this talk first we explore the link between braces over a field an regular subgroups of the affine group and we prove that with this connection we can construct a family of regular subgroups of the affine group such that they have trivial intersection with the translation group. Then we extend this result to the connection between skew braces and regular subgroups of the holomorph of a group. Finally, we introduce a suitable definition of the holomorph of a right group and extend the link between semi-braces and regular subgroups of this holomorph.

@incollection{talk_1,
  author = {Colazzo, I.},
  title = {Regular subgroups of the affine group},
  note = {University of Warsaw (Poland)},
  publisher = {Seminar Algebra},
  year = {2016-11-24},
  keywords = {seminar}
}

@incollection{talk_16,
  author = {Colazzo, I.},
  title = {The matched product of shelves},
  note = {Lecce (Italy), 25–28/06/2019},
  publisher = {Advances in Group Theory and Applications 2019},
  year = {2019-06-26},
  keywords = {contributed},
  file = {colazzoAgta2019.pdf}
}

This talk sits at the intersection between two areas: the solutions of the Yang-Baxter equation and self
distributive systems.
In the last years, the connection between self distributive systems and the solutions of the Yang-Baxter equation was highlighted. In [3], Lebed and Vendramin studied the (co)homology of left non-degenerate solutions. Additionally, they showed how to associate to any left non-degenerate solution a shelf operation that captures many of its properties: such as invertibility and involutivity. In [2] Lebed further deepened the link between a left non-degenerate solution and its associated shelf. In this talk, we investigate the matched product of solutions associated with right and left shelves [1]. First, we focus on the matched product of left shelves, and we prove that we can simplify the requirements to provide the matched product of solutions related to left shelves. Moreover, we show that we obtain the left non-degeneracy if and only if we start with left racks. Later, we treat solutions associated with right shelves. Here again, we show that the requirements to obtain a matched product system of solutions are straightforward. Furthermore, we present the matched product of one solution associated with a left shelf and the other associated with a right shelf; and we obtain natural conditions to check the properties of the matched product system. In all three cases, we prove that the structure shelf associated with the matched product solution does not depend on the action maps that give the matched product. Finally, we show that this is a general result, i.e., that the structure shelf associated with the matched product of left non-degenerate solutions does not depend on the actions.

References
[1] F. Catino, I. Colazzo, P. Stefanelli, The matched product of self-distributive systems, In preparation.
[2] V. Lebed, Applications of self-distributivity to Yang-Baxter operators and their cohomology, J. Knot Theory Ramifications 27 (11)
(2018) 1843012, 20.
[3] V. Lebed, L. Vendramin, Homology of left non-degenerate set-theoretic solutions to the Yang-Baxter equation, Adv. Math. 304 (2017)
1219–1261.

@incollection{talk_18,
  author = {Colazzo, I.},
  title = {Skew braces with non-trivial annihilator},
  note = {Lecce (Italy), 5–8/09/2017},
  publisher = {Advances in Group Theory and Applications 2017},
  year = {2017-09-07},
  keywords = {contributed},
  file = {AGTA2017Colazzo.pdf}
}

Skew braces are strictly link with non-degenerate (bijective) solution of the Yang-Baxter equation. In this talk, I will present a construction, the extension of skew braces, that allow us to describe all skew braces with non trivial annihilator. In particular, this construction is based on classical tools like 2-cocycles of groups and central group extensions. Finally, I describe all skew braces in terms of Hochschild product of skew braces.

@incollection{talk_19,
  author = {Colazzo, I.},
  title = {Semi-braces and the {Y}ang-{B}axter equation},
  note = {Spa (Belgium), 18–24/06/2017},
  publisher = {Groups, Rings and the Yang-Baxter equation},
  year = {2017-06-22},
  keywords = {contributed},
  file = {Spa2017.pdf}
}

Semi-braces and the Yang-Baxter equationIlaria ColazzoThe Yang-Baxter equation is a basic equation of the statistical mechanicsthat arose from Yang’s work in 1967 and Baxter’s one in 1972. Drinfeld in [2] posed the question of classifying the solutions of the Yang-Baxter equation,in particular those called set-theoretical.  This is a difficult task and manyauthors dealt with this problem.  In particular, several algebraic structureswere studied to answer this problem, such as groups, cycle sets, braces (forinstance, see [3], [4], [5]). Recently, a new generalization of braces,semi-brace, was introduced in [1]. In this talk, we describe how to obtain a solution of the Yang-Baxter equation  through  semi-braces. Furthermore,  we  show  which  properties  satisfythis kind of solutions.  Finally, we present a construction of solution of theYang-Baxter  equation  that  arises  from  the  matched  product  of  left  semi-braces.
References
[1] F. Catino, I. Colazzo, P. Stefanelli: Semi-braces and the Yang-Baxter equation, J. Algebra (2017), http://dx.doi.org/10.1016/j.jalgebra.2017.03.035 
[2] V. G. Drinfeld: On some unsolved problems in quantum group theory, in: Quantum Groups (Leningrad, 1990), Lecture Notes in Math.1510, Springer, Berlin, (1992), 1–8.
[3] P. Etingof, T. Schedler, A. Soloviev:Set-theoretical solutions to the quantumYang-Baxter equation, Duke Math. J.100(1999), 169–209.
[4] J.-H. Lu, M. Yan, Y.-C. Zhu:On the set-theoretical Yang-Baxter equation, DukeMath. J.,104(1) (2000) 1–18.
[5] W. Rump:Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra 307 (2007), 153–170.

