The Antwerp Algebra Colloquium

15 March 2024

• 14:00 - 15:00 — Solving Diophantine equations through algebra and geometry — Julian Lyczak (University of Antwerp)

In arithmetic geometry one studies polynomial equations and their solutions in rational numbers. There are three possible outcomes of such a study. Either there are no solutions, there are finitely many, or there are infinitely many. Generally, the outcome is predicted by studying the geometric object described by the polynomial equations.

In the case that there are infinitely many solutions a conjecture by Manin uses geometric input to quantify how complicated the solutions are. Here we consider a solution in rational numbers to be more complicated as the numerators and denominators grow.

On the other end, there are geometric and algebraic techniques, for example the Brauer-Manin obstruction, which in some cases can be used to explain the absence of rational solutions. It has been conjectured by Colliot-Thélène that if the system of polynomial equations is geometrically not too complicated, for example rational varieties and Fano varieties, these techniques suffice to give you a definitive answer on the existence of solutions.

In this talk I will explain the main ideas of these geometric and algebraic methods and how they feature in some my past and current projects.

8 March 2024

• 14:00 - 16:00 — (oo)-operads in chains: a tree-like approach. — Francesca Pratali (Université Sorbonne Paris Nord)

The first part of the talk is meant both as an introduction to operad theory and as a follow-up of previous talks about dg-categories. We will provide definition, motivations and examples of operads enriched in a symmetric monoidal category V, paying special attention to V being the category of spaces or that of chain complexes, and we introduce the notion of 'oo-operads in V', where we relax some of the defining axioms, asking that they hold only 'up to homotopy'. We talk about the study of the homotopy theory of oo-operads in V, and how this drastically depends on the category V. When V=Spaces, several equivalent models have been developed: our interest concerns Cisinki and Moerdijk's model, which generalizes the notion of complete Segal space and is based on a category of trees which extends the simplex category Delta. However, the same techniques are no longer applicable in the case of chain complexes, in the second part of the talk, we point out the issues brought by the linear case, and we present some techniques for solving them. In particular, we will explain how the tree-like approach can be applied to the linear case. We will discuss the combinatorics of trees and introduce a Segal-like condition for defining oo-operads in chains; we will see how these can be realized as certain coalgebras over a tree comonad, and what can be said at the homotopical level.

15 December 2023

• 14:30 - 16:15 — Computing linear relations between univariate integrals — Emre Sertöz (Leiden University)

The study of integrals of univariate algebraic functions (1-periods) provided the impetus to develop much of algebraic geometry and transcendental number theory. This old saga is now at a point of resolution. In 2022, Huber and Wüstholz gave a "qualitative description" of all linear relations with algebraic coefficients between 1-periods. New techniques for the determination of symmetries of complex tori allow us to develop algorithms to explicate these qualitative relations and to decide transcendence of 1-periods. This is work-in-progress with Joël Ouaknine (MPI SWS) and James Worrell (Oxford).

17 November 2023

• 15:00 - 16:00 — From Noncommutative Algebraic Geometry to Algebraic Topology — Wendy Lowen (University of Antwerp)

In the first part of the talk, we explain the basic idea of noncommutative algebraic geometry in which schemes are replaced by algebraic and categorical models for “noncommutative spaces” in the sense of Van den Bergh. We discuss how deformation theory allows to “quantize” spaces using different models like prestacks, abelian categories and dg categories. In the second part of the talk, motivated by certain drawbacks of the theory of dg categories, we propose the alternative framework of templicial modules, as introduced by Arne Mertens in his 2022 PhD thesis. These templicial (short for tensor-simplicial) modules are inspired by algebraic topology, and allow for an enhancement of Lurie’s dg nerve landing in quasi-categories in modules.

• 16:15 - 17:15 — Deformations of quasi-categories in modules — Violeta Borges Marques (University of Antwerp)

I will define deformations of templicial modules and show that two important subclasses of templicial modules - quasi-categories and deg-projectives - are stable under level-wise flat deformation.

