The Antwerp Algebra Colloquium takes place monthly. Each session consists of two one-hour-long colloquia sharing a common thread. The focus is set on learning the basics on different areas of algebra. The colloquia are thus meant to be accessible for researchers from all domains of algebra.

We will resume the colloquium in October 2022. The lectures take place in room M.G.016, in building G on Campus Middelheim. We stream the talks for those of us who want to follow online.

If you want to be informed about the upcoming sessions of the colloquium and get the corresponding links to the talks on Blackboard Collaborate, you can subscribe to our mailing list here.

Upcoming sessions

In the academic year 2022–2023, colloquiums are planned on 07/10, 04/11, 25/11, 16/12, 17/02, 10/03, 24/03, 21/04 and 26/05.
The speakers and topics are to be announced.

For updates or changes to the schedule, please subscribe to our mailing list here.

Previous sessions

You can find the speakers and abstracts of our previous editions below.

Academic year 2021–2022

29 April 2022

  • ​14:00 - 15:00 — The magic of ultraproducts: working with "almost all" Lara Verdijck (Universiteit Antwerpen)

In this talk, we will look at the ultraproduct construction, which is an algebraic construction with interesting model-theoretic properties. Using the Theorem of Łoś, the ultraproduct construction makes us able to prove statements for "almost all" structures of a certain class. The ultraproduct construction can be used in various ways in model theory and in algebra. During the talk, we will focus on the construction itself and show, by an example, that it is a very useful tool in algebra. For this purpose, we will look at the Ax-Kochen Theorem which is related to Artin's conjecture. Another important consequence of the ultraproduct construction is that it gives a proof of Gödel's Compactness Theorem, with a construction for the required model.

  • 15:15 - 16:15 — Steiner systems and Mathieu groups Sione Janssen Whiteman (Universiteit Antwerpen)

The Mathieu groups were the first of the sporadic groups to be discovered and they are often described as the automorphism groups of Steiner systems. In this talk I will give an introduction of Steiner systems, discuss some of their properties and work out an example of an extension of the Fano plane. I will also explain what is so special about the Mathieu groups and their specific Steiner systems and give the relation between the Steiner system S(5,8,24) and the Golay code.

22 April 2022

  • ​14:00 - 15:00 — Polynomial congruences, Igusa zeta functions and resolution of singularities Wim Veys (KU Leuven)

For a fixed polynomial over the integers, studying its numbers of zeros modulo varying integers m is a difficult number theoretical problem. This is related to properties of a certain p-adic integral, called Igusa zeta function. We present an approach using algebraic geometry, more precisely resolution of singularities.

  • 15:15 - 16:15 — Bernstein-sato polynomials and D-modules Robin van der Veer (KU Leuven)

In this talk I will give an introduction to the Bernstein-Sato polynomial and the theory of D-modules. These are important objects in modern singularity theory. If time permits I will explain the relation to monodromy of Milnor fibrations.

25 March 2022

  • ​14:00 - 15:00 — Free- and Amalgamated Products in Group Rings Doryan Temmerman (UHasselt)

In this talk, we will bridge the gap between the theory of Group Rings and Geometric Group Theory. We will briefly introduce a couple of conjectures that have been guiding the research in Group Rings over the last couple of decades. Using these conjectures as a motivation we will discuss how the amalgamated product of groups, a construction naturally appearing in Geometric Group Theory, warrants consideration within this domain. In doing so, my goal will be to provide you with an idea of the types of questions, and directions for solutions, that come up within the research field of Group Rings. Skipping the technical details, we will only hit the highlights of this beautiful interplay while strolling next to Bass-Serre Theory, Orders and Linear Groups.

  • 15:15 - 16:15 — An invitation to the theory of locally presentable categories Julia Ramos González (UCLouvain)

The family of locally presentable categories is sufficiently large to contain a myriad of examples of great interest, but small enough to fulfill a range of significant categorical properties. In particular, most of the categories in the toolkit of the working algebraist are locally presentable. For example, the categories of sets, magmas, semigroups, groups, rings, modules, graphs or relational structures are all locally presentable and in general any variety or quasi-variety of algebras is. The understanding of locally presentable categories thus provides us with a general understanding of the structural properties shared by all these categories. The aim of this talk is to present a brief introduction to the theory of locally presentable categories. First, we will provide the basic definitions and analyze several examples. We will then outline the main properties of locally presentable categories. We will conclude by analyzing the different presentations in which locally presentable categories can be found in nature.

