Schedule for: 17w5015 - Symmetries of Discrete Structures in Geometry

In this talk, we will show the following results: For every integer $n\geq 5$ and every weighting function $f:E(K_n)\to \{-1,1\}$ on the edges of the complete graph $K_n$ such that $|\sum_{e\in E(K_n)}f(e)|\leq n(n-1)/2-h(n)$, where $h(n)=2(n+1)$ if $n \equiv 0$ (mod $4$) and $h(n)=2n$ if $n \not\equiv 0$ (mod $4$), there is always a copy of $K_4$ in $K_n$ for which $\sum_{e\in E(K_4)}f(e)=0$, and this bound is sharp. However, if we consider any $K_k$ with $k\geq 2$, $k\neq 4$, this is not true anymore: we show that there are infinitely many values of $n$ such that there is a weighting function $f:E(K_n)\to \{-1,1\}$ with $\sum_{e\in E(K_n)}f(e)=0$ and such that $\sum_{e\in E(K_k)}f(e)\neq 0$ for every copy of $K_k$ in $K_n$. These results solve a problem raised by Caro and Yuster.\\ This is a joint work with Yair Caro and Amanda Montejano.

In this talk we give the notion of complete colorings in graphs, achromatic number, Knesser graphs and Steiner triple systems . Also, we explain how the Steiner triple systems solve the problem about the existence of complete colorations on Knesser graphs that attain the upper bound of the achromatic number, where the achromatic number of a graph G is the maximum integer value for the number of chromatic classes in a complete and proper coloring of G.

The Upper and Lower bound Theorems in convex geometry deal, respectively, with the maximum and minimum number of facets that a convex d-dimensional polytope with n vertices can have. In this talk we will offer bounds for such numbers for what we refer to as the projective case of this problem. Namely, we are interested in the following: Problem 1: Given a set of $n$ points in general position $X \subset \R^{d}$ what is the maximum number of facets that $conv(T(X))$ can have, among all the possible permisible projective transformations $T$ of $X$? Problem 2: Given a set of $n$ points in general position $X \subset \R^{d}$ what is the maximum number of vertices that $conv(T(X))$ can have (as the support of the convex hull), among all the possible permisible projective transformations $T$? We define the \textbf{projective class of a set of points} $X \subset \R^{d}$ as the set of all possible point configurations that are the image of $X$ under a permissible projective transformation, and denote it $[X].$ Mimicking the polytope notation, let $f_k([X])$ be \textbf{the maximum number of $k$--faces} that the convex hull of a point configuration in the class $[X]$ can have, i.e. $f_k([X])= max_{Y \in [X]} \{f_k(conv(Y))\}.$ Finally we denote as $f_k(n),$ the minimum $f_k([X])$ over all the possible configurations of $n$ points, i.e. $$f_k(n)=min_{X \subset \R^d, |X|=n}\{f_k([X])\}.$$ With this notation, Problems $\ref{P:affine}$ and $\ref{P:affine_v}$ and can be interpreted as the task of finding the value of $f_{d-1}(n)$ and $f_{0}(n)$ respectively. Both problems are natural generalizations of the well-known McMullen's problem: What is the maximum $n$ such that any set of $n$ points in general position, $X \subset \R^{d},$ can de mapped by a permisible projective transformation onto the vertices of a convex polytope? These two problems are closely related. Moreover, Problem $\ref{P:affine}$ has a direct application to the problem of counting the number of Radon partitions induced by a colouring.

A graph $X$ is {\em symmetric} if its automorphism group acts transitively on the arcs of $X$, and {\em $s$-arc-transitive} if its automorphism group acts transitively on the set of $s$-arcs of $X$. Furthermore, if the latter action is sharply-transitive on $s$-arcs, then $X$ is {\em $s$-arc-regular.} It was shown by Tutte (1947, 1959) that every finite symmetric cubic graph is $s$-arc-regular for some $s\leq 5$. Djokovi\v c and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of $X$. In this talk we consider the question of when a finite symmetric cubic graph can be a Cayley graph. We show that in five of the $17$ Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we use covers (and the `Macbeath trick') to show that in each of those last ten classes, there are infinitely many Cayley graphs, and infinitely many non-Cayley graphs. This research grew out of some recent discussions with Klavdija Kutnar and Dragan Maru{\v s}i{\v c}.