10 November 2023

• 15:00 - 16:45 — A new model for dg-categories — Elena Dimitriadis Bermejo (Université Toulouse III - Paul Sabatier)

What is a dg-category? Maybe some of you have heard the term in the corridors or at the coffee room. Well, if you have ever wondered, this might be the time to find out! In this talk, we will start with an introduction to dg-categories, with an explanation of their motivation and a look at their issues, and we will then continue with some more technical solutions to those issues. In that second part, we will introduce complete dg-Segal spaces, an alternative model to dg-categories based on Rezk's complete Segal spaces; we will construct a model structure on them, and explain how they are equivalent to dg-categories. If time allows, we will end up with a small introduction to our joint work with Violeta Marques Borges and Arne Mertens on the relationship between complete dg-Segal spaces and dg-Segal categories.

20 October 2023 (cancelled)

• 14:00 - 16:15 — Irreducible symplectic varieties via Prym fibrations — Sasha Viktorova (KU Leuven)

In this talk we present a construction of irreducible symplectic varieties (which are singular analogues of Hyperkähler manifolds). We start by fixing a K3 surface S with an antisymplectic involution i. For a choice of a smooth ample curve C on the quotient S/i, one can construct the corresponding compactified relative Prym variety. The variety P is a (singular) Largangian fibration over the linear system of C. By the work of Arbarello, Saccà and Ferretti, we know that if S/i is an Enriques surface, then P is an irreducible symplectic variety. Inspired by this result and an earlier work of Markushevich and Tikhomirov, we investigate the situation when S/i is a rational surface and find sufficient conditions to ensure that P is an irreducible symplectic variety. This is a joint work in progress with E. Brakkee, C. Camere, A. Grossi, L. Pertusi and G. Saccà.

29 September 2023

• 14:00 - 15:45 — Cliques in Representation Graphs of Quadratic Forms — Nico Lorenz (Ruhr-Universität Bochum)

Given a quadratic form q over an F-vector space V and an element a of F, we can construct a graph as follows: Take V to be the set of vertices and draw an edge between two vertices x, y if and only if q(x - y) = a. Combinatorial invariants of these graphs have been studied for decades now. The most famous problem on such graphs is the Hadwiger-Nelson problem asking for the chromatic number of the graph over the reals with a=1 and q the sum of two squares form.

We give a brief survey on this problem and consider a related problem, that of finding cliques of maximal size, i.e. subsets of pairwise adjacent vertices. We use an algebraical description of this problem to calculate the clique number and the number of maximal cliques for quadratic forms over certain fields.

9 June 2023

• 14:00 - 15:00 — Sums of squares in function fields— Karim Johannes Becher (Universiteit Antwerpen)
  

The Pythagoras number of a commutative ring K is the smallest natural number n such that every sum of squares of elements of K is a sum of n squares of elements of K, or infinity, if no such number n exists.  The study of this ring invariant, denoted by p(K), is a very classical topic of algebra, and it has in particular been a crucial incentive for the development of the theory of quadratic forms over fields. Nevertheless, very basic questions on Pythagoras numbers of fields remain unsolved until today. For example, very little is known about the growth of the Pythagoras number under field extensions.

In my talk, I want to present this problem and some known results. This will include the following result obtained in a joint work by Nicolas Daans, David Grimm, Gustavo Manzano-Flores, Marco Zaninelli and myself. In the late 1970s, Eberhard Becker had characterised the fields K for which p(K(X))=2 holds. Based on an intriguing local-global principle for quadratic forms over function fields due to Vlerë Mehmeti, we have now been able to show that the condition that p(K(X))=2 implies the upper bound p(F) ≤ 5 for function fields in one variable F/K, and furthermore the bound p(K(X,Y)) ≤ 8.