11 March 2022

  • ​14:00 - 15:00 — Embracing the generic prime ideal: the tale of an enchanting mathematical phantom — Ingo Blechschmidt (Universität Augsburg) (video)

Is there a number which squares to minus one? Back in the old days, the answer was straightforward: Of course not. But √-1 obtruded its effects so convincingly that we embraced a broader notion of existence, passing from the reals to the complex numbers.

The talk relates a higher-order variant of this tale, the tale of the generic prime ideal of any given commutative ring. Like the imaginary unit, it does not exist in a strict, narrow sense; it only comes about by passing to a larger universe. And also like the imaginary unit, the generic prime ideal unlocks a variety of techniques motivating its study. One particular such is: Without loss of generality, any reduced ring is a field.

The talk will explain how to bring the generic prime ideal into existence and illustrate its merits by showcasing applications rangingfrom dimension criteria for injective and surjective linear maps to Grothendieck's generic freeness theorem.

The generic prime ideal has close ties to mathematical logic and sheaf theory, but the talk assumes no background on either of these subjects.

  • 15:15 - 16:15 — Introduction Type Theory: Another foundation of mathematics Kobe Wullaert (TU Delft) (video)

In 1874, (naive/Cantorian) set theory was introduced by G. Cantor which revolutionized the world of mathematics. However, not long after that, this theory gave rise to several contradictions, likeRussell’s paradox. A well-know solution is to work in Zermelo-Fraenkel set theory, but this is not the only formal system in which one can formalize mathematics. In this talk, the audience is introduced totype theory which is another formal system (introduced by B. Russell in 1903) which was also designed/introduced to overcome the paradoxes of naive set theory.

Although (Zermelo-Fraenkel) set theory is widely used, there are some unpleasant features which type theory does not satisfy. Everything in set theory is a set, the set of natural numbers is defined (very cumbersomely) as {∅, {∅}, {{∅}}, · · · }, a function is defined as a subset of the powerset, etc. However, an ordinary mathematician never uses these specific definitions and we take these notions as primitive, instead of "special cases". In type theory, everything is a type and we have some (primitive) type constructors from which every mathematical structure can be specified as a type together with rules which expresses how the elements should look like instead of giving a concrete construction. As a consequence, type theory can be formalized in a computer as a programming language (which are called proof assistents).

In this talk, we start by introducing what types are and two perspectives which one can take to reason about types (i.e.Curry–Howard correspondence and Homotopy type theory). Then we go in detail how some of the most common type constructors are defined and how these correspond with the different perspectives on types. In the rest of the talk, it is illustrated how one formalizes mathematics using type theory, how such a formalization of mathematics can be implemented using a proof assistent (by the Curry–Howard correspondence) and lastly, we dive a little bit deeper in the realm on Homotopy type theory with the purpose of giving some intuition between type theory (∞-)category theory.

18 February 2022

  • ​14:00 - 15:00 — The Hopf category of Frobenius algebras Paul Großkopf (Université Libre de Bruxelles) (video)

The complete classification of Topological Quantum Field Theories (short: TQFTs) is still an open question and has only been fully understood for dimensions one and two. Whereas in 1D these are characterized by vectorspaces, (commutative) Frobenius algebrafully characterize the two dimensional case. The fact that the endomorphism space of a finite dimensional vector space (matrix algebra) forms a Frobenius algebra gives rise to the assumption that there is a link between consecutive dimensions. This talk, we explore generalized homomorphism spaces (more precisely, Sweedler’s universal measuring coalgebras [Swe]) between Frobenius algebras and using these we will show that Frobenius algebras can be organized in a Hopf category (in the sense of [BCV]), multi-object analogue to a Hopf algebra. This is a generalization of the well-known fact than any non-zero homomorhpism of Frobenius algebras is an isomorphism. Since Hopf algebras and Hopf categories are strongly linked to the 3D TQFTs, thisresult indicates it might be possible to construct 3D TQFTs out of 2D ones.

[BCV] E. Batista, S. Caenepeel and J. Vercruysse, Hopf categories, Algebr. Represent. Theory, 19 (2016), 1173–1216.
[Swe] Sweedler, M.E. Hopf Algebras. W. A. Benjamin New York, 1969.

