*Abstract*: Let be an element of a finite group and let be the subgroup generated by all the right Engel values over . In the case when is soluble we prove that if, for some , the Fitting height of is equal to , then belongs to the th Fitting subgroup . For nonsoluble , it is proved that if, for some , the generalized Fitting height of is equal to , then belongs to the generalized Fitting subgroup with depending only on and , where is the product of primes counting multiplicities. It is also proved that if, for some , the nonsoluble length of is equal to , then belongs to a normal subgroup whose nonsoluble length is bounded in terms of and . Earlier similar generalizations of Baer’s theorem (which states that an Engel element of a finite group belongs to the Fitting subgroup) were obtained by the first two authors in terms of left Engel-type subgroups.

Abstract: The structure of reductive *p*-adic groups arises from the interaction of Euclidean geometry and the arithmetic of *p*-adic fields. Reeder and Yu have built on this interaction to give a construction of certain “epipelagic” representations. Their construction has many benefits, but set of vectors that may be used as input for the construction is not well understood. I will talk about on-going work to classify this set, as well as its relationship to Vinberg–Levy theory of graded Lie algebras in characteristic *p*.

Title: Tree-homogeneous graphs

Abstract: Let $X$ be a class of graphs. A graph $\Gamma$ is $X$-homogeneous if every graph isomorphism $\varphi:\Delta_1\to \Delta_2$ between induced subgraphs $\Delta_1$ and $\Delta_2$ of $\Gamma$ such that $\Delta_1\in X$ extends to an automorphism of $\Gamma$. For example, if $X=\{K_1\}$, then $X$-homogeneity is vertex-transitivity, and if $X=\{K_2\}$, then $X$-homogeneity is arc-transitivity. A graph is \textit{tree-homogeneous} if it is $X$-homogeneous where $X$ is the class of trees. We will discuss some recent progress on classifying the finite tree-homogeneous graphs, as well as some connections with certain highly symmetric incidence geometries called partial linear spaces.

]]>Abstract: Ramanujan graphs were first considered by Lubotzky, Phillips, Sarnak to

get graphs with optimal spectral properties. In our days the theory of expander graphs

and, in particular, Ramanujan graphs is well developed, but the questions is what is the best definition of a higher-dimensional expander is still wide open. There are several approaches, suggested by Gromov, Lubotzky, Alon and others, but the cubical complexes were not much investigated from this point of view.

In this talk I will give new explicit examples of cubical Ramanujan complexes and discuss possible developments.

]]>

After reaching the woods, we followed the path leading to Hartsholme Country Park, famous for its lakes and the White Bridge.

Upon leaving the park, we headed back to the city via the scenic route along the Catchwater dyke (here one has great views of the cathedral), before joining River Witham and following the river back to the university.

]]>Talk title: Representations of *p*-adic groups via graded Lie algebras

Abstract: The structure of reductive *p*-adic groups arises from the interaction of Euclidean geometry and the arithmetic of *p*-adic fields. Reeder and Yu have built on this interaction to give a construction of certain “epipelagic” representations. Their construction has many benefits, but set of vectors that may be used as input for the construction is not well understood. I will talk about on-going work to classify this set, as well as its relationship to Vinberg–Levy theory of graded Lie algebras in characteristic *p*.