International Mathematics Research Notices - Advance Access

Virasoro Constraints and Descendant Hurwitz-Hodge Integrals

Jiang, Y., Tseng, H.-H. — Fri, 20 Nov 2009 03:13:46 PST

Virasoro constraints are applied to degree zero Gromov–Witten theory of weighted projective stacks P(1, N) and P(1, 1, N) to obtain formulas of descendant cyclic Hurwitz– Hodge integrals in higher genera.

Foliations of Hyperbolic Space by Constant Mean Curvature Hypersurfaces

Coskunuzer, B. — Tue, 17 Nov 2009 06:33:18 PST

We show that the constant mean curvature hypersurfaces in Hⁿ⁺¹ spanning the boundary of a star-shaped C^1,1 domain in Sⁿ(Hⁿ⁺¹) give a foliation of Hⁿ⁺¹. We also show that if is a closed codimension-1 C^2, submanifold in Sⁿ(Hⁿ⁺¹) bounding a unique constant mean curvature hypersurface _H in Hⁿ⁺¹ with _H = for any H (–1,1), then the constant mean curvature hypersurfaces {_H} foliate Hⁿ⁺¹.

Spectral Characterization of Poincare-Einstein Manifolds with Infinity of Positive Yamabe Type

Guillarmou, C., Qing, J. — Mon, 16 Nov 2009 07:17:54 PST

In this paper, we give a sharp spectral characterization of conformally compact Einstein manifolds with conformal infinity of positive Yamabe type in dimension n + 1 > 2. More precisely, we prove that the largest real scattering pole of a conformally compact Einstein manifold (X, g) is less than n/2 – 1 if and only if the conformal infinity of (X, g) is of positive Yamabe type. If this positivity is satisfied, we also show that the Green function of the fractional conformal Laplacian P() on the conformal infinity is nonnegative for all [0,2].

Quadratic Algebras and Integrable Chains

Soloviev, F. — Fri, 13 Nov 2009 08:28:29 PST

Using Krichever–Phong's universal formula, we show that a multiplicative representation linearizes Sklyanin quadratic brackets for a multi-pole Lax function with a spectral parameter. The spectral parameter can be either rational or elliptic. As a by-product, we obtain an extension of a Sklyanin algebra in the elliptic case. Krichever–Phong's formula provides a hierarchy of symplectic structures, and we show that there exists a non-trivial cubic bracket in Sklyanin's case.

Geometric Gamma Values and Zeta Values in Positive Characteristic

Chang, C.-Y., Papanikolas, M. A., Yu, J. — Thu, 12 Nov 2009 07:20:46 PST

In analogy with values of the classical Euler -function at rational numbers and the Riemann -function at positive integers, we consider Thakur's geometric -function evaluated at rational arguments and Carlitz -values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the -function and from the Euler–Carlitz relations and Frobenius p-th power relations of the -function.

Anisotropic Young Diagrams and Infinite-Dimensional Diffusion Processes with the Jack Parameter

Olshanski, G. — Thu, 12 Nov 2009 03:20:06 PST

We construct a family of Markov processes with continuous sample trajectories on an infinite-dimensional space, the Thoma simplex. The family depends on three continuous parameters, one of which, the Jack parameter, is similar to the beta parameter in random matrix theory. The processes arise in a scaling limit transition from certain finite Markov chains, the so called up-down chains on the Young graph with the Jack edge multiplicities. Each of the limit Markov processes is ergodic and its stationary distribution is a symmetrizing measure. The infinitesimal generators of the processes are explicitly computed; viewed as self-adjoint operators in the L² spaces over the symmetrizing measures, the generators have a purely discrete spectrum which is explicitly described. For the special value 1 of the Jack parameter, the limit Markov processes coincide with those of the recent work by Borodin and the author (Probability Theory and Related Fields 144 (2009), 281–318). In the limit, as the Jack parameter goes to 0, our family of processes degenerates to the one-parameter family of diffusions on the Kingman simplex studied long ago by Ethier and Kurtz in connection with some models of population genetics. The techniques of the article are essentially algebraic. The main computations are performed in the algebra of shifted symmetric functions with the Jack parameter and rely on the concept of anisotropic Young diagrams due to Kerov.