  • 15:15 - 16:15 — From Hochschild to Gerstenhaber-Schack, as told through drawings Lander Hermans (Universiteit Antwerpen) (video)

​In his foundational work Gerstenhaber defined a dgLie bracket on the Hochschild complex of an associative algebraA and showed it to control the deformations of A through the Maurer–Cartan equation. Algebraic geometry motivates the natural question whether a similar story exists for presheaves of associative algebras.In this talk I will explain how the Gerstenhaber–Schack complex fulfils this role, yet also motivates to generalize from presheaves to prestacks (i.e. pseudofunctors) as the correct objects to start with. By means of elementary drawings of rectangles, I will explain both the new differential on the GS-complex, established by Dinh Van–Lowen, and the new L-infinity-structure we recently obtained, thus completing the story: the higher Lie brackets on the GS complex control the deformations of prestacks through the generalized MC equation.

26 November 2021

  • 14:00 - 15:00 — Sums of squares in function fields of curves over R((t1)) . . . ((tn)) Gonzalo Manzano-Flores (Universidad de Santiago de Chile, University of Antwerp)

Ernst Witt showed in 1934 that every sum of squares is a sum of two squares in every algebraic function field F over the real numbers R (equivalently, the pythagoras number of F denoted by p(F), is equal to 2). It is a natural question whether we can bound p(F) also for algebraic function fields over a field with properties similar to R. For example, we can consider the case where the base field K is a hereditarily pythagorean field, such as Kn = R((t1))...((tn)), for some natural number n.

Consider an algebraic function field F/Kn. It was shown by J. Van Geel, K. Becher and D. Grimm that 2≤p(F)≤3 and that G(F)=(ΣF2)*/(F2+F2)* is a finite group, which controls the failure of a sum of squares to be a sum of 2 squares. It is natural to ask for the precise value of p(F) and for the size of G(F). In this talk, I will put the focus on algebraic function fields of the form F = Kn(X)/sqrt(f), where f ∈ Kn[X] is a square-free polynomial (the function field of the hyperelliptic curve C : Y2 = f(X)) and I will give some examples. Taking g ∈ N such that deg(f) = 2g+1 or deg(f) = 2g+2 (the genus of F/Kn) I will show that |G(F)| ≤ 2n(g+1) indicating that this bound is optimal.

  • 15:15 - 16:15 — Representation theory and pseudo-differential operators. New perspectives.Duván Cardona (Ghent University) (video)

The theory of pseudo-differential operators was born in the 1960s as a versatile technique to analyze several problems in partial differential equations.One of its more remarkable applications is the  role that it had  in the proof of the Atiyah-Singer index theorem, a celebrated result of the geometric analysis that interpolate several results of Topology: the Riemann-Roch theorem, the Hirzebruch signaturetheorem, etc. In this lecture we will give an introduction to the theory of pseudo-differential operators and we will discuss its connections with the representation theory of compact Lie groups.  The interplay of both theories provides a modern perspectiveon the field that was discovered some years ago by M. Ruzhansky and V. Turunen.

5 November 2021

  • 14:00 - 15:00 — Model-theoretic tools for the busy algebraist Nicolas Daans (Universiteit Antwerpen) (video)

Oftentimes in mathematics, especially algebra, statements are made which are inherently of a finite nature, even if they apply to infinite objects. In such situations, model-theoretic techniques can often provide an insightful and elegant way to argue why such statements hold, or why they carry over from some (infinite) objects to others. The goal of this talk is to introduce some of these techniques in a way that is accessible and ready-to-use for algebraists, and apply them to prove a classical result from algebra/algebraic geometry: the Ax-Grothendieck Theorem.

  • 15:15 - 16:15 — A sketch of semi-Galois theoryMorgan Rogers (Università degli Studi dell'Insubria) (video)

We explain why classical Galois correspondences should be thought of as an equivalence of categories. That perspective enables us to construct an analogue of Galois theory where the Galois group is replaced by a monoid, and subgroups are replaced by right congruences on the monoid; this is "semi-Galois theory".

