

Titles and abstracts 
 John van Bon:
Locally \( s \)arc transitive graphs with \( s \geq 4 \)
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
\( s \geq 4 \).
 Arjeh Cohen:
Extremal elements in Lie algebras and related buildings
In this talk, I will review recent developments on the correspondence between Lie algebras spanned by extremal elements and buildings.
 Theo Grundhöfer:
Nearfields
This is a survey on (finite or infinite) nearfields, with emphasis on
classification results.
 Jonathan Hall:
Reflections on triality
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
3transposition groups and edgetransitive
graphs, topics to which Richard Weiss has
contributed much through the years.
 JunMuk Hwang:
Prolongations of infinitesimal linear automorphisms of projective varieties
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 KobayashiNagano. In our jointworks 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.
 Luke Morgan:
The Weiss Conjecture and inspirations thereof
In 1979 Richard Weiss made a beautiful conjecture:
There exists a function \(f\) on the natural numbers such that,
for each \(G\)vertextransitive locallyprimitive graph \(X\), the order of a vertexstabiliser 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 vertextransitive graphs with
local actions of certain flavours, such as locallyquasiprimitive graphs.
This lead to the Praeger Conjecture of 2000, and for locallysemiprimitive graphs, the PotocnikSpigaVerret 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.
 Bernhard Mühlherr:
Moufang trees
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.
 Parimala:
Zero cycles of degree one versus rational points
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.
 Chris Parker:
Isolated \(p\)minimal subgroups in finite groups and their application
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 MeierfrankenfeldStellmacherStroth programme to understand groups of local characteristic \(p\) will be discussed.
This describes joint work with Ulrich Meierfrankenfeld and Peter Rowley.
 Holger Petersson:
An algebraic formalism for the octonionic structure of the \(E_8\)lattice
A nonorthogonal version of the classical CayleyDickson construction, generalizing an earlier more restrictive approach due to
GaribaldiPetersson [2], will be presented over arbitrary commutative rings.
Its input consists of a conic algebra \(B\) (i.e., a nonassociative 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,
commutativeassociative 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 nonsingular,
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.
References
[1] H. S. M. Coxeter, Integral Cayley numbers, Duke Math. J. 13 (1946), 561578.
[2] S. Garibaldi and H.P. Petersson, Wild Pfister forms over Henselian fields, K theory, and conic division algebras,
J. Algebra 327 (2011), 386465.
[3] K. McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 (1985), no. 1, 85109.
 Gopal Prasad:
Structure and classification of pseudoreductive groups
In my talk, I will give a brief summary of the structure of pseudoreductive groups studied in the monograph "Pseudoreductive groups, 2nd edition, Cambridge U. Press (2015)" by Conrad, Gabber and Prasad, and describe their classification obtained in the monograph "Classification of pseudoreductive groups, Annals of Math Studies, No 191 (2015)" by Conrad and Prasad.
 Nicolas Radu:
A locally nonDesarguesian \(\tilde{A}_2\)building admitting a uniform lattice
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 BruhatTits (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 nonDesarguesian finite planes,
and nevertheless shares an important property with the BruhatTits \(\tilde{A}_2\)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.
 Yoav Segev:
Idempotents inducing a \(\mathbb{Z}_2\)grading on nonassociative algebras and their corresponding involutions
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 \(x(x1)(x1/2)\),
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.
 Bernd Stellmacher:
A ZJTheorem for non\(p\)stable finite groups
In this talk a variation of Glauberman's ZJTheorem for finite groups \(G\) is discussed.
There are three major differences to Glauberman's result:
 \(G\) does not need to be \(p\)stable.
 Rather than \(\operatorname{Syl}_p(G)\) we use \(\operatorname{Bau}_p(G)\), the conjugacy class of \(p\)central Baumann subgroups.
 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.
 Gernot Stroth:
Graphs
Some thoughts about joint work with Richard starting in the 80's.
 JeanPierre Tignol:
Outer automorphisms of algebraic groups
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 QuéguinerMathieu, 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 GaribaldiPetersson.
 Vladimir Trofimov:
On the Weiss Conjecture
(Unfortunately, Vladimir Trofimov will not be able to participate in the conference.)
 Hendrik Van Maldeghem:
The point of geometry, adapted from Jacques Tits
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.
 Rebecca Waldecker:
Towards a \( Z^*_3 \)Theorem
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.