The Dual Quantum Group for the Quantum Group Analog of the Normalizer of SU(1, 1) in

Groenevelt, W., Koelink, E., Kustermans, J. — Wed, 04 Nov 2009 09:00:19 PST

The quantum group analog of the normalizer of SU(1, 1) in is an important and nontrivial example of a noncompact quantum group. The general theory of locally compact quantum groups in the operator algebra setting implies the existence of the dual quantum group. The first main goal of this article is to give an explicit description of the dual quantum group for this example involving the quantized enveloping algebra . It turns out that does not suffice to generate the dual quantum group. The dual quantum group is graded with respect to commutation and anticommutation with a suitable analog of the Casimir operator characterized by an affiliation relation to a von Neumann algebra. This is used to obtain an explicit set of generators. Having the dual quantum group the left regular corepresentation of the quantum group analog of the normalizer of SU(1, 1) in is decomposed into irreducible corepresentations. Upon restricting the irreducible corepresentations to -representation one finds combinations of the positive and negative discrete series representations with the strange series representations as well as combinations of the principal unitary series representations. The detailed analysis of this example involves the analysis of special functions of basic hypergeometric type and, in particular, some results on these special functions are obtained, which are stated separately. This article is split into two parts: the first part gives almost all of the statements and the results, and the statements of this part are independent of the second part. The second part contains the proofs of all the statements.

Parabolic and Levi Subalgebras of Finitary Lie Algebras

Dan-Cohen, E., Penkov, I. — Mon, 02 Nov 2009 01:04:10 PST

Let be a locally reductive complex Lie algebra that admits a faithful countable-dimensional finitary representation V. Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of , , , and finite-dimensional simple Lie algebras. A parabolic subalgebra of is any subalgebra that contains a maximal locally solvable (that is, Borel) subalgebra. Building upon the work by Dimitrov and the authors of the present article [4, 8], we give a general description of parabolic subalgebras of in terms of joint stabilizers of taut couples of generalized flags. The main differences with the Borel subalgebra case are that the description of general parabolic subalgebras has to use both the natural and conatural modules, and that the parabolic subalgebras are singled out by further "trace conditions" in the suitable joint stabilizer. The technique of taut couples can also be used to prove the existence of a Levi component of an arbitrary subalgebra of . If is splittable, we show that the linear nilradical admits a locally reductive complement in . We conclude the article with descriptions of Cartan, Borel, and parabolic subalgebras of arbitrary splittable subalgebras of .

On the Milnor Fibers of Sandwiched Singularities

Nemethi, A., Popescu-Pampu, P. — Fri, 30 Oct 2009 07:26:59 PDT

The sandwiched singularities are those rational surface singularities that dominate birationally smooth surface singularities. de Jong and van Straten showed that one can reduce the study of the deformations of a sandwiched singularity to the study of deformations of a one-dimensional object, a so-called decorated plane curve singularity. In particular, the Milnor fibers corresponding to their various smoothing components may be reconstructed up to diffeomorphisms from those deformations of associated decorated curves that have only ordinary singularities. Part of the topology of such a deformation is encoded in the incidence matrix between the irreducible components of the deformed curve and the points that decorate it, well defined up to permutations of columns. Extending a previous theorem of ours, which treated the case of cyclic quotient singularities, we show that the Milnor fibers that correspond to deformations whose incidence matrices are different up to permutations of columns are not diffeomorphic in a strong sense. This gives a lower bound on the number of Stein fillings of the contact boundary of a sandwiched singularity.