8 October 2021

  • 14:00 - 15:00 — An elementary introduction to (infinity,1)-categories via quasi-categories ​— Arne Mertens (Universiteit Antwerpen) (video)

Intuitively, infinity-categories should consist of objects, (1-)morphisms between the objects, 2-morphisms between the 1-morphisms, 3-morphisms between the 2-morphisms, etc such that these n-morphisms can be composed associatively with unit. Formalizing this idea in a way that is still manageable has proven to be difficult however, and is still a very active field of research. Today, many different equivalent models have been established for so-called (infinity,1)-categories, that is infinity-categories where the n-morphisms are all invertible for n > 1. By far the most developed of these models is that of quasi-categories. These were first introduced and researched by André Joyal and heavily expanded upon by Jacob Lurie. In this talk I'll formally define quasi-categories and give an overview of their most basic aspects. In particular, I will touch on the homotopy hypothesis, mapping spaces and limits and colimits in a quasi-category, and conclude with some examples.

Academic year 2020–2021

31 May 2021

Antwerp Algebra Colloquium on topics on cohomology:

  • 14:00 - 15:00 — Cohomological invariants of n-dimensional quadratic forms in I3Simon Rigby (Universiteit Gent)

​I will define cohomological invariants and talk about some techniques for classifying the mod 2 cohomological invariants of an algebraic group, with an emphasis on certain groups whose cohomology is meaningful in quadratic form theory (such as orthogonal groups, Spin groups, and even and extended Clifford groups). I will show how one can classify the invariants of 14-dimensional quadratic forms with trivial discriminant and Clifford algebra (and also the invariants of the group Spin14). This is a particularly interesting case, because the problem was solved some time ago in dimensions ≤ 12 and it becomes prohibitively difficult in dimensions > 14.

  • 15:15 - 16:15 — An introduction to spectraMatt Booth (Universiteit Antwerpen) (video)

Spectra are objects from algebraic topology that record information about stable homotopy theory (i.e. those properties of homotopy groups that are invariant under the suspension functor). I'll give a quick introduction to spectra and outline the relationship they have to generalised cohomology theories.

Spectra are at once both simpler and more algebraic than topological spaces. In particular, the discovery of categories of `highly structured' spectra in the 1990s allows one to do commutative and homological algebra with spectra: one can talk about ring spectra, modules over them, tensor products and internal homs, etc. I'll talk about spectral algebra, and time permitting outline some applications, like spectral algebraic geometry or topological Hochschild theory.

28 May 2021

Antwerp Algebra Colloquium session in the framework of the Master Program at UA:

  • ​14:00 - 15:00 — Elementary topoi as categorical models of set theoryKobe Wullaert (Universiteit Antwerpen)

The goal of this talk is to introduce the audience to the theory of elementary topoi. These are categories which generalize the category of sets (and functions), not only in the sense of having the same universal constructions, but they also become models of the set theory of Zermelo-Fraenkel. By this I mean that certain morphisms in a topos form the abstract formulas in such a theory, called the internal logic, and after de fining when such formulas are valid, it can be shown that the axioms of (intuitionistic) propositional- and predicate logic and set theory are valid.

The central notion of a topos is that of a subobject classifi er. This is a particular kind of object which plays the role of a two-element set by allowing each subobject to be described by a unique morphism into that object (just as a subset is uniquely de fined by a function into a two-element set). A formula then also corresponds with a morphism into this object, so to form all formulas of (intuitionistic) propositional logic, the subobject classifi er needs to have a specifi c kind of partial order structure, that of a Heyting algebra. The fundamental theorem of topoi allows us the interpret the quanti fiers.

21 May 2021

Antwerp Algebra Colloquium on internal groupoids:

  • 14:00 - 15:00 – Internal groupoids in semi-abelian categories Marino Gran (Université catholique de Louvain) (video)

Since their introduction twenty years ago, semi-abelian categories [1] have attracted a lot of interest, as they are useful to study some fundamental exactness properties the categories of groups, Lie algebras, compact groups and crossed modules have in common. In this talk I shall explain some simple ideas of this area of categorical algebra, with a special emphasis on the role of internal groupoids in semi-abelian categories. These structures are closely connected to commutators, central extensions and non-abelian homology. The category of groupoids in a semi-abelian category contains various interesting non-abelian torsion theories. It can also be seen as the exact completion of its subcategory of equivalence relations, as it follows from a general characterization of the semi-localizations of semi-abelian categories [2]. A couple of results concerning the internal structures in the semi-abelian category of cocommutative Hopf algebras will also be considered [3].

