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Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture,…
We introduce a notion of highly Kummer-faithful fields and study its relationship with the notion of Kummer-faithful fields. We also give some examples of highly Kummer-faithful fields. For example, if $k$ is a number field of finite degree…
For the cluster category of a hereditary or a canonical algebra, equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an…
We classify Artin-Schreier extensions of valued fields with non-trivial defect according to whether they are connected with purely inseparable extensions with non-trivial defect, or not. We use this classification to show that in positive…
In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature…
We explicitly present expansions of the complex field which are models of the theories of green points in the multiplicative group case and in the case of an elliptic curve without complex multiplication defined over $\mathbb{R}$. In fact,…
We introduce the notion of differential largeness for fields equipped with several commuting derivations (as an analogue to largeness of fields). We lay out the foundations of this new class of "tame" differential fields. We state several…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
Inspired by the theory of desirable gambles that is used to model uncertainty in the field of imprecise probabilities, I present a theory of desirable things. Its aim is to model a subject's beliefs about which things are desirable. What…
We use the theory of resultants of polynomials to study the stability of an arbitrary polynomial over a finite field, that is, the property of having all its iterates irreducible. This result partially generalises the quadratic polynomial…
In arXiv:1104.4441 it was shown that any 1-quasi-hereditary algebra affords a particular basis which is related to a given partial order on the set of simple modules. We show that the modules generated by these basis-elements are also…
We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the {\em total degree} and the…
The aim of this note is a classification of all nice and all inductively factored reflection arrangements. It turns out that apart from the supersolvable instances only the monomial groups $G(r,r,3)$ for $r \ge 3$ give rise to nice…
Let K be a number field and A an abelian variety over K. We are interested in the following conjecture of Morita: if the Mumford-Tate group of A does not contain unipotent Q-rational points then A has potentially good reduction at any…
Binary trees are fundamental objects in models of evolutionary biology and population genetics. Here, we discuss some of their combinatorial and structural properties as they depend on the tree class considered. Furthermore, the process by…
We try a conceptual analysis of inheritance diagrams, first in abstract terms, and then compare to "normality" and the "small/big sets" of preferential and related reasoning. The main ideas are about nodes as truth values and information…
We introduce several axioms which may or may not hold for any given subgraph of the directed graph of all organisms (past, present and future) where edges represent biological parenthood, with the simplifying background assumption that life…
Patterns on numerical semigroups are multivariate linear polynomials, and they are said to be admissible if there exists a numerical semigroup such that evaluated at any nonincreasing sequence of elements of the semigroup gives integers…
We prove special cases of a general conjecture: If an invertible field theory admits a projectively topological boundary theory, then it has finite order in the abelian group of invertible field theories. One can substitute `gapped' for…
The theory of Hubbard trees provides an effective classification of non-linear post-critically finite polynomial maps from \C to itself. This note will extend this classification to the case of maps from a finite union of copies of \C to…