Let \( \Delta \) be a connected graph, without loops or multiple edges, such that each vertex has valency at least 3. Let \( G \leq \operatorname{Aut}(\Delta) \) with \( \lvert G_z \rvert \lt \infty \), for all \( z \in \Delta \). The group \( G \) is said to act locally \( s \)-arc transitive if, for all \( z \in \Delta \), the group \( G_z \) acts transitively on the set of \( s \)-arcs originating at \( z \). In this case the graph is called locally \( (G,s) \)-arc transitive. We will discuss some results about the structure of the amalgam of adjacent vertex stabilizer for locally \( (G,s) \)-arc transitive graphs when \( s \geq 4 \).

In this talk, I will review recent developments on the correspondence between Lie algebras spanned by extremal elements and buildings.

This is a survey on (finite or infinite) nearfields, with emphasis on classification results.

We present material motivated by work of Jacques Tits on triality. This does not come from his famous IHES article but instead a second paper which presents a graph theoretic version of triality. The approach is directly related to 3-transposition groups and edge-transitive graphs, topics to which Richard Weiss has contributed much through the years.

Prolongations of a linear Lie algebra are studied by E. Cartan and irreducible linear Lie algebras with nonzero prolongations are classified in the works of Cartan and Kobayashi-Nagano. In our joint-works with Ngaiming Mok and Baohua Fu, we have extended this classification to Lie algebras of linear automorphisms of nonsingular projective varieties. We will survey these results and report on a new development on contact prolongations of symplectic Lie algebras.

In 1979 Richard Weiss made a beautiful conjecture: There exists a function \(f\) on the natural numbers such that, for each \(G\)-vertex-transitive locally-primitive graph \(X\), the order of a vertex-stabiliser is bounded by \( f(d) \), with \(d\) the valency of \(X\). Although the conjecture remains open in full generality, many mathematicians have been attracted to work on this problem, and in several interesting cases it has been shown that the conjecture is true. As well proofs of (cases of) the statement, Richard's conjecture also spurred people to understand vertex-transitive graphs with local actions of certain flavours, such as locally-quasiprimitive graphs. This lead to the Praeger Conjecture of 2000, and for locally-semiprimitive graphs, the Potocnik-Spiga-Verret Conjecture of 2012. At this point the graphs faded to the background and general questions about semiprimitive permutation groups were raised. These turned out to be quite difficult, but one can say something. I'll give an overview of some of these recent results and describe what we know on the various conjectures, that is, an overview of the wonderful mathematics that was inspired by Richard's Conjecture.

In 2002, the classification of Moufang polygons was completed by Jacques Tits and Richard Weiss. It is natural to ask to which extent one can hope for similar results in the case of Moufang trees. In my talk I will give an overview about what is known concerning this question. Furthermore, I will present a result on Moufang trees of prime order which has been obtained recently in joint work with Matthias Grüninger and Max Horn.

There are general questions concerning when the existence of zero cycles of degree one implies the existence of rational points on homogeneous spaces under connected linear algebraic groups. This question for principal homogeneous spaces was raised by Serre and is open. We relate this question to certain norm principles and explain some recent progress in this direction.

Isolated \(p\)-minimal subgroups will be defined and consequences of their existence will be given. The determination of all isolated rank one \(p\)-minimal subgroups in almost simple groups will be presented. An application to the Meierfrankenfeld-Stellmacher-Stroth programme to understand groups of local characteristic \(p\) will be discussed.

This describes joint work with Ulrich Meierfrankenfeld and Peter Rowley.

A non-orthogonal version of the classical Cayley-Dickson construction, generalizing an earlier more restrictive approach due to Garibaldi-Petersson [2], will be presented over arbitrary commutative rings. Its input consists of a conic algebra \(B\) (i.e., a non-associative algebra of degree \(2\) in the sense of McCrimmon [3]), a scalar \(\mu\) in the base ring and a linear form \(s\) on \(B\), measuring the deviation of the construction from being orthogonal, while its output is again a conic algebra. Conditions in terms of the input parameters that are necessary and sufficient for the output to be alternative, associative, commutative-associative or flexible, respectively, are described. Under comparatively mild restrictions on the base ring, a general procedure is presented leading to an output algebra that is non-singular, even though the input algebra is not. As a special case, this procedure yields various realizations of the Coxeter octonions [1] living on the \(E_8\)-lattice over the integers.

