Achievable ranks of intersections of finitely generated free groups

We answer a question due to A. Myasnikov by proving that all expected ranks occur as the ranks of intersections of finitely generated subgroups of free groups.Comment: 4 pages, 4 figure

Quantum Pin Codes

arXiv: 1906.11394We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a vast generalization of quantum color codes and Reed-Muller codes. A lot of the structure and properties of color codes carries over to pin codes. Pin codes have gauge operators, an unfolding procedure and their stabilizers form multi-orthogonal spaces. This last feature makes them interesting for devising magic-state distillation protocols. We study examples of these codes and their properties

Vector bundles and torsion free sheaves on degenerations of elliptic curves

In this paper we give a survey about the classification of vector bundles and torsion free sheaves on degenerations of elliptic curves. Coherent sheaves on singular curves of arithmetic genus one can be studied using the technique of matrix problems or via Fourier-Mukai transforms, both methods are discussed here. Moreover, we include new proofs of some classical results about vector bundles on elliptic curves.Comment: 39 pages, 5 figure

Abelian Gauge Fluxes and Local Models in F-Theory

We analyze the Abelian gauge fluxes in local F-theory models with G_S=SU(6) and SO(10). For the case of G_S=SO(10), there is a no-go theorem which states that for an exotic-free spectrum, there are no solutions for U(1)^2 gauge fluxes. We explicitly construct the U(1)^2 gauge fluxes with an exotic-free bulk spectrum for the case of G_S=SU(6). We also analyze the conditions for the curves supporting the given field content and discuss non-minimal spectra of the MSSM with doublet-triplet splitting.Comment: 43 pages, 15 tables; typos corrected, reference adde

Some Homological Properties of Lattice Ideals

The interplay of algebra and combinatorics is fruitful in both fields: combinatorics provides algebraic structures with tractable realizations, while algebra underpins combinatorial objects with a rigorous framework. Pioneered by Hochster and Stanley, interest in combinatorial commutative algebra has grown rapidly, often including techniques from simplicial topology and convex geometry. This thesis presents two main results that combine commutative algebra and combinatorics. The first result considers the Cohen–Macaulayness of a lattice ideal and its associated toric ideal. Despite the deep algebraic connection between these two ideals, we produce infinitely many examples, in every codimension, of pairs where one of these ideals is Cohen–Macaulay but the other is not. The second result describes the free resolution of the ground field over the quotient ring by a specific type of lattice ideal, that defining a rational normal 2-scroll. This chapter also includes a computation of the Betti numbers of the ground field when resolved over the ring coming from an arbitrary rational normal k-scroll

Quantum Pin Codes

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called $\ell$-orthogonal spaces meaning that the joint overlap between any $\ell$ stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties