TALKS / ABSTRACTS

• Christoph Möller [Martin-Luther-Universität Halle-Wittenberg]
  Title: Low fixity action on groups
  Abstract: In my talk, I will classify all pairs of finite groups $(G,A)$ where $A$ is a non-trivial subgroup of $\mathrm{Aut(G)}$ which acts with fixity at most $4$ on $G$, that is, every non-trivial automorphism in $A$ centralizes at most four elements of $G$. This is an extension of Thompson's work on fixed-point-free automorphisms and solves a problem which naturally arises when analyzing finite permutation groups in which non-trivial elements have at most four fixed points. If time allows, I will also present a related problem about finite simple groups that act with low fixity on one of their conjugacy classes -- a problem that is related to questions about pointed Hopf algebras.
  
  
• Gabriel de Area Leao Souza [Universidad de València]
  Title: Brauer’s Height Zero conjecture and Galois automorphisms
  Abstract: Perhaps one of the most important problems in the representation theory of finite groups is Brauer’s Height Zero conjecture, recently solved in 2022. One particular case of this conjecture states that a Sylow $p$-subgroup is abelian if and only if all characters in the principal $p$-block of $G$ have $p'$-degree. In this talk, we show that, in fact, it suffices to consider only some characters in the principal block; namely, those which are fixed by a certain group of Galois automorphisms $\mathcal{J}$. We then use this strengthening to obtain a result that can be understood as a Galois version of the celebrated Itô-Michler theorem.
  This is joint work with A. Moretó and N. Rizo.
  
  
• Alice Niemeyer [RWTH Aachen]
  Title: The Journey Created by an Algorithm to Recognise Finite Classical Groups
  Abstract: In 2011, at the Fifth de Brún Workshop in Galway, Ireland, Ákos Seress asked Cheryl Praeger and the speaker to estimate the proportion of certain elements in finite classical groups. This question marked the beginning of a journey to design and analyse a second-generation algorithm for the constructive recognition of finite classical groups in their natural representations. Although the first step of the algorithm is surprisingly straightforward, the theoretical problems it generated were many. In this talk, I will highlight a few of the challenges that arose along the way.
  
  
• Marco Praderio Bova [Technische Universität Dresden]
  Title: Inductive construction of fusion systems
  Abstract: Fusion systems are categories that, in a sense, generalize the p-local structure of finite groups and have found numerous applications in both algebra and topology. They are defined with the idea in mind that their morphisms should be "realized" as conjugation by elements in a finite group. However, such finite group does not always exist. When this happens we call the fusion system "exotic". Ever since the fundation of the theory of fusion systems, the search for exotic ones has been an active topic of research. One of the most, if not the most, successful method developed to build new fusion systems (in particular exotic ones) is what we call the "arboreal method" which exploits the homotopy theoretic properties of fusion systems. Despite its success, this method fails to build the Benson-Solomon fusion systems (the only known family of exotic fusion system at the prime $2$), which notably enough admit a similar construction. 
  Time permitting we will explain how this approach can be used to prove the sharpness conjecture (at least in some cases).
  This is joint ongoing work with Remí Molinier.
  
  
• Pratyush Gautam [University of Manchester]
  Title: Fusion systems related to sporadic groups
  Abstract: Fusion systems are closely related to groups where we focus only on conjugation (fusion) coming from the group to its $p$-subgroups. Fusion systems related to sporadic groups have been studied throughout, and many exotic fusion systems have been found on a Sylow $p$-subgroup of a sporadic group. I will discuss the classification of fusion systems on sporadic groups, focusing in particular on $p$ odd. This should give some idea as to why some $p$-groups support exotic fusion systems, and why some others do not! We will also go through an improved algorithm that allows us to complete the classification for large sporadic groups.
  
  
• Max Gheorghiu [HHU Düsseldorf]
  Title: Poincaré duality for profinite groups via condensed mathematics
  Abstract: Poincaré duality is a condition providing isomorphisms from the cohomology groups to certain homology groups of a manifold or a group. We establish Poincaré duality in unprecedented generality for profinite groups, which are (topological) groups assembled from finite groups in a particular manner. A great part of the talk is dedicated to introducing our primary proof technique, namely a novel promising theory called condensed mathematics.
  
  
• Torben Wiedemann [RPTU Kaiserslautern-Landau]
  Title: Constructing $F_4$-graded groups over rings
  Abstract: While the precise formulation of the following statement is somewhat intricate, it is known that algebraic $k$-groups of type $F_4$ are strongly related to octonion algebras over $k$, even when $k$ is not a field but an arbitrary commutative ring. It was shown by Wiedemann in earlier work that any abstract group (which is not assumed to be algebraic!) with a root group grading by the root system $F_4$ is "defined over" what is called a "conic algebra" over a commutative ring. The class of conic algebras generalises that of octonion algebras (most importantly by removing the non-degeneracy condition on the norm of an octonion algebra). It is however not evident that every conic algebra $C$ permits an $F_4$-graded group that is defined over $C$. In this talk, we present a construction of an $F_4$-graded Lie algebra from an arbitrary conic algebra $C$ over a commutative ring and show that its automorphism group contains an abstract $F_4$-graded group that is defined over $C$.
  This is joint work with Tom De Medts.