Deformation of Complete Intersections in the Plane

We study deformations of zero-dimensional complete intersectionsin the plane, and prove the following results. (1) Two complexnon-singular curves intersecting at r points with multiplicitiesd₁,...,d_r can be deformed into curves intersecting (at somepoints) with multiplicities d'₁,...,d'_s which are arbitraryprescribed partitions of the numbers d₁,...,d_r. (2) Two realcurves intersecting with multiplicity at most 2 at each of theirreal common points can be deformed so that all real multipleintersection points split into real simple intersection points.1991 Mathematics Subject Classification 14M10, 14P05.     

Shadows and intersections in vector spaces

We prove a vector space analog of a version of the Kruskal-Katona theorem due to Lovász. We apply this result to extend Frankl's theorem on r-wise intersecting families to vector spaces. In particular, we obtain a short new proof of the Erd?s-Ko-Rado theorem for vector spaces.     

The Problem of Topologies of Grothendieck and the Class of Fréchet T-Spaces

We examine the interpolation with periodic polynomial splines of degree d and defect r (d ≦ r) on equidistant partitions of the real axis and generalize known results for r = 0. We prove necessary and sufficient conditions for the existence and a certain L²-stability of the interpolants as well as their approximation properties in the scale of the periodic SOBOLEV spaces.     

On the proportion of permutations of order a multiple of the degree

We study permutations of a set of size n for which the orderis a multiple of n. We prove that, for large n, most such elementslie in one of two families. The first family consists of thosepermutations with a single very large cycle of order dividingn and includes the n-cycles, and the second consists of permutationsfor which the cycles of length dividing n have total lengthsignificantly less than n. This work was inspired by the algorithmicproblem of fast recognition of large symmetric groups actingprimitively on subsets.     

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier–Mukai transforms for coherentsystems consisting of a vector bundle over an elliptic curveand a subspace of its global sections, showing that these transformsare indexed by positive integers. We prove that the naturalstability condition for coherent systems, which depends on aparameter, is preserved by these transforms for small and largevalues of the parameter. By means of the Fourier–Mukaitransforms we prove that certain moduli spaces of coherent systemscorresponding to small and large values of the parameter areisomorphic. Using these results we draw some conclusions aboutthe possible birational type of the moduli spaces. We provethat for a given degree d of the vector bundle and a given dimensionof the subspace of its global sections there are at most d differentpossible birational types for the moduli spaces.     

The Convex Hull of Extremal Vector-Valued Continuous Functions

Let T be a completely regular space and X a strictly convexn-dimensional real space. We prove that every continuous functionfrom T into the closed unit ball of X can be expressed as anaverage of eight continuous functions from T into the sphereof X if and only if dim (T) n–1, where dim(T) denotesthe covering dimension of T. The proof we give can be used toprove the same fact, without hypotheses on T, when X is infinite-dimensional,although in this case it has been proved recently that a betterresult can be obtained.     

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Let Y be a reduced irreducible projective curve of arithmeticgenus g 2 with at most ordinary nodes as singularities. Fora subsheaf F of rank r', degree d' of a torsion-free sheaf Eof rank r, degree d, let s(E,F) = r'd-rd'. Define s_r'(E) = mins(E,F), where the minimum is taken over all subsheaves of Eof rank r'. For a fixed r', s_r' defines a stratification ofthe moduli space U(r,d) of stable torsion-free sheaves of rankr, degree d by locally closed subsets U_r',s. We study the nonemptinessand dimensions of the strata. We show that the general elementin each nonempty stratum is a vector bundle and it has onlyfinitely many (respectively unique) maximal subsheaves of rankr' for s r'(r-r')(g – 1) (respectively s < r'(r-r')(g– 1)). We prove that the tensor product of two generalstable vector bundles on an irreducible nodal curve Y is nonspecial.     

Harmonic manifolds with some specific volume densities

We show that a noncompact, complete, simply connected harmonic manifold (M ^d, g) with volume densityθ _m(r)=sinh^d-1 r is isometric to the real hyperbolic space and a noncompact, complete, simply connected Kähler harmonic manifold (M ^2d, g) with volume densityθ _m(r)=sinh^2d-1 r coshr is isometric to the complex hyperbolic space. A similar result is also proved for quaternionic Kähler manifolds. Using our methods we get an alternative proof, without appealing to the powerful Cheeger-Gromoll splitting theorem, of the fact that every Ricci flat harmonic manifold is flat. Finally a rigidity result for real hyperbolic space is presented.     

