Showing papers in "International Mathematics Research Notices in 2003"

Moments and cumulants of polynomial random variables on unitarygroups, the Itzykson-Zuber integral, and free probability

Abstract: We consider integrals on unitary groups U d of the form $\int_{{\text{?U}}_d}U_{i_1\text{?}j_1}\cdots U_{i_q\text{?}j_q}U_{{j^{'}}_1{i^{'}}_1}^\ast\cdots U_{{j^{'}}_{q^{'}}{i^{'}}_{q^{'}}}^{\ast} dU$ . We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows to rederive an asymptotic expansion as d → ∞. Using this, we rederive and strengthen a result of asymptotic freeness due to Voiculescu. We then study large d asymptotics of matrix-model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular, we give an explicit formula for ${\text{lim}}_{d\rightarrow\infty}{(\partial^n/\partial z^n\text{?})}d^{-2}\text{?log}\int_{{\text{U}}_d}e^{zd\text{?Tr}{(XUYU^\ast)}}dU\vert_{z=0}$ , assuming that the normalized traces d −1 Tr(X k ) and d −1 Tr (Y k ) converge in the large d limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

Calabi quasimorphism and quantum homology

Abstract: We prove that the group of area-preserving diffeomorphisms of the 2-sphere admits a non-trivial homogeneous quasimorphism to the real numbers with the following property. Its value on any diffeomorphism supported in a sufficiently small open subset of the sphere equals to the Calabi invariant of the diffeomorphism. This result extends to more general symplectic manifolds: If the symplectic manifold is monotone and its quantum homology algebra is semi-simple we construct a similar quasimorphism on the universal cover of the group of Hamiltonian diffeomorphisms.

Module categories over the Drinfeld double of a finite group

Abstract: Let C be a (semisimple abelian) monoidal category. A module category over C is a (semisimple abelian) category M together with a functor C×M → M and an associativity constraint (= natural isomorphism of two composition functors C × C × M → M) satifying natural axioms, see [18]. In physics, one is interested in the case when C is a category of representations of some vertex algebra V and irreducible objects of M are interpreted as boundary conditions for the conformal field theory associated to V, see [3, 9]. Thus, it is interesting from a physical point of view for a given monoidal category C to classify all possible module categories over C (it is known that in many interesting cases, the list of answers is finite, see [18]). This problem is also of mathematical interest, for example, the module categories with just one isomorphism class of irreducible objects are exactly the same as the fiber functors C → Vec, see [18]. It is known that, for a fusion categories of ŝl(2) at positive integer levels, the module categories are classified by ADE Dynkin diagrams, see [12, 17, 18]. In this paper, we consider another class of examples, known in physics as holomorphic orbifold models (see, e.g., [7, 11]). Let G be a finite group. It is well known that the monoidal structures (= associativity constraints) on the category Vec of G-graded vector spaces with the usual functor of tensor product are classified by the group H3(G,C∗), see [13]. For ω ∈ H3(G,C∗), let Vecω denote the corresponding monoidal category. Let D(G,ω) be the Drinfeld center

Finite-dimensional representations of rational Cherednik algebras

Abstract: A complete classification and character formulas for finite-dimensional irreducible representations of the rational Cherednik algebra of type A is given. Less complete results for other types are obtained. Links to the geometry of affine flag manifolds and Hilbert schemes are discussed.

Topological robotics: Motion planning in projective spaces

Abstract: In this paper, we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point. One wants to bring the line from its initial position A to a final position B by a continuous motion in space. The ultimate goal is to construct a motion planning algorithm which will perform this task once the initial position A and the final position B are presented. This problem becomes hard when the dimension of the space is large. Any such motion planning algorithm must have instabilities, that is, the motion of the system will be discontinuous as a function of A and B. These instabilities are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently in [6, 7]. With any path-connected topological space X, one associates in [6, 7] a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. The motion planning problem of moving a line in R n+1 reduces to a topological problem of calculating the topological complexity of the real projective space TC(RP n ), which we tackle in this paper. We compute the number TC(RP n ) for all n ≤ 23 (see

