Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities

We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.Mathematics Subject Classification (2000): 14E15, 16Rxx, 16S38, 18E30The first author was partially supported by the Nuffield Foundation grant NAL/00625/G and by the University of Washingtons Milliman Fund. The second author was partially supported by an NSF grant DMS-0070560. Both authors are grateful to the Edinburgh Mathematical Society and the Leverhulme Trust for support.Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Global Euler obstruction and polar invariants

Let be a purely dimensional, complex algebraic singular space. We define a global Euler obstruction Eu(Y) which extends the Euler-Poincaré characteristic in case of a nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of global polar invariants. Partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

On cyclic bra nched coverings of torus knots

We prove that then-fold cyclic coverings of the 3-sphere branched over the torus knotsK(p,q), p>q2 (i.e. the Brieskorn manifolds in the sense of [12]) admit spines corresponding to cyclic presentations of groups ifp1 (modq). These presentations include as a very particular case the Sieradski groups, first introduced in [14] and successively obtained from geometric constructions in [4], [9], and [15]. So our main theorem answers in affirmative to an open question suggested by the referee in [14]. Then we discuss a question concerning cyclic presentations of groups and Alexander polynomials of knots.Work Performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and partially supported by the Ministero per la Ricerca Scientifica e Tecnologica of Italy Within the projectsGeometria Reale e Complessa andTopologia and by the Korean Science and Engineering Foundation.     

Unitary periods,Hermitian forms and points on flag varieties

Let E be an imaginary quadratic extension of of class number one. We examine certain representation numbers associated to Hermitian forms over E , which involve counting integral points on flag varieties. G. Chinta is partially supported by the NSF and by a Humboldt Research Fellowship.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

Classification of 4-dimensional homogeneous D’Atri spaces

The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold (M, g) satisfying the first odd Ledger condition is said to be of type . The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers by Podesta-Spiro and Bueken-Vanhecke (which are mutually complementary). The authors started with the corresponding classification of all spaces of type , but this classification was incomplete. Here we present the complete classification of all homogeneous spaces of type in a simple and explicit form and, as a consequence, we prove correctly that all homogeneous 4-dimensional D’Atri spaces are locally naturally reductive. The first author’s work has been partially supported by D.G.I. (Spain) and FEDER Project MTM 2004-06015-C02-01, by a grant AVCiTGRUPOS03/169 and by a Research Grant from Ministerio de Educación y Cultura. The second author’s work has been supported by the grant GA ČR 201/05/2707 and it is part of the research project MSM 0021620839 financed by the Ministry of Education (MŠMT).     

Collapsed Riemannian manifolds with bounded diameter and bounded covering geometry

We study the class ofn-Riemannian manifolds in the title such that the torsion elements in the fundamental group have a definite bound on their orders. Our main result asserts the existence of a kind of generalized Seifert fiber structure onM ⁿ , for which the fundamental group of fibers injects into that ofM ⁿ . This provides a necessary and sufficient topological condition for a manifold to admit a sufficiently collapsed metric in our class. Among other consequences we obtain a strengthened version of the gap conjecture in this context.The work of the first author is partially supported by NSF Grant DMS 9303999. The work of the second author is supported by MSRI through NSF grant DMS 9022140 and partially supported by NSF Grant DMS 9204095.     

Sum-free sets in abelian groups

We show that there is an absolute constant δ>0 such that the number of sum-free subsets of any finite abelian groupG is

Dirac Cohomology of some Harish-Chandra Modules

In this paper we determine the Dirac cohomology of certain irreducible Harish-Chandra modules of a semisimple connected Lie group G with finite center: irreducible finite-dimensional modules and unitary A (λ) modules. We also comment on the relationship to (, K)-cohomology. The first named author is partially supported by research grants from RGC of Hong Kong SAR and NSF of China. The second named author is partially supported by the National Natural Science Foundation of China (10501025), Liu Hui Center for Applied Mathematics and Youth Teachers Foundation of Tianjin University. The third named author is partially supported by a grant from the Ministry of Science, Education and Sports of the Republic of Croatia.     

On solvable minimally transitive permutation groups

We investigate properties of finite transitive permutation groups in which all proper subgroups of G act intransitively on . In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups. Work partially supported by M.I.U.R. and London Mathematical Society.     

