Complex projective structures with maximal number of Möbius transformations

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely? curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.     

The work of Gian-Carlo rota on invariant theory

The purpose of this paper is twofold: first, to explain Gian-Carlo Rotas work on invariant theory; second, to place this work in a broad historical and mathematical context. Rotas work falls under three specific cases: vector invariants, the invariants of binary forms, and the invariants of skew-symmetric tensors. We discuss each of these cases and show how determinants and straightening play central roles. In fact, determinants constitute all invariants in the vector case; for binary forms and skew-symmetric tensors, they constitute all invariants when invariants are represented symbolically. Consequently, we explain the symbolic method both for binary forms and for skew-symmetric tensors, where Rota developed generalizations of the usual notion of a determinant. We also discuss the Grassmann algebra, with its two operations of meet and join, which was a theme which ran through Rotas work on invariant theory almost from the very beginning.To the memory of Gian-Carlo Rota     

On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

The minimum rank problem: A counterexample

We provide a counterexample to a recent conjecture that the minimum rank over the reals of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank of the graph over the reals is strictly smaller than the minimum rank of the graph over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of R.     

On a class of projectively Ricci-flat Finsler metrics

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover, we construct some nontrivial examples on such Finsler metrics.     

On the distribution of points in projective space of bounded height

In this paper we consider the uniform distribution of points in compact metric spaces. We assume that there exists a probability measure on the Borel subsets of the space which is invariant under a suitable group of isometries. In this setting we prove the analogue of Weyl's criterion and the Erdös-Turán inequality by using orthogonal polynomials associated with the space and the measure. In particular, we discuss the special case of projective space over completions of number fields in some detail. An invariant measure in these projective spaces is introduced, and the explicit formulas for the orthogonal polynomials in this case are given. Finally, using the analogous Erdös-Turán inequality, we prove that the set of all projective points over the number field with bounded Arakelov height is uniformly distributed with respect to the invariant measure as the bound increases.

     

Degree bounds for syzygies of invariants

Suppose that G is a linearly reductive group. Good degree bounds for generators of invariant rings were given in (Proc. Amer. Math. Soc. 129 (4) (2001) 955). Here we study minimal free resolutions of invariant rings. For finite linearly reductive groups G it was recently shown in (Adv. Math. 156 (1) (2000) 23, Electron Res. Announc. Amer. Math. Soc. 7 (2001) 5, Adv. Math. 172 (2002) 151) that rings of invariants are generated in degree at most the group order |G|. In characteristic 0 this degree bound is a classical result by Emmy Noether (see Math. Ann. 77 (1916) 89). Given an invariant ring of a finite linearly reductive group G, we prove that the ideal of relations of a minimal set of generators is generated in degree at most ?2|G|.     

On holomorphic immersions into K(a)hler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f: X → M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f)measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form.In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.     

The nullity theorem for principal pivot transform

We generalize the nullity theorem of Gustafson (1984) [8] from matrix inversion to principal pivot transform. Several special cases of the obtained result are known in the literature, such as a result concerning local complementation on graphs. As an application, we show that a particular matrix polynomial, the so-called nullity polynomial, is invariant under principal pivot transform.     

Interlacing eigenvalues on some operations of graphs

In this paper, we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix, standard Laplacian, and normalized Laplacian. Also, we explore some applications of this interlacing.     

Projective lines over Jordan systems and geometry of Hermitian matrices

Any set of σ-Hermitian matrices of size n×n over a field with involution σ gives rise to a projective line in the sense of ring geometry and a projective space in the sense of matrix geometry. It is shown that the two concepts are based upon the same set of points, up to some notational differences.     

Beltrami定理的推广

本文研究了Berwald流形之间的射影对应.利用Berwald流形上Weyl射影曲率张量的射影不变性,证明了当n>2时,与射影平坦的Berwald流形射影对应的黎曼流形M~n是常曲率流形,从而推广了Beltrami定理.     

A NOTE ON POLYNOMIAL RINGS OVER VON NEUMANN REGULAR RINGS

Abstract

Let R be an associative ring with 1. It is well known (see [1], [2]) that if R is commutative, then R is Yon Neumann regular (VNR) <=> the polynomial ring S = R[x] is semihereditary. While one of these implications is true in the general case, it is known that a polynomial ring over a regular ring need not be semihereditary (see [3]). In [4] we showed that a ring R is VNR <=> aS + xS is projective for each a ε R. In this note we sharpen this result and use it to show that if c is the ring epimorphism from R[x] to R that maps each polynomial onto its constant term, then R is Yon Neumann regular <=> the inverse image (under c) of each principal (right, left) ideal of R. is a principal (right. left) ideal of R[x] generated by a regular element. (Here an element is regular if and only if it is a non zero-divisor).     

Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups

We consider mixing ^d-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.     

Conformal geometry of surfaces in Lorentzian space forms

We study the conformal geometry of an oriented space-like surface in three-dimensional Lorentzian space forms. After introducing the conformal compactification of the Lorentzian space forms, we define the conformal Gauss map which is a conformally invariant two parameter family of oriented spheres. We use the area of the conformal Gauss map to define the Willmore functional and derive a Bernstein type theorem for parabolic Willmore surfaces. Finally, we study the stability of maximal surfaces for the Willmore functional.Dedicated to Professor T.J. WillmoreSupported by an FPPI Postdoctoral Grant from DGICYT Ministerio de Educación y Ciencia, Spain 1994 and by a DGICYT Grant No. PB94-0750-C02-02     

On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

Unary operations with long pre-periods

It is well known that the congruence lattice ConA of an algebra A is uniquely determined by the unary polynomial operations of A (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let A be a finite algebra with |A|=n. If Imf=A or |Imf|=1 for every unary polynomial operation f of A, then A is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f:A→A is not a permutation then A⊃Imf and there is a least natural number λ(f) with Imf^λ(f)=Imf^λ(f)+1. We consider unary operations with λ(f)=n-1 for n?2 and λ(f)=n-2 for n?3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.     

Subsheaves of the cotangent bundle

For any smooth projective variety, we study a birational invariant, defined by Campana which depends on the Kodaira dimension of the subsheaves of the cotangent bundle of the variety and its exterior powers. We provide new bounds for a related invariant in any dimension and in particular we show that it is equal to the Kodaira dimension of the variety, in dimension up to 4, if this is not negative.     

On the spectral geometry for the Jacobi operators of harmonic maps into a quaternionic projective space

We characterize both invariant and totally real immersions into the quaternionic projective space by the spectra of the Jacobi operator. Also, we study spectral characterization of harmonic submersions when the target manifold is the quaternionic projective space.