• 15:15 - 16:15 — Internal categorical structures in (weakly) Mal'tsev categories — Nadja Egner (UCLouvain)

The notions of reflexive graph, (small) category and groupoid can be internalized in every category with pullbacks. It is well known that the category of crossed modules is equivalent to that of internal categories in the category Grp of groups. Moreover, one can show that any internal reflexive graph in Grp admits at most one structure of internal category and that any internal category in Grp actually is an internal groupoid. The composition of two morphisms and the inverse of a morphism in an internal groupoid in Grp are given by means of the Mal'tsev operation p(x,y,z) = x - y + z in the algebraic theory of groups. These results hold more generally in any Mal'tsev category and in any Mal'tsev variety of universal algebras, respectively. In a weakly Mal'tsev category, it is still true that any internal reflexive graph admits at most one structure of internal category. However, an internal category can fail to be an internal groupoid. Examples of weakly Mal'tsev categories are given by any Mal'tsev category as the categories of groups, topological group, rings, Lie algebras and the dual of any topos. Furthermore, the categories of distributive lattices and commutative monoids with cancellation yield weakly Mal'tsev categories. A recent result of Pierre-Alain Jacqmin, Nelson Martins-Ferreira and the speaker is a syntactic characterization of weakly Mal'tsev varieties.

26 May 2023

• 14:00 - 15:00 — Lonely Runner Conjecture for function fields — Wout Wijnants (Universiteit Antwerpen)

In this talk we will present an analogue of the famous Lonely Runner Conjecture for function fields over a finite field. We will then transform the problem to a covering problem using some basic tools from linear algebra, and finally make use of so-called sunflowers to give a proof for a slightly weaker statement. The talk is based on work by Sam Chow and Luka Rimanic.​

21 April 2023​

• 14:00 - 15:00 — Classification of torsion-free abelian groups of finite rank — Cédric Aïd (Universiteit Antwerpen)

In this talk we will introduce the classification problem of torsion-free abelian groups of finite rank, as well as provide a proof for groups of rank 1. We will then give a known invariant for groups of higher rank and compare it to the rank 1 case. If time permits, we will also discuss some relevant results from descriptive set theory about the complexity of this problem.​

• 15:15 - 16:15 — Breaking the Supersingular Isogeny Diffie-Hellman protocol — Wouter Castryck (KU Leuven) (video)
  

Finding an explicit isogeny between two given isogenous elliptic curves over a finite field is considered a hard problem, even for quantum computers. In 2011 this led Jao and De Feo to propose a key exchange protocol that became known as SIDH, short for Supersingular Isogeny Diffie-Hellman. The security of SIDH does not rely on a pure isogeny problem, due to certain "auxiliary" elliptic curve points that are exchanged during the protocol (for constructive reasons). In 2017 SIDH was submitted to the NIST standardization effort for post-quantum cryptography, and since then it has attracted a lot of attention. Early July, it advanced to the fourth round. In this talk I will discuss a break of SIDH that was discovered in collaboration with Thomas Decru about three weeks later. The attack uses isogenies between abelian surfaces and exploits the aforementioned auxiliary points, so it does not break the pure isogeny problem. It allows for a full key recovery at the highest security level in a few hours. As time permits, I will also discuss some more recent improvements and follow-up work due to Maino et al. and Robert.

24 March 2023

• 15:15 - 16:15 — The Algebra of Aristotelian Diagrams — Lorenz Demey (KU Leuven) (video)
  