Double Affine Hecke Algebras and Bispectral Quantum Knizhnik-Zamolodchikov Equations

van Meer, M., Stokman, J. — Thu, 29 Oct 2009 03:21:58 PDT

We use the double affine Hecke algebra of type GL_N to construct an explicit consistent system of q-difference equations, which we call the bispectral quantum Knizhnik–Zamolodchikov (BqKZ) equations. BqKZ includes, besides Cherednik’s quantum affine KZ equations associated to principal series representations of the underlying affine Hecke algebra, a compatible system of q-difference equations acting on the central character of the principal series representations. We construct a meromorphic self-dual solution of BqKZ which, upon suitable specializations of the central character, reduces to symmetric self-dual Laurent polynomial solutions of quantum KZ equations. We give an explicit correspondence between solutions of BqKZ and solutions of a particular bispectral problem for Ruijsenaars’ commuting trigonometric q-difference operators. Under this correspondence, becomes a self-dual Harish-Chandra series solution ⁺of the bispectral problem. Specializing the central character as above, we recover from ⁺the symmetric self-dual Macdonald polynomials.

Statistics for Traces of Cyclic Trigonal Curves over Finite Fields

Bucur, A., David, C., Feigon, B., Lalin, M. — Tue, 27 Oct 2009 09:34:08 PDT

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q + 1 independent random variables taking the value 0 with probability 2/(q + 2) and 1, e^{2 i/3}, e^{4 i/3} each with probability q/(3(q + 2)). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.

Leafwise Coisotropic Intersections

Gurel, B. Z. — Fri, 23 Oct 2009 09:29:56 PDT

We establish the leafwise intersection property for closed, coisotropic submanifolds in an exact symplectic manifold satisfying natural additional assumptions.

Correspondance de Langlands et Fonctions L des carres Exterieur et Symetrique

Henniart, G. — Fri, 23 Oct 2009 09:29:55 PDT

Soient p un nombre premier, F une extension finie de Q_p, et un caractère additif non trivial de F. La correspondance de Langlands donne une bijection ↦ () entre les représentations –semisimples de dimension n du groupe de Weil–Deligne de F, à isomorphisme près, et les représentations lisses irréductibles de , à isomorphisme près. Pour certaines représentations r du groupe dual de , on sait associer, par voie globale, à une représentation lisse irréductible de , des facteurs L(, r, s) et (, r, s, ). On conjecture l’égalité L( (), r, s) = L (r , s), et de même pour les facteurs , quand est une représentation de dimension n du groupe de Weil–Deligne de F. Répondant à une question de D. Jiang et D. Soudry, nous prouvons que si r = ² ou , on a L( (), r, s) = L (r , s) et ( (), r, s, ) = (r , s, ), où est une racine de l’unité.

(Langlands correspondence and L–functions for the exterior and symmetric squares) Let p be a prime number, F a finite extension of Q_p and a non trivial additive character of F. The Langlands correspondence is a bijection ↦ () between -semisimple degree n representations of the Weil–Deligne group of F, up to isomorphism, and smooth irreducible representations of , up to isomorphism. For some representations r of the dual group of , local-global methods attach factors L(, r, s) and (, r, s, ) to any smooth irreducible representation of . Conjecturally we have L( (), r, s) = L (r , s), and similarly for the -factors, when is a degree n representation of the Weil–Deligne group of F.

Microlocal Study of Lefschetz Fixed-Point Formulas for Higher-Dimensional Fixed Point Sets

Matsui, Y., Takeuchi, K. — Sun, 11 Oct 2009 21:18:50 PDT

We introduce new Lagrangian cycles that encode local contributions of Lefschetz numbers of constructible sheaves into geometric objects. We study their functorial properties and apply them to Lefschetz fixed-point formulas with higher-dimensional fixed-point sets.

A Generalization of the Chebyshev Polynomials and Nonrooted Posets

Tomie, M. — Fri, 09 Oct 2009 09:49:32 PDT

In this article we give a generalization of the Chebyshev polynomials of the first kind. Then we describe a Möbius function of the generalized subword order over P_s . These results give the affirmative answer for the conjecture proposed in [A. Björner and B. Sagan, "Rationality of the Möbius function of the composition poset," Theoretical Computer Science 359, no. 1–3 (2006): 282–98.] and [B. Sagan and V. Vatter, "The Möbius function of the composition poset," Journal of Algebraic Combinatorics 24, no. 2 (2006): 117–36].

