Mini Courses
John Baez: How to count using categories
Abstract: Many questions in enumerative combinatorics ask us to count the number of structures of some sort on an n-element set, as a function of n. But what is a “structure of some sort”? This is made precise by Joyal’s theory of species. This theory also clarifies a simple yet powerful technique for counting structures on finite sets, namely generating functions. We shall see that the category of species is the “free 2-rig on one generator”, and that combinatorially interesting operations on generating functions arise from this fact. Thus, a large chunk of enumerative combinatorics can be seen as a branch of categorified ring theory. We hope to strike a balance between abstract theory and fun concrete examples of counting problems.
Răzvan Diaconescu: Model-theoretic Universal Algebra
Universal algebra is the meta-level of classical higher algebra. It has also important applications in computer science, especially in algebraic specification. In this course, we will first introduce the basic concepts (signatures, universal algebras, homomorphisms, congruences, etc.) and prove the First Isomorphism Theorem.
Then we will introduce the logic side of universal algebras, which is the equational logic. With all these we will prove the Birkhoff Variety theorem, which says that varieties of universal algebras are axiomatizable by sets of universally quantified equations, the Birkhoff Completeness Theorem for equational logic, and develop the initial semantics for quasi-varieties (classes of universal algebras defined by conditional equations). In the final part of the course, we will present several extensions of traditional universal algebras, such as universal algebras with partial operations, many-sorted algebras, and preordered algebras. All these extensions are highly relevant in computer science.
Cédric Lecouvey: An overview on the representation theory of the symmetric group : foundations and perspectives
Abstract: This mini-course is an introduction to the representation theory of the symmetric group. We will start by recalling some basic facts about the representation theory of finite groups over the complex numbers and its historical connections with the determination of the normal basis in Galois field extensions. Next, we will focus on the case of the symmetric group, describe how its characters are connected to symmetric polynomials and summarise the construction of its irreducible representations via Specht modules and partition combinatorics. If time permits, we will finally illustrate, through some examples, the complexity of this representation theory in positive characteristics.
Götz Pfeiffer: Algorithmic Aspects of Finite Groups and their Representations
Abstract: According to the classification of finite simple groups, a finite simple group is either cyclic, alternating, of Lie type, or one of 26 so-called sporadic simple groups. Theorems about finite groups can often be reduced to the simple case, and the proof then might require detailed investigation of individual groups. In these lectures we will present and discuss some fundamental algorithmic tools that are useful in finite group theory and representation theory, with applications to some sporadic groups and to the finite Coxeter groups that control the structure and representation theory of finite groups of Lie type.
Invited Talks
Hara Charalambous: Graded Free Resolutions in Commutative Algebra: classical results and newer trends
Abstract: Commutative Algebra, an important and active area of research in Mathematics, studies commutative rings, their ideals and modules. Commutative Algebra has recently seen extraordinary developments: long standing conjectures have been proven and new connections to different areas of mathematics have emerged. In particular, the study of free resolutions is a central and beautiful topic, providing a method for describing the structure of modules with connections to Algebraic Geometry, Number Theory, Combinatorics, Computational Algebra, Invariant Theory, Mathematical Physics, and other fields. In this talk we will discuss invariants of minimal free resolutions, recent developments and trends.
Pantelis Eleftheriou: O-minimality and combinatorics
Abstract: We will discuss some recent connections between model theory (a branch of mathematical logic), linear algebra, and extremal graph theory (a branch of combinatorics). More precisely, we will look at the Zarankiewicz problem, which asks for an upper bound on the number of edges of a hypergraph, assuming it has no complete sub-hypergraphs of a given size. We will see how the upper bounds can be improved if we further assume that the class of hypergraphs we consider is “definable” in certain o-minimal structures. In particular, we will survey those bounds when the hypergraphs are semialgebraic (respectively, semilinear), that is, boolean combinations of polynomial (respectively, linear) equlities and inequalities. This is mostly a survey talk with some recent results obtained together with Aris Papadopoulos.
Stavros Garoufalidis: Habiro cohomology
Abstract: We will explain in broad strokes and with examples some recent arithmetic structures on 3-dimensional topological quantum field theory, ultimately related to motivic cohomology. Joint work with Peter Scholze, Campbell Wheeler and Don Zagier.
Dimitra Kosta: Markov bases of toric ideals: one of the first connections between statistics and commutative algebra
Abstract: In the first part of the talk, I will give an introduction on Markov bases of toric ideals, which are one of the first connections between statistics and commutative algebra. In the second part of the talk, I will discuss recent work on strongly robust toric ideals. A toric ideal is called strongly robust when the Graver basis is a minimal system of generators. In the talk, I will explain how to build a strongly robust simplicial complex which determines the strongly robust property of toric ideals. I will then discuss our results on the strongly robust property in the case of monomial curves as well as codimension 2 toric ideals. This is joint work with A. Thoma and M. Vladoiu.
Till Mossakowski (online): Algebra and logic meet deep learning
Abstract: Reference ontologies play an essential role in organising knowledge in
the life sciences and other domains. They are built and maintained
manually. Since this is an expensive process, many reference ontologies
only cover a small fraction of their domain. We develop techniques that
enable the automatic extension of the coverage of a reference ontology
by extending it with entities that have not been manually added yet. The
extension shall be faithful to the (often implicit) design decisions by
the developers of the reference ontology. While this is a generic
problem, our use case addresses the Chemical Entities of Biological
Interest (ChEBI) ontology with classes of molecules, since the chemical
domain is particularly suited to our approach. ChEBI provides
annotations that represent the structure of chemical entities (e.g.,
molecules and functional groups).
We show that classical machine learning approaches can outperform
ClassyFire, a rule-based system representing the state of the art for
the task of classifying new molecules, and is already being used for the
extension of ChEBI. Moreover, we develop RoBERTa and Electra transformer
neural networks that achieve even better performance. In addition, the
axioms of the ontology can be used during the training of prediction
models as a form of semantic loss function. Furthermore, we show that
ontology pre-training can improve the performance of transformer
networks for the task of prediction of toxicity of chemical molecules.
Finally, we show that our model learns to focus attention on more
meaningful chemical groups when making predictions with ontology
pre-training than without, paving a path towards greater robustness and
interpretability. This strategy has general applicability as a
neuro-symbolic approach to embed meaningful semantics into neural networks.