I will describe how a sieve method can be used to establish the Hasse principle for the variety
$$f(t)=N(x_1,\ldots,x_k),$$
where $f$ is an irreducible cubic and $N$ is a norm form for a number field satisfying certain hypotheses.
Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties.
This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.
I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez:
For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.
Let F be a field. A symplectic alternating algebra over F
consists of a symplectic vector space V over F with a non-degenerate
alternating form that is also equipped with a binary alternating
product · such that the law (u·v, w)=(v·w, u) holds. These algebraic
structures have arisen from the study of 2-Engel groups but seem also
to be of interest in their own right with many beautiful properties.
We will give an overview with a focus on some recent work on the
structure of nilpotent symplectic alternating algebras.
By using the Weil-Gel'fand-Zak transform of Faddeev's quantum dilogarithm,
Andersen and Kasheav have proposed a new state-integral model for the
Andersen--Kashaev TQFT, where the circle valued state variables live on
the edges of oriented levelled shaped triangulations. I will look at a
couple of examples which give an idea of how the theories are coupled.