In my talk, I will give a brief summary of the structure of pseudo-reductive groups studied in the monograph "Pseudo-reductive groups, 2nd edition, Cambridge U. Press (2015)" by Conrad, Gabber and Prasad, and describe their classification obtained in the monograph "Classification of pseudo-reductive groups, Annals of Math Studies, No 191 (2015)" by Conrad and Prasad.

An \(\tilde{A}_2\)-building is a Euclidean building all of whose rank \(2\) residues are generalized \(3\)-gons, i.e. incidence graphs of projective planes. When the building is Bruhat-Tits (over a local field), those residues are classical planes (over a finite field). This talk will report the discovery of an \(\tilde{A}_2\)-building whose rank \(2\) residues are non-Desarguesian finite planes, and nevertheless shares an important property with the Bruhat-Tits \(\tilde{A}_2\)-buildings, namely the existence of a discrete group acting with finitely many orbits. The question of existence of such buildings was raised by Kantor in 1984.

Jordan algebras \(J\) of characteristic not \(2\) sometimes have a set of idempotents \((e^2=e)\) that generate \(J\) such that their adjoint map \(\operatorname{ad}_e: u \mapsto ue\) \((u\in J)\) has the minimal polynomial \(x(x-1)(x-1/2)\), and with additional restrictions on products of elements in the eigenspaces of \(\operatorname{ad}_e\) (for each \(e\)). Generalizing these properties (not only of such Jordan algebras) Hall, Rehren, Shpectorov (HRS) introduced "Axial algebras of Jordan type". In my talk I will present structural results on axial algebras of Jordan type \(1/2\) (a case which was not dealt with in HRS), I will discuss their idempotents \(e\), the corresponding "Miyamoto involutions" \(\tau(e)\) and the group that these involutions generate.

This is joint work with J. Hall and S. Shpectorov.

In this talk a variation of Glauberman's ZJ-Theorem for finite groups \(G\) is discussed. There are three major differences to Glauberman's result:

1. \(G\) does not need to be \(p\)-stable.
2. Rather than \(\operatorname{Syl}_p(G)\) we use \(\operatorname{Bau}_p(G)\), the conjugacy class of \(p\)-central Baumann subgroups.
3. Our result has the same shape as the Local \(C(G,T)\)-Theorem since one has to allow exceptions (Baumann blocks). However, these exceptions can be described precisely.

This is joint work with G. Parmeggiani and Ulrich Meierfrankenfeld.

Some thoughts about joint work with Richard starting in the 80's.

For an absolutely simple linear algebraic group that is simply connected or adjoint, the action of the automorphism group of the Dynkin diagram on the Tits class provides an obstruction to the existence of outer automorphisms. This talk will report on a joint work with Anne Quéguiner-Mathieu, in which we give examples where outer automorphisms do not exist even though the Tits class obstruction vanishes. This settles in the negative a conjecture of Garibaldi-Petersson.

(Unfortunately, Vladimir Trofimov will not be able to participate in the conference.)

In this talk we review the influence of Jacques Tits on the modern Incidence Geometry. In fact, Tits can be considered the founder of that field, and we explain how this came about. In the very same piece of work of Tits, he introduced the (geometric version of the) Magic Square, and we join those two historical facts by explaining some recent results on the incidence geometric side of the Square. Of course, a link with Richard Weiss will not be missing.

The topic of this talk is the ongoing joint work with Imke Toborg on the \(Z^*_3\)-Theorem. A special case has been treated by Peter Rowley, and a generalisation of Glauberman's \(Z^*\)-Theorem for all odd primes is already known and can be proven by direct application of the Classification of Finite Simple Groups. However, we are interested in arguments that reveal how strongly closed subgroups or isolated elements influence the local structure of a finite group. Therefore we are now concentrating on the prime \(3\), understanding special cases and working our way towards a \(Z^*_3\)-Theorem.