Rodrigues' Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations

Duran, A. J. — Thu, 08 Oct 2009 06:34:24 PDT

We develop a method to find Rodrigues’ formulas for orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n. Using it, we produce Rodrigues’ formulas for three illustrative examples of arbitrary size.

On Virasoro Constraints for Orbifold Gromov-Witten Theory

Jiang, Y., Tseng, H.-H. — Wed, 07 Oct 2009 05:58:32 PDT

Virasoro constraints for orbifold Gromov–Witten theory are described. These constraints are applied to the degree zero, genus zero orbifold Gromov–Witten potentials of the weighted projective stacks , , and to obtain formulas of descendant cyclic Hurwitz–Hodge integrals.

Borel-Weil Theory for Root Graded Banach-Lie Groups

Muller, C., Neeb, K.-H., Seppanen, H. — Mon, 05 Oct 2009 23:48:28 PDT

In this article, we introduce (weakly) root graded Banach–Lie algebras and corresponding Lie groups as natural generalizations of group like for a Banach algebra A or groups like C(X,K) of continuous maps of a compact space X into a complex semisimple Lie group K. We study holomorphic induction from holomorphic Banach representations of so-called parabolic subgroups P to representations of G on holomorphic sections of homogeneous vector bundles over G/ P. One of our main results is an algebraic characterization of the space of sections which is used to show that this space actually carries a natural Banach structure, a result generalizing the finite dimensionality of spaces of sections of holomorphic bundles over compact complex manifolds. We also give a geometric realization of any irreducible holomorphic representation of a (weakly) root graded Banach–Lie group G and show that all holomorphic functions on the spaces G/ P are constant.

Upper Bounds on L-Functions at the Edge of the Critical Strip

Li, X. — Sat, 03 Oct 2009 22:23:44 PDT

On the (Ir)rationality of Kontsevich Weights

Felder, G., Willwacher, T. — Fri, 25 Sep 2009 14:34:55 PDT

We compute the weight of a particular Kontsevich graph in deformation quantization. Up to rationals, the result is (3)²/⁶. Hence, it is probably not true that all Kontsevich weights are rational.

Topology of Locally Conformally Kahler Manifolds with Potential

Ornea, L., Verbitsky, M. — Fri, 25 Sep 2009 07:56:32 PDT

Locally conformally Kähler (LCK) manifolds with potential are those which admit a Kähler covering with a proper, automorphic, global potential. The existence of a potential can be characterized cohomologically as vanishing of a certain cohomology class, called the Bott–Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with potential can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic to a smooth elliptic fibration over a Kähler orbifold. We show that the pluricanonical condition on LCK manifolds introduced by G. Kokarev is equivalent to vanishing of the Bott–Chern class. This gives a simple proof of some of the results on topology of pluricanonical LCK manifolds, discovered by Kokarev and Kotschick.

Actions of the Derived Group of a Maximal Unipotent Subgroup on G-Varieties

Panyushev, D. I. — Wed, 23 Sep 2009 03:06:19 PDT

Let U be a maximal unipotent subgroup of a connected semisimple group G and U' the derived group of U. We study actions of U' on affine G-varieties. First, we consider the algebra of U' invariants on G/ U. We prove that is a polynomial algebra of Krull dimension 2r, where r = rk(G). A related result is that, for any simple finite-dimensional G-module V, is a cyclic U/ U'-module. Second, we study "symmetries" of Poincare series for U'-invariants on affine conical G-varieties. The results we obtain are very similar to those for the algebras of U-invariants. Third, we obtain a classification of simple G-modules V with polynomial algebras of U'-invariants (for G simple).

Discrete Subgroups of Generated by Triangular Matrices

Benoist, Y., Oh, H. — Sun, 20 Sep 2009 18:13:43 PDT

Based on the ideas in some recently uncovered notes of Selberg [14] on discrete subgroups of a product of ’s, we show that a discrete subgroup of generated by lattices in the upper and lower triangular subgroups is an arithmetic subgroup and hence a lattice in .