The aim of this talk is to provide a broad introduction to Aristotelian diagrams, with a special focus on their algebraic aspects. The talk consists of four main parts. The first two parts are introductory in nature, while the last two present ongoing research. In the first part, I will sketch the historical and philosophical context of Aristotelian diagrams. We will meet some interesting examples of such diagrams, coming from over a millennium ago as well as from contemporary AI research. We will also ask the fundamental question: why are Aristotelian diagrams used so widely in the first place? In the second part, I will show that Aristotelian diagrams can be studied most naturally in the context of Boolean algebra. We will discuss the topics of logic-sensitivity and bitstring semantics, and consider notions of Aristotelian and Boolean isomorphism. This theoretical machinery will be illustrated with a case study on the 14th-century philosopher John Buridan. In the third part, I will present a concrete open problem concerning the interface between logic-sensitivity and bitstring semantics. In particular, we will discuss a case study about logic-sensitivity which looks very elegant if the bitstrings are viewed are purely algebraic entities; however, if they are viewed as semantically meaningful representations of logical formulas, the case study seems to break down. In the fourth and final part, I will discuss how Aristotelian diagrams give rise to various categories, depending on the notion of morphism that we equip them with. There are two major criteria here: on the one hand, the categorically defined notions (e.g. isomorphism) should, as much as possible, be in line with previous work; on the other hand, the categories that arise should have a rich internal structure (e.g., initial/terminal objects, products, etc.). 

10 March 2023

• 14:00 - 15:00 — Birational automorphisms of Hilbert schemes of points on K3 surfaces — Pietro Beri (Jussieu-Paris) 

Hilbert schemes of points on K3 surfaces are an important family of varieties, since they share important properties with K3 surfaces themselves: they are the first example, in higher dimension, of hyper-Kähler manifolds, which in turn appear naturally in the celebrated Beauville-Bogomolov's decomposition Theorem (1983). Some unique features of hyper-Khäler manifolds allow a precise classification of the birational automorphisms of Hilbert schemes of points on algebraic K3 surfaces of Picard rank one; this has been done recently in a joint work with Al. Cattaneo. In this talk, after an introduction of the objects in play, we will present our classification.This kind of result does not give any hint about a geometric description of the involution; typically, finding such a description is a complex problem. We will present some famous geometric construction and some ways to build up new ones.

• 15:15 - 16:15 — Kodaira dimension of moduli spaces of hyperkähler varieties — Emma Brakkee (Universiteit Leiden)

Hyperkähler varieties are much studied objects in algebraic geometry, for instance because they can be seen as building blocks for other varieties. Their moduli spaces are objects whose points represent hyperkähler varieties in a natural way. In a series of papers from 2007-2011, Gritsenko, Hulek and Sankaran proved an important result about the birational geometry of these moduli spaces: that many of them have maximal Kodaira dimension. In this talk I will explain these results, and present new results about the Kodaira dimension of many more types of hyperkähler moduli spaces. If time permits, I will say some words about the proof strategy. This is joint work with I. Barros, P. Beri and L. Flapan.

24 February 2023

• 14:30 - 15:30 — High Dimensional Expanders — Inga Valentiner-Branth (Universiteit Gent) (video)
  

 Expander Graphs are sparse graphs that are highly connected. Apart from their combinatorial definition, they can be identified by spectral properties of their adjacency matrix. The first constructions of families of expanders involve group theoretic techniques around Kazhdan’s property (T). On the other hand, concrete examples of expander graphs are used in many areas including complexity theory, design of robust computer networks, and the theory of error-correcting codes.
Simplicial complexes are a way of generalizing the notion of graphs to higher dimension, by not just looking at vertices and edges but also triangles, tetrahedrons etc. Naturally, one is interested in a generalization of the notion of expander graphs to simplicial complexes. Unfortunately, generalizing the different, but equivalent, definitions of expander graphs leads to non-equivalent notions of High Dimensional Expanders.
In this talk, I will introduce the concept of expander graphs, motivate why they are interesting from an algebraists point of view and introduce some of the definitions of High Dimensional Expanders. If time permits, I will give an idea of the construction that our research group hopes to use to construct high dimensional expanders.