On a map between two K3 surfaces associated to a net of quadrics

We study the geometry of the birational map between an intersection of a net of quadrics in that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S ₆ is discriminant of a net of quadrics in such that S ₆ is isomorphic to the intersection of this net iff a certain configuration of rational curves on S₆ is weakly even. Received: 14 September 2005 Suported by the DFG Schwerpunktprogramm ‘Global methods in complex geometry’. The first named author is partially supported by the KBN Grant No. 1 P03A 008 28. The second named author is partially supported by the KBN Grant No. 2 P03A 016 25.     

<Emphasis Type="Italic">BMO</Emphasis> spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality

We identify the dual space of the Hardy-type space related to the time independent Schrödinger operator =–+V, with V a potential satisfying a reverse Hölder inequality, as a BMO-type space . We prove the boundedness in this space of the versions of some classical operators associated to (Hardy-Littlewood, semigroup and Poisson maximal functions, square function, fractional integral operator). We also get a characterization of in terms of Carlesson measures.Mathematics Subject Classification (2000): 35J10, 42B35, 42B30This research was partially supported by the European Commission, within the IHP Network HARP 2002-2006, contract number HPRN-CT-2001-00273-HARP. Second author also supported by Programa Ramón y Cajal and grant BMF2001-0189 MCyT (Spain). The first and the fifth author supported by grants 5P03A05020 and 5P03A02821 from KBN (Poland) and by Foundation of Polish Sciences Subsidy 3/99. The third and the fourth author supported by grant MCyT BMF2002-04013-C02-02.     

A globally and quadratically convergent affine scaling method for linearℓ 1 problems

Recently, various interior point algorithms related to the Karmarkar algorithm have been developed for linear programming. In this paper, we first show how this interior point philosophy can be adapted to the linear ₁ problem (in which there are no feasibility constraints) to yield a globally and linearly convergent algorithm. We then show that the linear algorithm can be modified to provide aglobally and ultimatelyquadratically convergent algorithm. This modified algorithm appears to be significantly more efficient in practise than a more straightforward interior point approach via a linear programming formulation: we present numerical results to support this claim.This paper was presented at the Third SIAM Conference on Optimization, in Boston, April 1989.Research partially supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under grant DE-FG02-86ER25013.A000, by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University, and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.Research partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute, Cornell University and by the Computational Mathematics Program of the National Science Foundation under grant DMS-8706133.     

Dense graphs and edge reconstruction

By a well-known result of Nash-Williams if a graphG is not edge reconstructible, then for all ,|A||E(G)| mod 2 we have a permutation ofV(G) such thatE(G)E(G)=A. Here we construct infinitely many graphsG having this curious property and more than edges.Research (partially) supported by Hungarian National Foundation for Scientific Research Grant No.T016389.     

On the value distribution of a certain type of differential polynomials

Some quantitative estimations on the value distribution of the functionafg ⁿ(n2) are obtained, whereg is a linear differential polynomial inf anda is a small function off.The research was partially supported by a U.G.C grant of Hong Kong     

Desingularizations of the moduli space of rank 2 bundles over a curve

Let X be a smooth projective curve of genus g3 and M₀ be the moduli space of rank 2 semistable bundles over X with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan [NR78], Seshadri [Ses77] and Kirwan [Kir86b] respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwans desingularization to Seshadris, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of M₀ are terminal and the plurigenera are all trivial. As an application, we compute the Betti numbers of the cohomology of Seshadris desingularization in all degrees. This generalizes the result of [BS90] which computes the Betti numbers in low degrees. Another application is the computation of the stringy E-function (see [Bat98] for definition) of M₀ for any genus g3 which generalizes the result of [Kie03].Young-Hoon Kiem was partially supported by KOSEF R01-2003-000-11634-0 and SNU; Jun Li was partially supported by NSF grants.Mathematics Subject Classification (2000): 14H60, 14F25, 14F42     

Stability for hypersurfaces of constant mean curvature with free boundary

The partitioning problem for a smooth convex bodyB ³ consists in to study, among surfaces which divideB in two pieces of prescribed volume, those which are critical points of the area functional.We study stable solutions of the above problem: we obtain several topological and geometrical restrictions for this kind of surfaces. In the case thatB is a Euclidean ball we obtain stronger results.Antonio Ros is partially supported by DGICYT grant PB91-0731 and Enaldo Vergasta is partially supported by CNPq grant 202326/91-8.