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.

The lectures take place in room **M.G.016**, in building G on Campus Middelheim, but we plan to stream the talks for those 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

For this academic year, the different sessions are all planned on Friday afternoon, on the following days: 08/10, 05/11, 26/11, 17/12, 18/02, 11/03, 25/03, 22/04 and 20/05. The schedule is subject to change, so please subscribe to our mailing list here.

## Previous sessions

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

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Academic year 2021–2022

**8 October 2021**

In the first session of the new academic year, Arne Mertens gave an introduction to quasi-categories.

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

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.

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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 I*^{3}—**Simon 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 Spin_{14}). 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 spectra*—**Matt 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 theory*—**Kobe 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 defining 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 classifier. 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 defined 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 classifier needs to have a specific kind of partial order structure, that of a Heyting algebra. The fundamental theorem of topoi allows us the interpret the quantifiers.

**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): G_{1} → G_{0} x G_{0}. 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 Topology*–**Bruno 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 theorems*–**Pedro 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 F_{4}, E_{6}, E_{7}, E_{8}, and their classical subgeometries.

- 15:15 - 16:15 –
*Mustafin degenerations: Between applied and arithmetic geometry*–**Marvin 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.