• 15:45 - 16:45 — The curvature problem, two ways — Alessandro Lehmann (Universiteit Antwerpen)
  

I will begin by recalling the classical notion of infinitesimal deformation of an algebra, and how it is related to Hochschild cohomology. I will then explain how parallel concepts for differential graded algebras can be defined, and what is the curvature problem in this setting. Finally, I will sketch the theory of formal moduli problems and discuss how the curvature problem emerges in that context.

16 December 2022

• 14:00 - 15:00 — Comparing additive and monoidal categorification via Quantum Affine Algebras — Geoffrey Janssens (UCLouvain)

Given an affine simple Lie algebra g, one can consider its universal enveloping U(g). Following Drinfeld and Jimbo, the latter can be quantised which will result in the so-called quantum affine algebra U_q(g). In this talk we will be interested in its representation theory. An important tool for this, is the canonical basis of Lusztig whose multiplicative properties are given by a combinatorial model called a cluster algebra. The first half of the talk will recall these concepts. In the second half we will focus on two main methods, an additive and a monoidal, to categorify such algebras. More precisely, we aim to give an impression of some conjectural connections between them, triggered by  recent work of Ryo Fujita.

• 15:15 - 16:15 — Diagram categories of Brauer type — Sigiswald Barbier (Universiteit Gent)

Diagram categories are a special kind of tensor categories that can be represented using diagrams. In this talk I will give an introduction to categories represented using Brauer diagrams. In particular I will explain the relation with the Brauer algebra and how the categorical framework can be applied to representation theory of the corresponding algebra. 

25 November 2022

• 14:00 - 15:00 — Graphical calculus for quantum vertex operators — Hadewijch De Clercq (Universiteit Gent)

Graphical calculus provides a diagrammatic framework for performing topological computations with morphisms in ribbon categories. This amounts to a functorial identification of such morphisms with oriented diagrams colored by a ribbon category. The category of finite-dimensional representations of a quantum group serves as a motivating example. In this talk I will first introduce the fundamentals of the graphical calculus for ribbon categories. Then I will explain how this can be extended to categories with less structure. This will allow us to include also infinite-dimensional quantum group modules, and to provide graphical presentations for special morphisms such as quantum vertex operators and dynamical R-matrices. I will demonstrate the potential of this approach by graphically deriving certain q-difference equations for twisted trace functions of N-point quantum vertex operators. These include the dual q-KZB and Macdonald equations first obtained by Etingof and Varchenko, as well as some generalizations. This talk is based on joint work with Nicolai Reshetikhin (UC Berkeley) and Jasper Stokman (University of Amsterdam).

•  15:15 - 16:15 — An introduction to (categorical) resolutions of singularities. — Timothy De Deyn (Vrije Universiteit Brussel)

 I will give a gentle introduction to resolutions of singularities and their categorical incarnations. We will start with some birational geometry, observe that not all varieties are created equal and contemplate a bit about what a space actually is. If time permits, I will say some words on my ongoing work with M. Van den Bergh on constructing certain categorical resolutions.

4 November 2022

• 14:00 - 15:00 — Erdős–Ko–Rado problems in finite geometry — Jozefien D'haeseleer (Universiteit Gent)

One of the classical problems in extremal set theory is to determine the size of the largest families of pairwise non-disjoint subsets in a given set. Erdös, Ko and Rado investigated this problem and classified the largest examples of these families. In the last decades, this problem, originated in set theory, was generalized to many other structures such as permutation groups, integer partitions, and projective spaces. In this presentation I will give an overview of the known Erdös–Ko–Rado theorems in set theory and projective geometry. In this last setting I will also give a classification result.

• 15:15 - 16:15 — Enriched Segal categories as models for weakly enriched categories — Violeta Borges Marques (Universiteit Antwerpen) (video)
  

In this talk I want to explain why Segal categories can be seen as weakly (or “up to homotopy”) enriched categories. First, I will use two guiding examples - topological categories and the fundamental oo-groupoid of a topological space - to arrive at the definition of Segal category. Then I will focus on two key features that justify the claim in the title of the talk. On the last slides, I will shed some light on my current topic of research, a generalisation of Segal categories.

