When is a ring of torus invariants a polynomial ring?

Let ρ:T→GL(V) be a finite dimensional rational representation of a torus over an algebraically closed fieldk. We give necessary and sufficient conditions on the arrangement of the weights ofV within the character lattice ofT for the ring of invariants,k[V] ^T , to have a homogeneous system of parameters consisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions fork[V] ^T to be a polynomial ring (Theorem 5.8 and Theorem 5.10). Research supported in part by NSERC Grant OGP 137522     

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

Vassiliev invariants of type one for links in the solid torus

In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal type one invariant, and we show that the generalized Aicardi's invariant restricted to n-component links in the solid torus with zero winding number for each component is equal to an invariant we define using the universal cover of the solid torus. We also define and study a geometric invariant for n-component links in the solid torus. We give a lower bound on this invariant using the type one invariants, which are easy to calculate, which helps in computing this geometric invariant, which is usually hard to calculate.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type

The problem of getting effective Fredholm conditions for operators with bihomogeneous kernels reduces to the question of invertibility for families of operators with homogeneous kernels and to the calculation of homotopy invariants for spaces of Fredholm and invertible operators of that type. The purpose of the present paper is to study integral operators with homogeneous kernels of compact type in L _p (? ⁿ ), 1 < p < +??. The classes of homotopy equivalence for the spaces of Fredholm and invertible operators in the C*-algebra of pair operators with homogeneous kernels of compact type are calculated by means of operator K-theory.     

Heegaard–Floer homologies of (+1) surgeries on torus knots

We compute the Heegaard–Floer homology of $S^{3}_{1}(K)$ (the (+1) surgery on the torus knot T _p,q ) in terms of the semigroup generated by p and q, and we find a compact formula (involving Dedekind sums) for the corresponding Ozsváth–Szabó d-invariant. We relate the result to known knot invariants of T _p,q as the genus and the Levine–Tristram signatures. Furthermore, we emphasize the striking resemblance between Heegaard–Floer homologies of (+1) and (?1) surgeries on torus knots. This relation is best seen at the level of τ functions.     

Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

Frobenius splitting and invariant theory

We consider varieties over an algebraically closed field k of characteristicp>0. Given a linear representation of a reductive group, we prove that the ring of invariants is F-regular provided the associated projective quotient is Frobenius-split, the twisting sheaves are Cohen-Macaulay (C-M), and a mild technical condition is met. As an example of how this can be used, we show that the ring of invariants (under the adjoint action of SL (3)) ofg copies ofM ₃ is C-M. (HereM ₃ denotes the vector space of 3×3 matrices over k andp>3.) The method of proof involves an induction, and is potentially of wide applicability. As a corollary we obtain that the moduli space of rank 3 and degree 0 bundles on a smooth projective curve of genusg is C-M.     

Monomial modules and graded Betti numbers

Let K be a field, S = K[x ₁,…, x _n ], the polynomial ring over K, and let F be a finitely generated graded free S-module with homogeneous basis. Certain invariants, such as the Castelnuovo-Mumford regularity and the graded Betti numbers of submodules of F, are studied; in particular, the componentwise linear submodules of F are characterized in terms of their graded Betti numbers.     

Dual topology of the Heisenberg motion groups

Let ? _n be the (2n + 1)-dimensional Heisenberg group, and let T ⁿ be the n-dimensional torus acting on ? _n by automorphisms. In this paper, we describe the space of admissible coadjoint orbits of the Heisenberg motion group G _n = T ⁿ ? ? _n and we determine the topology of this space. We show that the bijection between the unitary dual ? _n of G _n and its admissible coadjoint orbit space is a homeomorphism.     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

Protecting convex sets

A point-setS is protecting a collection F =T ₁,T ₂,..., _n ofn mutually disjoint compact sets if each one of the setsT _i is visible from at least one point inS; thus, for every setT _i ∈F there are points xS andy T _i such that the line segment joining x to y does not intersect any element inF other thanT _i . In this paper we prove that [2(n-2)/3] points are always sufficient and occasionally necessary to protect any family F ofn mutually disjoint compact convex sets. For an isothetic family F, consisting ofn mutually disjoint rectangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary to protect it. IfF is a family of triangles, [4n/7] points are always sufficient. To protect families ofn homothetic triangles, [n/2] points are always sufficient and [n/2] points are sometimes necessary.     

Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ=(λ₁,…,λk) with λ₁?n there is a Schubert variety X(λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ(λ) of X(λ). The main result of the present paper is an explicit formula for the restriction of the class σ(λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring .     

Jordan isomorphisms of upper triangular matrix rings

Let R be a 2-torsionfree ring with identity 1 and let T_n(R), n ? 2, be the ring of all upper triangular n × n matrices over R. We describe additive Jordan isomorphisms of T_n(R) onto an arbitrary ring and generalize several results on this line.     

Universal classes of nuclear Köthe spaces with a continuous norm

We study the existence of universal and quotient universal spaces in the class of nuclear Köthe spaces with a continuous norm. It is shown that no countable set of these spaces has all L_f(a, r) spaces, ?∞ < r ? 0, as subspaces or those with 0 < r ? ∞ as quotients. On the other hand, the quotients and subspaces of (s) with basis constitute a universal and a quotient universal class, respectively.     

Invariants and rings of quotients of <Emphasis Type="Italic">H</Emphasis>-semiprime <Emphasis Type="Italic">H</Emphasis>-module algebras satisfying a polynomial identity

We consider an action of a finite-dimensional Hopf algebra H on a PI-algebra. We prove that an H-semiprime H-module algebra A has a Frobenius artinian classical ring of quotients Q, provided that A has a finite set of H-prime ideals with zero intersection. The ring of quotients Q is an H-semisimple H-module algebra and a finitely generated module over the subalgebra of central invariants. Moreover, if algebra A is a projective module of constant rank over its center, then A is integral over its subalgebra of central invariants.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

On linear systems with quasiperiodic coefficients and bounded solutions

For a discrete dynamical system ω _n =ω₀+αn, where a is a constant vector with rationally independent coordinates, on thes-dimensional torus Ω we consider the setL of its linear unitary extensionsx _n+1=A(ω₀+αn)x _n , whereA (Ω) is a continuous function on the torus Ω with values in the space ofm-dimensional unitary matrices. It is proved that linear extensions whose solutions are not almost periodic form a set of the second category inL (representable as an intersection of countably many everywhere dense open subsets). A similar assertion is true for systems of linear differential equations with quasiperiodic skew-symmetric matrices.