Related papers: Stack structures on GIT quotients parametrizing hy…

We construct new moduli spaces of quiver representations with multiplicities, i.e. over rings of truncated power series. This includes moduli of framed representations and analogues of Nakajima quiver varieties. Our construction relies on…

We define syzygy points of projective schemes, and introduce a program of studying their GIT stability. Then we describe two cases where we have managed to make some progress in this program, that of polarized K3 surfaces of odd genus, and…

An invariant I of quasiprojective K-varieties X with values in a commutative ring R is "motivic" if I(X)= I(Y)+I(X\Y) for Y closed in X, and I(X x Y)=I(X)I(Y). Examples include Euler characteristics chi and virtual Poincare and Hodge…

Consider a finite-dimensional real vector space equipped with a finite group acting unitarily on it. We address the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our approach is based on…

Let G be a compact Lie group. We build a tower of G-spectra over the suspension spectrum of the space of linear isometries from one G-representation to another. The stable cofibres of the maps running down the tower are certain interesting…

Whenever graphs admit equitable partitions, their quotient graphs highlight the structure evidenced by the partition. It is therefore very natural to ask what can be said about two graphs that have the same quotient according to certain…

In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We…

We consider generalized gradients in the general context of $G$-structures. They are natural first order differential operators acting on sections of vector bundles associated to irreducible $G$-representations. We study their geometric…

A noncommutative-geometric formalism of framed principal bundles is sketched, in a special case of quantum bundles (over quantum spaces) possessing classical structure groups. Quantum counterparts of torsion operators and Levi-Civita type…

We present a proof of Thue-Siegel-Roth's Theorem (and its more recent variants, such as those of Lang for number fields and that "with moving targets" of Vojta) as an application of Geometric Invariant Theory (GIT). Roth's Theorem is…

We define numerical link-homotopy invariants of link maps of any number of components, which naturally generalize the Kirk invariant. The Kirk invariant is a link-homotopy invariant of 2-component link maps given by linking numbers of loops…

We present a stack model for breaking down the complexity of entanglement-based quantum networks. More specifically, we focus on the structures and architectures of quantum networks and not on concrete physical implementations of network…

We study a global invariant for path structures. The invariant is obtained as a secondary invariant from a Cartan connection on a canonical bundle associated to a path structure. It is computed in examples which are defined in terms of…

The transparent way for the invariant (Hamiltonian) description of equivariant localization of the integrals over phase space is proposed. It uses the odd symplectic structure, constructed over tangent bundle of the phase space and permits…

This note proposes a new notion of a gradient-like vector field and discusses its implications for the theory of Stein and Weinstein structures.

We introduce a spherical variant of Milnor's classifying construction for diffeological groups, based on quadratic normalization of barycentric coordinates. This construction gives rise to a contractible diffeological space endowed with…

Let $X/k$ be a noetherian scheme over a field $k$ of characteristic 0, such that the residue field at its closed points are algebraic extensions of $k$. Let ${\mathfrak g}_{X/k}\subset T_{{X/k}}$ be an ${\mathcal O}_{X}$-submodule of the…

The problem of bi-equivariant extension of continuous maps of binary $G$-spaces is considered. The concept of a structural map of distributive binary $G$-spaces is introduced, and a theorem on the bi-equivariant extension of structural maps…

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\G$ be a connected semisimple group scheme over $X$. Under certain hypothesis we prove the equality of two numbers associated…

We examine which of the compact connected Lie groups that act transitively on spheres of different dimensions leave the unique spin structure of the sphere invariant. We study the notion of invariance of a spin structure and prove this…