Related papers: Five not-so-easy pieces:open problems about vertex…
At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…
We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory.
In this piece, we examine one variant of the infamous 15 Tile Puzzle and develop a mathematical backing behind why it is unsolvable. Using concepts of permutations, bijectivity, and cycle transpositions, we not only prove how to model this…
These are notes of a series of lectures on mirror symmetry and topological string theory given at the Mathematical Sciences Center at Tsinghua University. The N=2 superconformal algebra, its deformations and its chiral ring are reviewed. A…
We present a brief overview of Moonshine with an emphasis on connections to physics. Moonshine collectively refers to a set of phenomena connecting group theory, analytic number theory, and vertex operator algebras or conformal field…
This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and…
We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. The central problem is whether and how certain rings are (additively) generated by their units.…
We present some open problems and describe briefly some possible research directions in the emerging theory of Hardy spaces of Dirichlet series and their intimate counterparts, Hardy spaces on the infinite-dimensional torus. Links to number…
The article is devoted to microbundles over topological rings. Their structure, homomorphisms, automorphisms and extensions are studied. Moreover, compactifications and inverse spectra of microbundles over topological rings are…
PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with some…
This Thesis is devoted to the analysis of entanglement in relevant physical systems. Entanglement is the conducting theme of this research, though I do not dedicate to a single topic, but consider a wide scope of physical situations. I have…
Similarly to how the classical group ring isomorphism problem asks, for a commutative ring $R$, which information about a finite group $G$ is encoded in the group ring $RG$, the twisted group ring isomorphism problem asks which information…
We construct a representation of the blob algebra over a ring allowing base change to every interesting (i.e. non--semisimple) specialisation which, in quasihereditary specialisations, passes to a full tilting module.
We discuss several open problems in classical Knot Theory and we develop techniques that allow us to study them: Lagrangian tangles, skein modules and Burnside groups.
We propose and study a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between…
In this article we give a survey on open problems and conjectures concerning L^2-invariants. We cover the whole portfolio and not only certain aspects as they are considered in the previous more specialized (and within their scope more…
We discuss topological defect lines in holomorphic vertex operators algebras and superalgebras, in particular Frenkel-Lepowsky-Meurman Monster VOA $V^\natural$ with central charge $c=24$, and Conway module SVOA $V^{f\natural}$ with $c=12$.…
The "Ring Learning with Errors" (RLWE) problem was formulated as a variant of the "Learning with Errors" (LWE) problem, with the purpose of taking advantage of an additional algebraic structure in the underlying considered lattices; this…
This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some…
Various topics concerning the entanglement of composite quantum systems are considered with particular emphasis concerning the strict relations of such a problem with the one of attributing objective properties to the constituents. Most of…