Related papers: Configuration spaces of tori

The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular the following results are proved: a) If L is an algebraic subgroup of a connected real algebraic group G such that…

The automorphism groups ${\rm Aut\,}A_n$ and ${\rm Aut\,}W_n$ of the polynomial algebra $A_n=C[x_1,x_2,\cdots, x_n]$ and the rank $n$ Witt algebra $W_n={\rm Der\,}A_n$ are studied in this paper. It is well-known that ${\rm Aut\,}A_n$ for…

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of…

For various values of n, d, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. In the…

The total homology of the loop space of the configuration space of ordered distinct n points in R^m has a structure of a Hopf algebra defined by the 4-term relations if m>2. We describe a relation of between the cohomology of this loop…

The mapping torus of an endomorphism \Phi of a group G is the HNN-extension G*_G with bonding maps the identity and \Phi. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated…

We classify spin structures on the noncommutative torus, and find that the noncommutative n-torus has 2^n spin structures, corresponding to isospectral deformations of spin structures on the commutative n-torus. For n>3 the classification…

We establish faithful tropicalisation for point configurations on algebraic tori. Building on ideas from enumerative geometry, we introduce tropical scaffolds and use them to construct a system of modular fan structures on the tropical…

A covariant functor from the category of mapping tori to a category of AF-algebras is constructed; the functor takes continuous maps between such manifolds to stable homomorphisms between the corresponding AF-algebras. We use this functor…

We give a complete classification to when a finite group of outer automorphisms preserves a bi-order on a non-abelian free group and bi-orderable surface groups. We also give another new criterion for an outer automorphism of $F_n$ induced…

In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that…

Let $D$ be a bounded logarithmically convex complete Reinhardt domain in $\mathbb{C}^n$ centered at the origin. Generalizing a result for the one-dimensional case of the unit disk, we prove that the $C^*$-algebra generated by Toeplitz…

We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with…

The soft tori constitute a continuous deformation, in a very precise sense, from the commutative C*-algebra C(T^2) to the highly non-commutative C*-algebra C*(F_2). Since both of these C*-algebras are known to have a separating family of…

We show that for $n\geq 2$ there exists an exact Lagrangian submanifold $L$ in the cotangent bundle $T^*\mathbb{T}^n$ of the $n$-dimensional torus $\mathbb{T}^n$ such that $L$ is symplectically but not Hamiltonian isotopic to the zero…

A toric manifold is a compact non-singular toric variety equipped with a natural half-dimensional compact torus action. A torus manifold is an oriented, closed, smooth manifold of dimension $2n$ with an effective action of a compact torus…

Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm…

We announce the following result and give several applications: A Hamiltonian $T$-space (for $T$ a torus) with isolated fixed points is cobordant to a disjoint union of weighted projective spaces which are constructed from its fixed point…

An irreducible algebraic variety $X$ is rigid if it admits no nontrivial action of the additive group of the ground field. We prove that the automorphism group $\text{Aut}(X)$ of a rigid affine variety contains a unique maximal torus…

Given a subvariety $V$ of the complex algebraic torus ${\mathbb G}_{\rm m}^n$ defined by polynomials of total degree at most $d$ and a power map $\phi: {\mathbb G}_{\rm m}^n \to {\mathbb G}_{\rm m}^n$, the points ${\bf x}$ whose forward…