References:

[1] G. Janelidze, L. Marki and W. Tholen, Semi-abelian categories, J. Pure Appl. Algebra 168 (2002) 367-386.

[2] M. Gran and S. Lack, Semi-localizations of semi-abelian categories, J. Algebra 454 (2016) 206-232.

[3] M. Gran, F. Sterck and J. Vercruysse, A semi-abelian extension of a theorem by Takeuchi, J. Pure Appl. Algebra 223 (2019) 4171-4190.

  • ​15:15 - 16:15 – Orbispace Mapping Objects: Exponentials and Enrichment! Dorette Pronk (Dalhousie University) (video)

Orbifolds are defined like manifolds, by local charts. Where manifold charts are open subsets of Euclidean space, orbifold charts consist of an open subset of Euclidean space with an action by a finite group (thus allowing for local singularities). This affects the way that transitions between charts need to be described, and it is generally rather cumbersome to work with atlases. It has been shown in [Moerdijk-P] that one can represent orbifolds by groupoids internal to the category of manifolds, with etale structure maps and a proper diagonal, I.e., combined source-target map (s,t): G1 → G0 x G0. We have since generalized this notion further to orbispaces, represented by proper etale groupoids in the category of Hausdorff spaces. Two of these groupoids represent the same orbispace if they are Morita equivalent.  However, Morita equivalences are generally not pseudo-invertible in this 2-category, so we consider the bicategory of fractions with respect to Morita equivalences. ​

For a pair of paracompact locally compact orbigroupoids G and H, with G orbit-compact, we want to study the mapping groupoid [G, H] of arrows and 2-cells in the bicategory of fractions. The question we want to address is how to define a topology on these mapping groupoids to obtain mapping objects for the bicategory of orbispaces. This question was addressed in [Chen], but not in terms of orbigroupoids, and with only partial answers.

We will present the following results:

1. When the orbifold G is compact, we define a topology on [G,H] to obtain a topological groupoid OMap(G, H), which is Morita equivalent to an orbigroupoid. To obtain a Morita equivalent orbigroupoid, we need to restrict ourselves to so-called admissible maps to form AMap(G,H), and Orbispaces(K × G, H) is equivalent to Orbispaces(K, AMap(G, H)).So AMap(G,H) is an exponential object in the bicategory of orbispaces.

2. We will also show that AMap(G,H) thus defined provides the bicategory of orbit-compact orbispaces with bicategorical enrichment over the bicategory of orbispaces: composition can be given as a generalized map (an arrow in the bicategory of fractions) of orbispaces.

In this talk I will discuss how this work extends the work done by Chen and I will show several examples. This is joint work with Laura Scull.

References:

[Chen] Weimin Chen, On a notion of maps between orbifolds I: function spaces, Communications in Contemporary Mathematics 8 (2006), pp. 569-620.

[Moerdijk-P] I. Moerdijk, D.A. Pronk, Orbifolds, sheaves and groupoids, K-Theory 12 (1997), pp. 3-21.

23 April 2021

Antwerp Algebra Colloquium on higher structures:

  • 14:00 - 15:00 – Higher Structures in Algebra, Geometry, and TopologyBruno Vallette (Université Sorbonne Paris Nord) (slides, video)

The goal of this talk will be to introduce the audience with higher structures (homotopy algebras, operadic calculus and higher categories). These structures and the way to work efficiently with them were extensively studied over the past 30 years. We will show that their consideration is mandatory to formulate and to establish some open conjectures in Algebra, Geometry, and Topology.

  • 15:15 - 16:15 – Endofunctors and Poincaré-Birkhoff-Witt theoremsPedro Tamaroff (Trinity College Dublin) (slides, video)

In joint work with V. Dotsenko, we developed a categorical framework for Poincaré-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures, and used methods of term rewriting for operads to obtain new PBW theorems, in particular answering a question of J.-L. Loday. Later, in joint work with A. Khoroshkin, we developed a formalism to study Poincaré–Birkhoff–Witt type theorems for universal envelopes of algebras over differential graded operads, motivated by the problem of computing the universal enveloping algebra functor on dg Lie algebras in the homotopy category. I will survey and explain the role homological algebra, homotopical algebra, and effective computational methods play in the main results obtained with both V. Dotsenko (1804.06485) and A. Khoroshikin (2003.06055), and mention further directions where these ideas have been or can be applied.