Visible actions on symmetric spaces

A visible action on a complex manifold is a holomorphic action that admits a J-transversal totally real submanifold S. It is said to be strongly visible if there exists an orbit-preserving anti-holomorphic diffeomorphism σ such that σ|_S = id. In this paper we prove that for any Hermitian symmetric space D = G/K the action of any symmetric subgroup H is strongly visible. The proof is carried out by finding explicitly an orbit-preserving anti-holomorphic involution and a totally real submanifold S. Our geometric results provide a uniform proof of various multiplicity-free theorems of irreducible highest weight modules when restricted to reductive symmetric pairs, for both classical and exceptional cases, for both finite- and infinite-dimensional cases, and for both discrete and continuous spectra.     

Real Paley-Wiener Theorems

The aim of this paper is to exhibit a real Paley–Wienerspace sitting inside the Schwartz space, and to give a quickand simple proof of a Paley–Wiener-type theorem. A simpleand elementary proof of a theorem postulated by H. H. Bang isalso given. 2000 Mathematics Subject Classification 42A38.     

A metric characterization of the irrationals via a group operation

Let S be a separable metric space with a compatible metric d that satisfies: For each point x ? S and each nonnegative real number r there exists a unique point y ? S such that d(x,y) = r.In this paper spaces that meet the above criterion are investigated. It is shown that, under the assumption of completeness, this metric property characterizes the space of irrationals.     

A vanishing theorem for modular symbols on locally symmetric spaces

A modular symbol is the fundamental class of a totally geodesic submanifold embedded in a locally Riemannian symmetric space , which is defined by a subsymmetric space . In this paper, we consider the modular symbol defined by a semisimple symmetric pair (G,G'), and prove a vanishing theorem with respect to the -component in the Matsushima-Murakami formula based on the discretely decomposable theorem of the restriction . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV. Received: December 8, 1996     

Optimal cubature in Besov spaces with dominating mixed smoothness on the unit square

We prove new optimal bounds for the error of numerical integration in bivariate Besov spaces with dominating mixed order r$r$. The results essentially improve on the so far best known upper bound achieved by using cubature formulas taking points from a sparse grid. Motivated by Hinrichs’ observation that Hammersley type point sets provide optimal discrepancy estimates in Besov spaces with mixed smoothness on the unit square, we directly study quasi-Monte Carlo integration on such point sets. As the main tool we prove the representation of a bivariate periodic function in a piecewise linear tensor Faber basis. This allows for optimal worst case estimates of the QMC integration error with respect to Besov spaces with dominating mixed smoothness up to order r<2$r<2$. The results in this paper are the first step towards sharp results for spaces with arbitrarily large mixed order on the d$d$-dimensional unit cube. In fact, in contrast to Fibonacci lattice rules, which are also practicable in this context, the QMC methods used in this paper have a proper counterpart in d$d$ dimensions.     

Even infinite-dimensional real Banach spaces

This article is a continuation of a paper of the first author [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about complex structures on real Banach spaces. We define a notion of even infinite-dimensional real Banach space, and prove that there exist even spaces, including HI or unconditional examples from [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] and C(K) examples due to Plebanek [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We extend results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] relating the set of complex structures up to isomorphism on a real space to a group associated to inessential operators on that space, and give characterizations of even spaces in terms of this group. We also generalize results of [V. Ferenczi, Uniqueness of complex structure and real hereditarily indecomposable Banach spaces, Adv. Math. 213 (1) (2007) 462–488] about totally incomparable complex structures to essentially incomparable complex structures, while showing that the complex version of a space defined by S. Argyros and A. Manoussakis [S. Argyros, A. Manoussakis, An indecomposable and unconditionally saturated Banach space, Studia Math. 159 (1) (2003) 1–32] provides examples of essentially incomparable complex structures which are not totally incomparable.     

Spaces of Harmonic Functions

It is important and interesting to study harmonic functionson a Riemannian manifold. In an earlier work of Li and Tam [21]it was demonstrated that the dimensions of various spaces ofbounded and positive harmonic functions are closely relatedto the number of ends of a manifold. For the linear space consistingof all harmonic functions of polynomial growth of degree atmost d on a complete Riemannian manifold Mⁿ of dimension n,denoted by H_d(Mⁿ), it was proved by Li and Tam [20] that thedimension of the space H₁(M) always satisfies dimH₁(M) dimH₁(Rⁿ)when M has non-negative Ricci curvature. They went on to askas a refinement of a conjecture of Yau [32] whether in generaldim H_d(Mⁿ) dimH_d(Rⁿ)for all d. Colding and Minicozzi made animportant contribution to this question in a sequence of papers[5–11] by showing among other things that dimH_d(M) isfinite when M has non-negative Ricci curvature. On the otherhand, in a very remarkable paper [16], Li produced an elegantand powerful argument to prove the following. Recall that Msatisfies a weak volume growth condition if, for some constantA and , (1.1) for all x M and r R, where V_x(r) is the volume of the geodesicball B_x(r) in M; M has mean value property if there exists aconstant B such that, for any non-negative subharmonic functionf on M, (1.2) for all p M and r > 0.     