Welschinger invariant and enumeration of real rational curves

Abstract: Welschinger’s invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using Mikhalkin’s approach which deals with a corresponding count of tropical curves. In particular, our estimate implies that, for any positive integer d, there exists a real rational curve of degree d through any collection of 3d − 1 real points in the projective plane, and, moreover, asymptotically in the logarithmic scale at least one third of the complex plane rational curves through a generic point collection are real. We also obtain similar results for curves on other toric Del Pezzo surfaces.

Symmetric polynomials vanishing on the shifted diagonals and Macdonald polynomials

Abstract: For each pair (k,r) of positive integers with r>1, we consider an ideal I^(k,r)_n of the ring of symmetric polynomials in n variables. The ideal I_n^(k,r) has a basis consisting of Macdonald polynomials P(x_1,...,x_n;q,t) at t^{k+1}q^{r-1}=1, and is a deformed version of the one studied earlier in the context of Jack polynomials. In this paper we give a characterization of I^(k,r)_n in terms of explicit zero conditions on the k-codimensional shifted diagonals of the form x_{2}=tq^{s_1}x_1,...,x_{k+1}=tq^{s_k}x_k. The ideal I^(k,r)_n may be viewed as a deformation of the space of correlation functions of an abelian current of the affine Lie algebra \hat{sl_r}. We give a brief discussion about this connection.

L2 stability of solitons for KdV equation

Abstract: We prove stability and asymptotic stability in L 2 of solitons for Korteweg-de Vries equation on the whole line from the use of the Miura transformation. Asymptotic stability of kinks for modified KdV equation is also considered.

On the rigidity for conformally compact Einstein manifolds

Abstract: In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions.

Omega results for the divisor and circle problems

Abstract: We obtain new omega results for the error terms in two classical lattice point problems. These results are likely to be the best possible.

Equilibrium distribution of zeros of random polynomials

Abstract: We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

Global carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems

Abstract: We consider a general second order elliptic equation with right-hand side f+∑j=0N∂fj∂xj∈H−1(Ω) where f,fj∈L2(Ω) and Dirichlet boundary condition g∈H1/2(Γ). We prove a global Carleman estimate for the solution y of this equation in terms of the weighted L2 norms of f and fj and the H1/2 norm of g. This estimate depends on two real parameters s and λ which are supposed to be large enough and is sharp with respect to the exponents of these parameters. This allows us to obtain, for example, sharper estimates on the pressure term in the linearized Navier–Stokes equations and it turns out to be very useful in the context of controllability problems. To cite this article: O.Y. Imanuvilov, J.-P. Puel, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 33–38.

The Wilson function transform

Abstract: Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function involved can be considered as a non-polynomial extension of the Wilson polynomial, and is therefore called a Wilson function. The two integral transforms are called a Wilson function transform of type I and type II. Furthermore, a few explicit transformations of hypergeometric functions are calculated, and it is shown that the Wilson function transform of type I maps a basis of orthogonal polynomials onto a similar basis of polynomials.

A supersingular K3 surface in characteristic 2 and the Leech lattice

Abstract: We construct a K3 surface over an algebraically closed field of characteristic 2 which contains two sets of 21 disjoint smooth rational curves such that each curve from one set intersects exactly 5 curves from the other set. This configuration is isomorphic to the configuration of points and lines on the projective plane over the finite field of 4 elements. The surface admits a finite automorphism group which contains the group of automorphisms of the plane which acts on the configuration of each set of 21 smooth rational curves, and the additional element of order 2 which interchanges the two sets. The Picard lattice of the surface is a reflective sublattice of an even unimodular lattice of signatuire (1,25) and the classes of the 42 curves correspond to some Leech roots in this lattice.

