In this talk, an overview will be presented about hom-algebra structures, with focus on foundations and recent advances on graded (color) quasi Lie algebras, quasi-hom Lie algebras, hom-Lie algebras and related hom-algebra structures. These interesting algebraic structures appear for example when discretizing the differential calculus as well as in constructions of differential calculus on non-commutative spaces.

Hypercubes are arguably one of the most fundamental and most studied objects in Combinatorics. They have many nice properties which are very useful in several applications in Computer Science, one these properties being Hamiltonian - a graph is said to be Hamiltonian if it contains a cycle which goes through every vertex exactly once.

In dimension 2, a domino is a $2\times 1$ rectangle. Domino tilings of quadriculated regions have been extensively studied, with several deep and famous results.

The corresponding problems in dimension 3 (or higher) appear to be almost without exception much harder. In dimension 2, it is known, for instance, that for any quadriculated disk any two tilings can be joined by a finite sequence of flips: a flip consists in lifting two adjacent dominos and placing them back after a quarter turn rotation.

We give an introduction on the Brauer-Manin obstruction to the Hasse principle and present some recent results on zero-cycles on certain varieties.

This talk will be about a project aiming to illustrate geometry through puzzles. The puzzles are played on surfaces, and have natural configuration graphs with a geometry of their own. These graphs are reminiscent of combinatorial graphs used in the study of moduli spaces of surfaces which can be visualised in similar ways.

The puzzles were created together with Paul Turner, and brought to life together with Mario Gutierrez and Reyna Juarez.

Let $k$ be an algebraicallly closed field of characteristic $p\geq 0$ and let $G$ be a linear algebraic group of rank $\ell\geq 1$ over $k$. Let $V$ be a rational finite-dimensional $kG$-module and let $V_g(\mu)$ denote the eigenspace corresponding to the eigenvalue $\mu\in k^*$ of $g \in G$ on $V$. We set $\nu_G(V)=\min\{\dim(V)-\dim(V_g(\mu))| g \in G \setminus Z(G), \mu \in k^*\}$.

Let $G$ be a subgroup of $S_n$. What can be said on the number of conjugacy classes of $G$, in terms of $n$?

I will review many results from the literature and give examples. I will then present an upper bound for the case where $G$ is primitive with nonabelian socle. This states that either $G$ belongs to explicit families of examples, or the number of conjugacy classes is smaller than $n/2$, and in fact, it is $o(n)$. I will finish with a few questions. Joint work with Nick Gill. 

A celebrated theorem of B. H. Neumann states that if $G$ is a group in which all conjugacy classes are finite with bounded size, then the derived group $G'$  is finite.  In this talk we will discuss a stronger version of  Neumann's result and some consequences for finite and profinite groups.  Based on a joint work with Pavel Shumyatsky.

This talk concerns numerical semigroups, i.e. cofinite submonoids $S$ of $\mathbb{N}$. Wilf's conjecture (1987) on numerical semigroups $S$ states that $e \cdot \ell \ge c$, where $e$ is the embedding dimension of $S$ and $\ell$ is the number of elements of $S$ which are smaller than its conductor $c$. We shall present the main ideas of a recently published proof of Wilf's conjecture in the particular case $e \ge m/3$, where $m$ is the smallest nonzero element of $S$. Asymptotically, most numerical semigroups seem to satisfy $e \ge m/3$ as the genus goes to infinity.

Using graphs as a tool to encode properties of groups is a well established approach to many problems nowadays.