Shortest billiard trajectories

In this paper we prove that any convex body of the d-dimensional Euclidean space (d ≥ 2) possesses at least one shortest generalized billiard trajectory moreover, any of its shortest generalized billiard trajectories is of period at most d + 1. Actually, in the Euclidean plane we improve this theorem as follows. A disk-polygon with parameter r > 0 is simply the intersection of finitely many (closed) circular disks of radii r, called generating disks, having some interior point in common in the Euclidean plane. Also, we say that a disk-polygon with parameter r > 0 is a fat disk-polygon if the pairwise distances between the centers of its generating disks are at most r. We prove that any of the shortest generalized billiard trajectories of an arbitrary fat disk-polygon is a 2-periodic one. Also, we give a proof of the analogue result for ε-rounded disk-polygons obtained from fat disk-polygons by rounding them off using circular disks of radii ε > 0. Our theorems give partial answers to the very recent question raised by S. Zelditch on characterizing convex bodies whose shortest periodic billiard trajectories are of period 2. K. Bezdek partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant.     

Unitary groups of nondegenerate Hermitian forms and the homotopy groups of pseudospheres

The Witt Extension Theorem states that the unitary group of a finite-dimensional vector space V equipped with a nondegenerate hermitian form acts transitively on the pseudosphere induced by the form. We provide a new, constructive proof of this result for finite-dimensional vector spaces V over R, C, or H. This constructive proof is then used to prove a similar result for the unitary group of a finitely generated free right module over an abelian AW^∗-algebra. The topology of these unitary groups is examined and as an application we determine the homotopy groups π₁ and π₂ of the induced real, complex, and quaternionic pseudospheres.     

Projective Extensions of Fields

Every field K admits proper projective extensions, that is,Galois extensions where the Galois group is a non-trivial projectivegroup, unless K is separably closed or K is a pythagorean formallyreal field without cyclic extensions of odd degree. As a consequence,it turns out that almost all absolute Galois groups decomposeas proper semidirect products. We show that each local field has a unique maximal projectiveextension, and that the same holds for each global field ofpositive characteristic. In characteristic 0, we prove thatLeopoldt's conjecture for all totally real number fields isequivalent to the statement that, for all totally real numberfields, all projective extensions are cyclotomic. So the realizabilityof any non-procyclic projective group as Galois group over Qproduces counterexamples to the Leopoldt conjecture.     

Total Colourings of Graphs

We prove that the TCC (Total Colouring Conjecture) is true forcomplete r-partite graphs, which extends a result of M. Rosenfeld.We also give an alternative, slightly simpler proof of an earlierresult (which says that the TCC is true for graphs having maximumdegree 3) obtained independently by M. Rosenfeld and N. Vijayaditya.     

Linear polarization constants of Hilbert spaces

This paper has been motivated by previous work on estimating lower bounds for the norms of homogeneous polynomials which are products of linear forms. The purpose of this work is to investigate the so-called nth (linear) polarization constant cn(X) of a finite-dimensional Banach space X, and in particular of a Hilbert space. Note that cn(X) is an isometric invariant of the space. It has been proved by J. Arias-de-Reyna [Linear Algebra Appl. 285 (1998) 395-408] that if H is a complex Hilbert space of dimension at least n, then cn(H)=nn/2. The same value of cn(H) for real Hilbert spaces is only conjectured, but estimates were obtained in many cases. In particular, it is known that the nth (linear) polarization constant of a d-dimensional real or complex Hilbert space H is of the approximate order dn/2, for n large enough, and also an integral form of the asymptotic quantity c(H), that is the (linear) polarization constant of the Hilbert space H, where dimH=d, was obtained together with an explicit form for real spaces. Here we present another proof, we find the explicit form even for complex spaces, and we elaborate further on the values of cn(H) and c(H). In particular, we answer a question raised by J.C. García-Vázquez and R. Villa [Mathematika 46 (1999) 315-322]. Also, we prove the conjectured cn(H)=nn/2 for real Hilbert spaces of dimension n?5. A few further estimates have been also derived.