@incollection{talk_20,
  author = {Colazzo, I.},
  title = {The algebraic structure of semi-brace},
  note = {Naples (Italy), 23–24/05/2017},
  publisher = {Young Researchers Algebra Conference 2017},
  year = {2017-05-23},
  keywords = {contributed},
  file = {Yrac2017.pdf}
}

Rump, in [4], introduced braces to study non-degenerate involutive solutions of the Yang-Baxter equation. Many aspects of this algebraic structure were studied and developed (see, for instance [2] and its bibliography). Recently, Guarnieri and Vendramin in [3] obtained a generalization of braces, skew braces, in order to construct non-degenerate bijective solutions of the Yang-Baxter equation. In this talk, we focus on semi-braces, a further generalization of braces introduced in [1] that allows us to construct new solutions, not necessarily bijective. In particular, we describe the structural aspects of a semi-brace and provide a clear characterization of this structure. Finally, we introduce suitable concepts of ideal and quotient structure of a semi-brace.

References
[1] F. Catino, I. Colazzo and P. Stefanelli: Semi-braces and the Yang-Baxter equation, accepted for
publication, J. Algebra.
[2] F. Cedó, E. Jespers and J. Okniński: Brace and Yang-Baxter equation, Comm. Math. Phys. 327(1)
(2014), 101–116.
[3] L. Guarnieri and L. Vendramin: Skew braces and the Yang-Baxter equation, accepted for publication
in Math. Comp. arXiv:1511.03171 (2015).
[4] W. Rump: Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007),
153–170.

@incollection{talk_2,
  author = {Colazzo, I.},
  title = {Radical braces and the {Y}ang-{B}axter equation},
  note = {Graz (Austria),  3–8/07/2016},
  publisher = {Conference on Rings and Polynomials},
  year = {2016-07-04},
  keywords = {contributed},
  file = {ColazzoGRAZ.pdf}
}

The Yang-Baxter equation is a basic equation of the statistical mechanics that arose from Yang’s work in 1967 and Baxter’s one in 1972. In 1992, Drinfeld [in Quantum Groups (Leningrad, 1990), Lecture Notes in Math.Springer, Berlin, 1992, 1–8, 1510] posed the question of finding all set-theoretic solutions of the Yang-Baxter equation. Later, in the seminal paper, Etingof, Schedler and Soloviev [Duke Math. J.,1999, 169–209, 100] laid the groundwork for the study of a particular class of these solutions, the non-degenerate involutive ones. From a radical ring we may construct a non-degenerate involutive solution of the Yang-Baxter equation. In 2007, Rump [J. Algebra, 2007, 153–170, 307] found a strict link between a generalization of radical rings, called radical braces, and the non-degenerate involutive solutions.
Later in this talk, we introduce more general algebraic structures, the skew braces and the semi-braces that further generalize radical rings and have a link with solutions of the Yang-Baxter equation that are not necessary involutive (see [L. Guarnieri, L. Vendramin, Skew braces and the Yang-Baxter equation, Accepted for publication in Math. Comp., 2015] and [F. Catino, I. Colazzo, P. Stefanelli, Semi-braces and the Yang-Baxter equation, in preparation] )

@incollection{talk_3,
  author = {Colazzo, I.},
  title = {Regular subgroups of an affine group},
  note = {Florence (Italy), 16-17/06/2016},
  publisher = {Group Theory in Florence: a meeting in honor of Guido Zappa},
  year = {2016-06-17},
  keywords = {contributed},
  file = {colazzoFIRENZE.pdf}
}

Finding all regular subgroups of an affine group is an open problem formalized by Liebeck, Praeger and Saxl in 2010 (see [5]). A systematic way to obtain regular subgroups is obtained by Catino and Rizzo in [3] in terms of radical braces over a field F , a generalization of radical algebras. On the other hand, there is an unexpected connection between radical braces and involutive non-degenerate solutions of Yang-Baxter equation (see [6]).
In this talk we are going to introduce constructions of radical braces over a field F (see [1] and [2]) that allows us to obtain systematic constructions of regular subgroups of the affine group. In particular, this approach allows to put in a more general context the regular subgroups constructed by Hegedűs in [4] and by Tamburini Bellani in [7].

References
[1] F. Catino, I. Colazzo, P. Stefanelli: On regular subgroups of the affine group, Bull.
Aust. Math. Soc. 91 (1) (2015) 76–85.
[2] F. Catino, I. Colazzo, P. Stefanelli: Regular subgroups of the affine group and asym-
metric product of radical braces, J. Algebra, 455 (2016) 164–182.
[3] F. Catino, R. Rizzo: Regular subgroups of the affine group and radical circle algebras,
Bull. Aust. Math. Soc. 79 (2009), 103–107.
[4] P. Hegedűs: Regular subgroups of the affine group, J. Algebra 225 (2) (2000) 740–742.
[5] M. Liebeck, C. Praeger, J. Saxl: Regular subgroups of primitive permutation groups,
Memoirs of the AMS 203 (2010), no. 952.
[6] W. Rump:Braces, radical rings, and the quantum Yang-Baxter equation. J. Algebra
307 (2007), 153–170.
[7] M.C. Tamburini Bellani: Some remarks on regular subgroups of the affine group Int.
J. Group Theory, 1 (2012), 17–23.

@incollection{talk_4,
  author = {Colazzo, I.},
  title = {The {A}symmetric {P}roduct of radical braces},
  note = {Porto Cesareo, Lecce (Italy), 16–19/06/2015},
  publisher = {Advances in Group Theory and Applications 2015},
  year = {2015-06-18},
  keywords = {contributed},
  file = {colazzoAGTA15.pdf}
}

In this talk, we are going to introduce a new radical brace construction which extended the semidirect product of two radical braces. In the case of braces over a field, we may obtain some regular subgroups of the affine group and this approach allows us to generalise the regular subgroups constructed by Hegedus in Regular Subgroups of the Affine Groups, J. Algebra 225 (2000), 740–742.