On the Hodge-Newton decomposition for split groups

Abstract: The main purpose of this paper is to prove a group-theoretic generalization of a theorem of Katz on isocrystals. Along the way we reprove the group-theoretic generalization of Mazur's inequality for isocrystals due to Rapoport-Richartz, and generalize from split groups to unramified groups a result of Kottwitz-Rapoport which determines when an affine Deligne-Lusztig subset of the affine Grassmannian is non-empty.

Uniform Asymptotics for Polynomials Orthogonal with Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles: Announcement of Results

Abstract: We compute the pointwise asymptotics of orthogonal polynomials with respect to a general class of pure point measures supported on finite sets as both the number of nodes of the measure and also the degree of the orthogonal polynomials become large. The class of orthogonal polynomials we consider includes as special cases the Krawtchouk and Hahn classical discrete orthogonal polynomials, but is far more general. In particular, we consider nodes that are not necessarily equally spaced. The asymptotic results are given with error bound for all points in the complex plane except for a finite union of discs of arbitrarily small but fixed radii. These exceptional discs are the neighborhoods of the so-called band edges of the associated equilibrium measure. As applications, we prove universality results for correlation functions of a general class of discrete orthogonal polynomial ensembles, and in particular we deduce asymptotic formulae with error bound for certain statistics relevant in the random tiling of a hexagon with rhombus-shaped tiles. The discrete orthogonal polynomials are characterized in terms of a a Riemann-Hilbert problem formulated for a meromorphic matrix with certain pole conditions. By extending the methods of [17, 22], we suggest a general and unifying approach to handle Riemann-Hilbert problems in the situation when poles of the unknown matrix are accumulating on some set in the asymptotic limit of interest.

Path model for a level-zero extremal weight module over a quantum affine algebra

Abstract: We give a path model for a level zero extremal weight module over a quantum affine algebra. By using this result, we prove a branching rule for an extremal weight module with respect to a Levi subalgebra. Furthermore, we also show a decomposition rule of Littelmann type for the concatenation of path models for an integrable highest weight module and a level zero extremal weight module in the case where the extremal weight is minuscule.

Periods and Igusa local zeta functions

Abstract: We show that the coefficients in the Laurent series of the Igusa local zeta functions I(s) = ∫ C fω are periods. This is proved by first showing the existence of functional equations for these functions. This will be used to show in a subsequent paper (by P. Brosnan) that certain numbers occurring in Feynman amplitudes (up to Gamma factors) are periods. We also give several examples of our main result, and one example showing that Euler’s constant γ is an exponential period.

Proper isometric actions of Thompson's groups on Hilbert space

Abstract: In 1965, Thompson defined the groups F, T , and V [8]. Thompson’s group V is the group of right-continuous bijections v of [0, 1] that map dyadic rational numbers to dyadic rational numbers, that are differentiable except at finitely many dyadic rational numbers, and such that, on each interval on which v is differentiable, v is affine with derivative a power of 2. The group F is the subgroup of V consisting of homeomorphisms. The group T is the subgroup of V consisting of those elements which induce homeomorphisms of the circle, where the circle is regarded as [0, 1] with 0 and 1 identified. It is a long-standing open question to determine whether F is amenable. The main theorem of this paper establishes that the groups F, T , and V all have the weaker property of a-T-menability. A theorem of Higson and Kasparov [4] states that every a-T-menable group satisfies the Baum-Connes conjecture with arbitrary coefficients, so Thompson’s groups F, T , and V satisfy the Baum-Connes conjecture as well. An isometric action of a discrete group G on a metric space X is proper if, for any x ∈ X and any bounded subset U of X, there are only finitely many elements of g that translate x inside U. A function f : V1 → V2 between two vector spaces is affine if it is the composition of a linear map followed by a translation.