Strange Duality for Verlinde Spaces of Exceptional Groups at Level One

Boysal, A., Pauly, C. — Thu, 17 Sep 2009 01:24:36 PDT

The moduli stack of principal E₈-bundles over a smooth projective curve X carries a natural divisor . We study the pullback of the divisor to the moduli stack , where P is a semisimple and simply connected group such that its Lie algebra Lie (P) is a maximal conformal subalgebra of Lie (E₈). We show that the divisor induces "Strange Duality"-type isomorphisms between the Verlinde spaces at level one of the following pairs of groups (SL (5), SL (5)), (Spin (8), Spin (8)), (SL (3), E₆), and (SL (2), E₇).

Isotropic Curvature and the Ricci Flow

Nguyen, H. T. — Sat, 12 Sep 2009 04:14:26 PDT

In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to four. In order to do so, we introduce a new technique to prove that curvature functions defined on the orthonormal frame bundle are preserved by the Ricci flow. At a minimum of such a function, we compute the first and second derivatives in the frame bundle. Using an algebraic construction, we can use these expressions to show that the nonlinearity is positive at a minimum. Finally, using the maximum principle, we can show that the Ricci flow preserves the cone of curvature operators with nonnegative isotropic curvature.

Elementary Combinatorics of the HOMFLYPT Polynomial

Chmutov, S., Polyak, M. — Thu, 03 Sep 2009 07:32:19 PDT

We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.

Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian

Ferapontov, E. V., Hadjikos, L., Khusnutdinova, K. R. — Thu, 03 Sep 2009 01:25:04 PDT

We investigate integrable second-order equations of the form which typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples include the Boyer–Finley equation , the potential form of the dispersionless Kadomtsev–Petviashvili (dKP) equation , the dispersionless Hirota equation , etc. The integrability is understood as the existence of an infinity of hydrodynamic reductions. We demonstrate that the natural equivalence group of the problem is isomorphic to Sp(6), revealing a remarkable correspondence between differential equations of the above type and hypersurfaces of the Lagrangian Grassmannian. We prove that the moduli space of integrable equations of the dispersionless Hirota type is 21-dimensional, and the action of the equivalence group Sp(6) on the moduli space has an open orbit.

Wegner Estimate and Level Repulsion for Wigner Random Matrices

Erdos, L., Schlein, B., Yau, H.-T. — Sat, 29 Aug 2009 04:41:40 PDT

We consider N x N Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order 1/ N. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales >> N^–1. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result [6]. We then show a Wegner estimate, i.e., that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.

The Infinitesimal Hopf Algebra and the Operads of Planar Forests

Foissy, L. — Thu, 27 Aug 2009 20:32:09 PDT

We introduce two operads based on the set of planar forests. With its usual product and two other products defined by different types of graftings, the algebra of planar rooted trees becomes an algebra over these operads. The compatibility with the infinitesimal coproduct of and these structures is studied. As an application, an inductive way of computing the dual basis of for its infinitesimal pairing is given. Moreover, three Cartier–Quillen–Milnor–Moore theorems are given for the operads of planar forests and a rigidity theorem for one of them.

Sigma Function as A Tau Function

Nakayashiki, A. — Wed, 26 Aug 2009 23:37:02 PDT

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n, s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the multivariate sigma function. This expression is in turn used to prove fundamental properties on the series expansion of the sigma function established in a previous article in a different method.

On the Special Values of Certain Rankin-Selberg L-Functions and Applications to Odd Symmetric Power L-Functions of Modular Forms

Raghuram, A. — Mon, 24 Aug 2009 03:45:18 PDT

We prove an algebraicity result for the central critical value of certain Rankin–Selberg L-functions for GL _n x GL _n–1. This is a generalization and refinement of the results of Harder [14], Kazhdan, Mazur, and Schmidt [23], and Mahnkopf [29]. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands' functoriality, one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blasius and Panchishkin on twisted L-values in some cases. These results, as in the above works, are, in general, based on a nonvanishing hypothesis on certain archimedean integrals.