In 1997 Graver and Watkins partitioned edge-transitive maps into 14 classes, distinguished by the quotient of each map by its full automorphism group. These classes include those of regular, chiral and just-edge-transitive maps. I shall describe the possible topological and combinatorial properties of the maps in these classes, including maps with boundary, together with the possibilities for their automorphism groups. In the case of finite simple groups, the latter builds on results of Nuzhin and others relevant to regular maps (with a small but important correction), and more recently of Leemans and Liebeck for chiral maps.

In a series of three papers, Monson and Schulte showed that, subject to some mild constraints, certain families of real reflection groups can be reduced modulo odd primes to yield finite string C-subgroups of orthogonal groups. Further, polytopes of any desired rank can be constructed this way. In this talk I will show that the latter is also true for orthogonal groups defined over any non-prime field of characteristic 2. Of course, the modular reduction method cannot be used for such groups, so the polytopes are constructed from scratch using analogues of reflections. This is a report on recent joint work with J.T. Ferrara and Dimitri Leemans.

In terms of their symmetries, not many geometric objects have received more attention than polytopes. Since ancient times, polytopes have been considered with great interest, not only for aesthetic reasons, but also for scientific reasons. More recently, the theory of abstract polytopes, which generalizes the concept of classical polytopes, has emerged as a powerful tool for studying symmetries. In the last decade, efforts to classify highly symmetric polytopes, including chiral and regular ones, have highly increased. Given a group Γ, a lot has been done in order to enumerate all polytopes of a certain type and which have Γ as their group of symmetries. Many results have been obtained for almost simple groups and atlases for small groups have been drawn up. However, not much is known about chiral polytopes, which are polytopes that have all possible rotational symmetries but no reflection. In this talk, we will define the basic notions necessary to understand the presentation. Then we will go through some techniques to prove the existence of chiral polytopes whose automorphism group is isomorphic to an almost simple group with socle PSL(2,q).

An abstract polytope is flat if every facet is incident on every vertex. In this talk, we prove that no chiral polytope has flat finite regular facets and finite regular vertex- figures. We then determine the three smallest non-flat regular polytopes in each rank, and use this to show that for $n \geq 8$, a chiral $n$-polytope has at least $48(n-2)(n-2)!$ flags.

We describe the smallest $C$-groups with complete diagram whose rank 3 residues are hypermaps of type $(n,n,n)$. It turns out that these $C$-groups are not hypertopes, indeed all rank 3 residues fail to be thin. We then focus on rank 3 regular hypertopes. A characterization of thiness will be given and some infinite families of ``small'' rank 3 regular hypertopes of type $(n,n,n)$ will arise. This is a joint work with Michael Giudici.

Generalized polygons are Bruhat-Tits buildings of rank two. They can also be defined in terms of their bipartite incidence graph, which has the property that the girth is twice the diameter. By the Feit-Higman Theorem (1964), the only finite generalized polygons are thin (two points on each line or two lines on each point) or the diameter is 3, 4, 6 or 8, corresponding to the finite projective planes, the generalized quadrangles, the generalized hexagons and the generalized octagons, respectively. In particular there are no generalized pentagons. An alternative way to generalize the pentagon was introduced by Simeon Ball et al. in [1]. In this talk I will discuss what we know about these incidence geometries. [1] S. Ball, J. Bamberg, A. Devillers and K. Stokes. An alternative way to generalize the pentagon. J. Combin. Des., 21:163–179, 2013. [2] T. S. Griggs and K. Stokes. On pentagonal geometries with block size 3, 4 or 5. Springer Proc. in Math. & Stat., 159:147–157, 2016.

Let G be a finite group and A a finite set. A cellular automaton is a transformation of the configuration space A^G that commutes with the natural action of G. We shall present some algebraic results on the monoid of cellular automata over A^G, and discuss some of the difficulties of extending these to a more general setting.

It is well known that if $G$ is an abelian group and $D=(G, Dev(c))$ is a $(v,k,\lambda)$-symmetric design for some k-subset $c$ of G, then $c$ is a $(v,k,\lambda)$-difference set in $G$. In this talk we will give a similar result for $(p^2, k, 2)$-biplanes with a primitive, flag-transitive automorphism group of affine type that characterizes them as some difference sets in $GF(p^2)$.

Given a combinatorial design $D$ and its incidence graph $I_D$, the embedding of $D$ on a surface $S$ is defined by a transformation on the embedding of the graph $I_D$ on $S$. In this talk will be introduce this concepts to describe the embedding of the possible bipalnes known so far.

We describe how the symmetries in configuration spaces for mass partition problems are the key to solving such problems. In particular, we will focus on the following problem: Given positive integers r and d, and d smooth finite measures in R^d, there is a partition of R^d into r convex sets that simultaneously splits each measure evenly.