(This is joint work with J. Delgado and M. Roy) For two subgroups of a group, $H_1, H_2\leq G$, we can look at the eight possibilities for the finite/non-finite generability of $H_1$, $H_2$, and $H_1\cap H_2$. For example, all eight are possible in a free non-abelian group except one of them, expressing the well-known fact that free groups are Howson: intersection of two finitely generated subgroups is again finitely generated.

The classification of simple modules for a simple Lie algebra $\mathfrak{g}$ seems beyond reach: it is complete only for $\mathfrak{sl}(2)$. However, some classes of simple $\mathfrak{g}$-modules are well understood, such the category of weight modules with finite dimensional weight spaces. Irreducible weight representations were classified due to the effort of S. Fernando and O. Mathieu.

We introduce the Jordan-strict topology on the multipliers algebra of a JB*-algebra. In case that a C*-algebra $A$ is regarded as a JB*-algebra, the J-strict topology of $M(A)$ is precisely the well-studied C*-strict topology. We prove that every JB*-algebra A is J-strict dense in its multipliers algebra $M(A)$, and that the latter algebra is J-strict complete.

In 2004, Fit-Florea and Matula presented an algorithm for computing the discrete logarithm modulo  $2^{k}$ with logarithmic base 3. The algorithm is suitable for hardware support of applications where fast arithmetic computation is desirable.

A transposed Poisson algebra  is a triple $(\mathcal{L},\cdot,[\cdot,\cdot])$ consisting of a vector space $\mathcal{L}$ with two bilinear operations $\cdot$ and $[\cdot,\cdot]$, such that

1. $(\mathcal{L},\cdot)$ is a commutative associative algebra;
2. $(\mathcal{L},[\cdot,\cdot])$ is a Lie algebra;
3. the "transposed" Leibniz law holds: $2z\cdot [x,y]=[z\cdot x,y]+[x,z\cdot y]$ for all 
$x,y,z\in \mathcal{L}$. 

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.