Abelian ideals of a Borel subalgebra and long positive roots

Abstract: Let b be a Borel subalgebra of a simple Lie algebra g. Let Ab denote the set of all Abelian ideals of b. It is easily seen that any a ∈ Ab is actually contained in the nilpotent radical of b. Therefore, a is determined by the corresponding set of roots. More precisely, let t be a Cartan subalgebra of g lying in b and let ∆ be the root system of the pair (g, t). Choose ∆, the system of positive roots, so that the roots of b are positive. Then a = ⊕γ∈Igγ, where I is a suitable subset of ∆ and gγ is the root space for γ ∈ ∆. It follows that there are finitely many Abelian ideals and that any question concerning Abelian ideals can be stated in terms of combinatorics of the root system. An amazing result of D. Peterson says that the cardinality of Ab is 2rk . His approach uses a one-to-one correspondence between the Abelian ideals and the so-called minuscule elements of the affine Weyl group Ŵ. An exposition of Peterson’s results is found in [5]. Peterson’s work appeared to be the point of departure for active recent investigations of Abelian ideals, ad-nilpotent ideals, and related problems of representation theory and combinatorics [1, 2, 3, 4, 5, 6, 7, 8]. We consider Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of Ab. Our goal is to study this poset structure. It is easily seen that Ab is a ranked poset; the rank function attaches to an ideal its dimension. It was shown in [8] that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of g. (For each simple Lie algebra, the maximal Abelian ideals were determined in [10].) This correspondence possesses a number of nice properties, but the very existence of it was demonstrated in

Classical and quantum scattering for a class of long range random potentials

Abstract: We prove an almost sure existence of the modified wave operators for a class of Schrodinger operators with random long range potentials. The assumed decay of potential at infinity places it beyond the threshold of the standard class of long range potentials as described in the work of Buslaev-Matveev, Alsholm-Kato, and Hormander. We develop an approach to quantum scattering which relies on averaging of potential over classical random trajectories. We also establish classical scattering for corresponding classical hamiltonians.

The Brauer group of analytic K3 surfaces

Abstract: We show that for complex analytic K3 surfaces any torsion class in H2(X,O∗ X) comes from an Azumaya algebra. In other words, the Brauer group equals the cohomological Brauer group. For algebraic surfaces, such results go back to Grothendieck. In our situation, we use twistor spaces to deform a given analytic K3 surface to suitable projective K3 surfaces, and then stable bundles and hyperholomorphy conditions to pass back and forth between the members of the twistor family. In analogy to the isomorphism Pic(X) ∼= H1(X,O∗ X), Grothendieck investigated in [8] the possibility of interpreting classes in H2(X,O∗ X) as geometric objects. He observed that the Brauer group Br(X), parameterizing equivalence classes of sheaves of Azumaya algebras on X, naturally injects into H2(X,O∗ X). It is not difficult to see that Br(X) ⊂ H2(X,O∗ X) is contained in the torsion part of H2(X,O∗ X) and Grothendieck asked: Is the natural injection Br(X ) ⊂ H (X ,O∗ X )tor an isomorphism? This question is of interest in various geometric categories, e.g. X might be a scheme, a complex space, a complex manifold, etc. It is also related to more recent developments in the application of complex algebraic geometry to conformal field theory. Certain elements in H2(X,O∗ X) have been interpreted as so-called B-fields, and those are used to construct super conformal field theories associated to Ricci-flat manifolds. Thus, understanding the geometric meaning of the cohomological Brauer group Br′(X) := H2(X,O∗ X)tor is also of interest for the mathematical interpretation of string theory and mirror symmetry. An affirmative answer to Grothendieck’s question has been given only in very few special cases: • If X is a complex curve, then H2(X,O∗ X) = 0. Hence, Br(X) = Br ′(X) = H2(X,O∗ X) = 0 (see [8, Cor.2.2] for the general case of a curve). • For smooth algebraic surfaces the surjectivity has been proved by Grothendieck [8, Cor.2.2] and for normal algebraic surfaces a proof was given more recently by Schroer [14]. • Hoobler [9] and Berkovich [3] gave an affirmative answer for abelian varieties of any dimension and Elencwajg and Narasimhan gave another proof for complex tori [6].