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Many physically important mechanical systems may be described with a Lie group $G$ as configuration space. According to the well-known Noether's theorem, underlying symmetries of the Lie group may be used to considerably reduce the…

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion…

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show…

Grassmann angles improve upon similar concepts of angle between subspaces that measure volume contraction in orthogonal projections, working for real or complex subspaces, and being more efficient when dimensions are different. Their…

In this work we compute the triangulated Grothendieck groups for each of the family of discrete cluster categories of Dynkin type $A_{\infty}$ as introduced by Holm-Jorgensen. Subsequently, we also compute the Grothendieck group of a…

A surprising 'converse to the polynomial method' of Aaronson et al. (CCC'16) shows that any bounded quadratic polynomial can be computed exactly in expectation by a 1-query algorithm up to a universal multiplicative factor related to the…

In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the…

Local symmetries of a non-expanding horizon has been investigated in the 1st order formulation of gravity. When applied to a spherically symmetric isolated horizon only a U(1) subgroup of the Lorentz group survives as residual local…

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group.…

We construct a gerbe over a complex reductive Lie group G attached to an invariant bilinear form on a maximal diagonalizable subalgebra which is Weyl group invariant and satisfies a parity condition. By restriction to a maximal compact…

In this paper we look at Grothendieck's work on classifying holomorphic bundles over the complex projective line. The paper is divided into $4$ parts. The first and second part we build up the necessary background to talk about vector…

We deal with the finite-dimensional mesh algebras given by stable translation quivers. These algebras are self-injective, and thus the stable categories have a structure of triangulated categories. Our main result determines the…

We construct invariants of birational maps with values in the Kontsevich--Tschinkel group and in the truncated Grothendieck groups of varieties. These invariants are morphisms of groupoids and are well-suited to investigating the structure…

This paper attempts to relate some ideas of Grothendieck in his Esquisse d'un programme and some of the recent results on 2-dimensional topology and geometry. Especially, we shall discuss the Teichm\"uller theory, the mapping class groups,…

Classification of finite dimensional representations of the q-deformed Heisenberg algebra $H_q(3)$ is made by the help of Clifford algebra of polynomials and generalized Grassmann algebra. Special attention is paid when $q$ is a primitive…

We investigate the complement of the discriminant in the projective space PSym^d C^{n+1} of polynomials defining hypersurfaces of degree d in P^n. Following the ideas of Zariski we are able to give a presentation for the fundamental group…

The symmetric group S_n acts on the skew polynomial ring A=k_{-1}[x_1,..., x_n] as permutations. For subgroups G of S_n we consider properties of the ring of invariants A^G. We show that A^G is always Artin-Schelter Gorenstein, and we…

We determine the Grothendieck ring of finite-dimensional comodules for the free Hopf algebra on a matrix coalgebra, and similarly for the free Hopf algebra with bijective antipode and other related universal quantum groups. The results turn…

Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…

To each finitely generated group $G$, we associate a quasi-isometric invariant called the \emph{Dehn spectrum} of $G$. If $G$ is finitely presented, our invariant is closely related to the Dehn function of $G$, but provides more information…