The dichromatic number and the diachromatic number, generalice the concepts of chromatic number and achromatic number. An {\bf{acyclic}} $k$-vertex-coloring of a digraph $D$ is vertex coloring using $k$ colors such that $D$ has no monochromatic cycles and a \emph{complete} $k$-vertex-coloring of a digraph $D$ is a vertex coloring using $k$ colors such that for every ordered pair $(i,j)$ of different colors, there is at least one arc $(u,v)$ such that $u$ has color $i$ and $v$ has color $j$. The dichromatic number $dc(D)$ and diachromatic number $dac(D)$ of $D$ are the smallest and the largest number of colors in a complete proper $k$-vertex-coloring of $D$, therefore $dc(D)\leq dac(D)$. \

We determine the achromatic and diachromatic numbers of some specific circulant graphs and digraphs and give general bounds for these two parameters on these graphs and digraphs. Also, we determine the achromatic index for circulant graphs of order $q^2+q+1$ using projective planes.

Cauchy, Dehn, and Alexandrov proved that arbitrary convex triangulated spheres are isostatic, and therefore any realization of these 3-connected planar graphs $G=(V,E)$, with $|E|=3|V|-6$, at generic positions of the vertices are also isostatic. Finbow and Whiteley provided another class, the class of 3-dimensional isostatic frameworks, the block and hole polyhedra, along with methods to verify generic rigidity that can be extended to other classes. These methods are based on tracking when a larger framework can be derived from a known small example using vertex splitting, an operation known to take a minimally generically rigid framework to a new minimally generically rigid framework with one more vertex.

There is another technique for adding a vertex to a rigid structure that preserves isostatic rigidity; spider splitting. In this talk we will present some preliminary results regarding the relationship between vertex-splitting and spider-splitting.

In the talk I plan to briefly survey some known results on the Voronoi conjecture and outline the proof of the conjecture in $\mathbb{R}^5$.

Based on a joint work with Alexander Magazinov.

Unconditionally symmetric convex body in $\mathbb{E}^d$ is a body that together with every its point with coordinates $(x_1, \dots, x_d)$ also contains all the points with coordinates $(\pm x_1, \dots, \pm x_d)$. In my talk I will show that 8 illumination directions are enough to completely illuminate the boundary of an unconditionally symmetric cap body of a ball in $\mathbb{E}^4$.

This work was generously supported at Universidad Nacional Aut\'{o}noma de M\'{e}xico (Morelia) by PAPIIT-UNAM grant \#IN100518 .

Unconditionally symmetric convex body in $\mathbb{E}^d$ is a body that together with every one of its point with coordinates $(x_1, \dots, x_d)$ also contains all the points with coordinates $(\pm x_1, \dots, \pm x_d)$. In my talk, I will show that 6 illumination directions are enough to completely illuminate the boundary of an unconditionally symmetric cap body of a ball in $\mathbb{E}^3$.

We will discuss connections with Asymptotic Convex Geometry and measure concentration, as well as with entropic methods, that allow for at least a subexponential improvement to the long-standing general upper bounds of Rogers for $N(n)$.

This is joint work with Han Huang, Boaz Slomka and Tomasz Tkocz.