5 March 2021

Antwerp Algebra Colloquium on combinatorics in algebra:

  • 14:00 - 15:00 – Buildings: what’s the point (set)?Anneleen De Schepper (Universiteit Gent) (slides)

It is my intention to give a gentle introduction to (spherical) buildings, from the viewpoint of incidence geometry.  This means that I will consider point-line geometries (Grassmannians) associated to them, which amounts to selecting one type of objects of the building as the 'points' (the 'lines' then can be deduced from this). An elementary axiom system was introduced by Cooperstein in the 70'ies exactly to capture the behaviour of such point-line geometries associated to spherical buildings, giving rise to the notion of 'parapolar spaces'.  I will mention some old and recent partial classification results in the theory of parapolar spaces, giving surprisingly easy descriptions of geometries related to the exceptional algebraic groups F4, E6, E7, E8, and their classical subgeometries.

  • 15:15 - 16:15 – Mustafin degenerations: Between applied and arithmetic geometryMarvin Anas Hahn (Max-Planck-Institut für Mathematik in den Naturwissenschaften) (video)

The study of degenerations is at the core of algebraic and arithmetic geometry. An especially interesting class of degenerations is provided by so-called "Mustafin degenerations", which may be studied via combinatorics in Bruhat-Tits buildings. Moreover, this class is accessible by computer algebra. In this talk, we will give an introduction to their beautiful theory and outline some of their applications in arithmetic and applied geometry. In particular, we will report on recent advances towards a p-adic Narasimhan-Seshadri theorem and a connection between computer vision and Hilbert schemes.

19 February 2021

Antwerp Algebra Colloquium at the intersection of algebra and logic:

  • 14:00-15:00 – An introduction to the theory of Borel complexity of classification problems – Julien Melleray (Université Lyon 1) (video)

A common theme throughout mathematics is to try and classify a class of objects according to some notion of isomorphism, and try to produce some nice invariants for this classification. Sometimes, nice invariants cannot exist; and one is led to the idea of comparing complexities of classification problems. Many such problems fit in a framework introduced by Friedman and Stanley in the eighties: Borel complexity of equivalence relations. I will explain this framework and give some examples related to classification of some classes of countable groups up to isomorphism (abelian groups, torsion-free abelian groups of finite ranks, locally finite simple groups...).

  • 15:15-16:15 – An introduction to o-minimality – Siegfried Van Hille (KU Leuven) (video)

In this talk we discuss various properties of o-minimal structures. The goal is to explain how the simple o-minimality property ensures a “tame” geometry, where tame is the idea envisaged by Grothendieck in his “Esquisse d’un progamme”. I will highlight some important algebraic and geometric concepts that are used to prove that a structure is o-minimal. Finally, if time permits, I will also discuss more recent developments in the study of rational points on transcendental sets.

18 December 2020  

Antwerp Algebra Colloquium on semigroups and lattices:

  • 14:00-15:00 - Two Dimensional Inverse Semigroups/Categories - Darien DeWolf (St. Francis Xavier University)

Double inverse semigroups were defined by Kock in 2007 to be sets equipped with two compatible inverse semigroup operations. However, we showed that such structures exhibit an Eckmann-Hilton property: such a double inverse semigroup is always proper (both operations are equal) and commutative. This talk will present double inverse semigroups instead as certain double categories, following the program of Brown’s double group theory. We will show that this approach is further justified by internalizing Cockett and Lack’s restriction categories.

  • 15:15-16:15 - Set-theoretic solutions of the Yang-Baxter equation and associated algebraic structures - Charlotte Verwimp (Vrije Universiteit Brussel) (video)

​The Yang-Baxter equation originates from papers by Yang and Baxter on quantum and statistical mechanics, and the search for solutions has attracted numerous studies both in mathematical physics and pure mathematics. As the study of arbitrary solutions is complex, Drinfeld proposed in 1992 to focus on the class of set-theoretic solutions. The goal is simple, find all set-theoretic solutions of the Yang-Baxter equation. One part of this research domain focuses on describing the algebraic structures that arise from set-theoretic solutions, for example the structure group and monoid. Another part attempts to find algebraic structures that provide set-theoretic solutions, like braces and skew lattices.

In this talk we try to give a glimpse in the world of set-theoretic solutions of the Yang-Baxter equation. We briefly introduce all notions mentioned above, give some background and motivation, and discuss important results.