Related papers: Open Problems in the Theory of Ample Fields
Differential central simple algebras are the main object of study in this survey article. We recall some crucial notions such as differential subfields, differential splitting fields, tensor products etc. Our main focus is on differential…
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including…
This is a survey article on the theory of lattice points in large planar domains and bodies of dimensions 3 and higher, with an emphasis on recent developments and new methods, including a lot of results established only during the last few…
Several open problems in algebraic logic are solved.
We give a survey on recent developments in the model theory of valued fields since the introduction of the notion of ``tame valued field'', and of the modifications and generalizations of this notion.
This review summarizes Effective Field Theory techniques, which are the modern theoretical tools for exploiting the existence of hierarchies of scale in a physical problem. The general theoretical framework is described, and explicitly…
In this article, we realize some groups as Galois groups over rational numbers and finite extension of rational numbers by studying right splitting of some exact sequences, Galois correspondence and algebraic operations on Galois…
Cosmology, high-energy physics and astrophysics are converging on the study of large-scale magnetic fields. While the experimental evidence for the existence of large-scale magnetization in galaxies, clusters and superclusters is rather…
The purpose of this note is to give a number of open problems on matching theory and their relation to the well-known results in this area. We also give a linear analogue of the acyclic matchings.
A systematic field theory is presented for charged systems. The one-loop level corresponds to the classical Debye-H\"uckel (DH) theory, and exhibits the full hierarchy of multi-body correlations determined by pair-distribution functions…
Enumeration of tilings is the mathematical study concerning the total number of coverings of regions by similar pieces without gaps or overlaps. Enumeration of tilings has become a vibrant subfield of combinatorics with connections and…
Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and the…
The study of the Lefschetz properties of Artinian graded algebras was motivated by the hard Lefschetz theorem for a smooth complex projective variety, a breakthrough in algebraic topology and geometry. Over the last few years, this topic…
We use class field theory to search for curves with many rational points over small finite fields. By going through abelian covers of curves of small genus we find a number of new curves. In particular, we settle the question of how many…
In this short note we present some remarks and conjectures on two of Erd\"os's open problems in number theory.
The wild McKay correspondence is a form of McKay correspondence in terms of stringy invariants that is generalized to arbitrary characteristics. It gives rise to an interesting connection between the geometry of wild quotient varieties and…
For each finite subgroup $G$ of $PGL_2(\mathbb{Q})$, and for each integer $n$ coprime to $6$, we construct explicitly infinitely many Galois extensions of $\mathbb{Q}$ with group $G$ and whose ideal class group has $n$-rank at least…
Deformation theory of complex manifolds is a classical subject with recent new advances in the noncompact case using both algebraic and analytic methods. In this note, we recall some concepts of the existing theory and introduce new notions…
We discuss some aspects of Extrapolation theory. The presentation includes many examples and open problems.
Hilbert's irreducibility theorem plays an important role in inverse Galois theory. In this article we introduce Hilbertian fields and present a clear detailed proof of Hilbert's irreducibility theorem in the context of these fields.