21 October 2022

• ​14:00 - 15:00 — Birational classification of moduli spaces — Ignacio Barros (Universität Paderborn)

I will survey the classical problem of determining the birational complexity of the moduli space of curves. We will review some elementary facts about these moduli spaces, talk about open problems, and if time permits I will present recent results joint with S. Mullane on the hyperelliptic locus. This will be a Colloquium style talk, so questions are strongly encouraged!

20 May 2022

15:00 - 16:00 — An introduction to Mal'tsev categories —Pierre-Alain Jacqmin (UCLouvain)

Mal'tsev categories turned out to be a central concept in categorical algebra. The simplicity and the beauty of the notion is revealed by the wide variety of characterizations of a markedly different flavour. Depending on the context, one can define Mal'tsev categories as those for which `any reflexive relation is an equivalence relation'; `any relation is difunctional'; `the composition of equivalence relations on the same object is commutative'; `each fibre of the fibration of points is unital' or `any sub-reflexive graph of an internal groupoid is itself a groupoid'. For a variety of universal algebras, these are also equivalent to the existence in its algebraic theory of a Mal'tsev operation, i.e., a ternary operation p(x,y,z) satisfying the axioms p(x,x,y)=y and p(x,y,y)=x. As examples of Mal'tsev categories, one can cite the categories of groups, topological groups, Heyting algebras, any abelian category and the dual of any topos. Mal'tsev categories is a good context in which to develop the theory of centrality of equivalence relations and some homological lemmas such as the denormalized 3x3 lemma. Recently, some embedding theorems have been established for Mal'tsev categories, similar to the Freyd-Mitchell embedding theorem for abelian categories.

16:15 - 17:15 — Multiplicative preprojective algebras for Dynkin quivers — Daniel Kaplan (UHasselt) (video)

Crawley-Boevey and Shaw defined the multiplicative preprojective algebra to understand Kac’s middle convolution and to solve the Deligne-Simpson problem. In Shaw’s thesis he noticed a curious phenomenon: for the D_4 quiver the multiplicative preprojective algebra (with parameter q=1) is isomorphic to the (additive) preprojective algebra if and only if the underlying field has characteristic not two. Later, Crawley-Boevey proved the multiplicative and additive preprojective algebras are isomorphic for all Dynkin quivers over the complex numbers. Recent work of Etgü-Lekili and Lekili-Ueda, in the dg-setting, sharpens the result to hold over fields of good characteristic, meaning characteristic not 2 in type D, not 2 or 3 in type E and not 2, 3, or 5 for E_8. Neither work constructs an explicit isomorphism.

13 May 2022

• 14:00 - 15:00 — Faithful ADE braid group actions on triangulated categories —Anya Nordskova (UHasselt)

• 15:15 - 16:15 — Bézout domains and Pythagoras numbers of fields — Marco Zaninelli (Universiteit Antwerpen) (video)
  

For a field F, the Pythagoras number of F is the minimal number m of squares such that any sum of squares in F is a sum of m squares in F. For several fields, it is possible to find a subring that is a semilocal Bézout domain, and which contains all the information about the sums of 2^n squares in the field. In the colloquium it will be shown how such subrings can be exploited in order to obtain an upper bound for the Pythagoras number of the form 2^n+1.

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) (video)
  

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) (video)
  

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)) . . . ((t_n)) — 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 K_n = R((t₁))...((t_n)), for some natural number n.

Consider an algebraic function field F/K_n. 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 = K_n(X)/sqrt(f), where f ∈ K_n[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)| ≤ 2^n(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 theory — Morgan 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.

31 May 2021

Antwerp Algebra Colloquium on topics on cohomology:

• 14:00 - 15:00 — Cohomological invariants of n-dimensional quadratic forms in I³ — 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 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 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 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 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 F₄, E₆, E₇, E₈, 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.