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The affinely regular polygons in certain planar sets are characterized. It is also shown that the obtained results apply to cyclotomic model sets and, additionally, have consequences in the discrete tomography of these sets.

For $E/F$ quadratic extension of local fields and $G$ a reductive algebraic group over $F$, the paper formulates a conjecture classifying irreducible admissible representations of $G(E)$ which carry a $G(F)$ invariant linear form, and the…

We study the problem of classifying the lines of the projective $3$-space $PG(3,q)$ over a finite field $GF(q)$ into orbits of the group $G=PGL(2,q)$ of linear symmetries of the twisted cubic $C$. A generic line neither intersects $C$ nor…

In this paper, we study the isomorphism problem for linear representations. A linear representation Tn*(K) of a point set K is a point-line geometry, embedded in a projective space PG(n+1,q), where K is contained in a hyperplane. We put…

Extended objects (defects) in Quantum Field Theory exhibit rich, nontrivial dynamics describing a variety of physical phenomena. These systems often involve strong coupling at long distances, where the bulk and defects interact, making…

By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with…

Let $L$ be the function field of a projective space ${\mathbb P}^n_k$ over an algebraically closed field $k$ of characteristic zero, and $H$ be the group of projective transformations. An $H$-sheaf ${\mathcal V}$ on ${\mathbb P}^n_k$ is a…

A linear system is a pair $(P,\mathcal{L})$ where $\mathcal{L}$ is a family of subsets on a ground finite set $P$ such that $|l\cap l^\prime|\leq 1$, for every $l,l^\prime \in \mathcal{L}$. The elements of $P$ and $\mathcal{L}$ are called…

If $E/F$ is a quadratic extension $p$-adic fields, we first prove that the $\mathrm{SL}_n(F)$-distinguished representations inside a distinguished unitary L-packet of $\mathrm{SL}_n(E)$ are precisely those admitting a degenerate Whittaker…

We define and study the set ${\mathcal E}(\rho)$ of end invariants of a $\SL(2,C)$ character $\rho$ of the one-holed torus $T$. We show that the set ${\mathcal E}(\rho)$ is the entire projective lamination space $\mathscr{PL}$ of $T$ if and…

Structures in low-dimensional topology and low-dimensional geometry -- often combined with ideas from (quantum) field theory -- can explain and inspire concepts in algebra and in representation theory and their categorified versions. We…

Quantum spin liquids can be faithfully represented and efficiently characterized within the framework of Projected Entangled Pair States (PEPS). Guided by extensive exact diagonalization and density matrix renormalization group…

We construct, for any finite commutative ring $R$, a family of representations of the general linear group $\mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $\mathrm{GL}_n$ over a finite field.

This paper studies pattern formations in coupled elliptic PDE systems governed by transparent boundary conditions. Such systems unify diverse areas, including inverse boundary problems (via a single passive/active boundary measurement),…

We investigate the topological and metric structure of the set of idempotent operators and projections which have prescribed diagonal entries with respect to a fixed orthonormal basis of a Hilbert space. As an application, we settle some…

Let K be a field of characteristic zero. We prove that images of a linear K-derivation and a linear K-E-derivation of the ring K[x 1 ,x 2 ,x 3 ] of polynomial in three variables over K are Mathieu-Zhao subspaces, which affirms the LFED…

Synthetic algebraic geometry is a new approach to algebraic geometry. It consists in using homotopy type theory extended with three axioms, together with the interpretation of these in a higher version of the Zariski topos, in order to do…

The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting)…

We study the homotopy type of the space $E(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Our main result is a simple description of the fundamental group, or motion group, of $E(L)$, and…

We construct a version of rational Symplectic Field Theory for pairs $(X,L)$, where $X$ is an exact symplectic manifold, where $L\subset X$ is an exact Lagrangian submanifold with components subdivided into $k$ subsets, and where both $X$…