Well-posedness for the Generalized Benjamin-Ono Equations with Arbitrary Large Initial Data in the Critical Space

Vento, S. — Wed, 19 Aug 2009 22:47:02 PDT

We prove the local well-posedness of the generalized Benjamin–Ono equations , k ≥ 4, in the scaling invariant spaces where s_k = 1/2 – 1/ k. Our results also hold in the nonhomogeneous spaces . In the case k = 3, local well-posedness is obtained in , s > 1/3.

Hodge Cohomology of Etale Nori Finite Vector Bundles

Cuong, T. T. — Wed, 19 Aug 2009 02:08:07 PDT

Étale Nori finite vector bundles are the bundles defined by representations of a finite étale group scheme in the usual way. In this article, we show that in many cases the dimensions of the Hodge cohomology groups of such a vector bundle and a twist of it by an automorphism of the ground field are the same. This generalizes to the higher rank case the result of Pink–Roessler [13, Proposition 3.5].

On the Geometric Origin of the Bi-Hamiltonian Structure of the Calogero-Moser System

Bartocci, C., Falqui, G., Mencattini, I., Ortenzi, G., Pedroni, M. — Tue, 18 Aug 2009 00:37:09 PDT

We show that the bi-Hamiltonian structure of the rational n-particle (attractive) Calogero–Moser system can be obtained by means of a double projection from a very simple Poisson pair on the cotangent bundle of . The relation with the Lax formalism is also discussed.

Nuclear Dimension and the Corona Factorization Property

Ng, P. W., Winter, W. — Mon, 17 Aug 2009 23:26:18 PDT

We show that the stabilizations of sufficiently noncommutative separable unital C^*-algebras with finite nuclear dimension have the corona factorization property.

Automorphic Properties of Generating Functions for Generalized Rank Moments and Durfee Symbols

Bringmann, K., Lovejoy, J., Osburn, R. — Sun, 16 Aug 2009 00:33:50 PDT

We define two-parameter generalizations of two combinatorial constructions of Andrews: the kth symmetrized rank moment and the k-marked Durfee symbol. We prove that three specializations of the associated generating functions are so-called quasimock theta functions, while a fourth specialization gives quasimodular forms. We then define a two-parameter generalization of Andrews' smallest parts function and note that this leads to quasimock theta functions as well. The automorphic properties are deduced using q-series identities relating the relevant generating functions to known mock theta functions.

The Generalized Injectivity Conjecture for Classical p -Adic Groups

Hanzer, M. — Thu, 13 Aug 2009 01:39:04 PDT

We prove that the conjecture posed by Casselman and Shahidi, which says that a generic subquotient of a generic standard representation of a connected quasi-split reductive p-adic group is a subrepresentation ("generalized injectivity conjecture"), is valid in the case of classical groups with the generalized rank one standard representation. The generalized rank one case that we prove in this article is the key step in proving the claim in general, for every generic standard representation of a classical group.

Kirillov-Reshetikhin Conjecture: The General Case

Hernandez, D. — Wed, 12 Aug 2009 09:30:21 PDT

We prove the Kirillov–Reshetikhin (KR) conjecture in the general case: for all twisted quantum affine algebras, we prove that the characters of KR modules solve the twisted Q-system [20] and we get explicit formulas for the character of their tensor products (the untwisted case was treated in [16, 33, 34]). The proof provides several new developments for the representation theory of twisted quantum affine algebras, in particular on the twisted Frenkel–Reshetikhin q-characters that we define (expected in [11, 12]) and on the parameterization of simple finite dimensional representations without Drinfeld presentation. We also prove the twisted T-system [30]. As an application, we get a proof of explicit (q)-characters formulas conjectured in various papers. We prove an isomorphism of Grothendieck rings between a twisted quantum affine algebra and the corresponding simply-laced quantum affine algebra.

Integration over Tropical Plane Curves and Ultradiscretization

Iwao, S. — Sun, 09 Aug 2009 00:59:11 PDT

In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemann surfaces.

Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

Beresnevich, V., Velani, S. — Wed, 05 Aug 2009 07:52:58 PDT

Let denote the set of -approximable points in . Under the assumption that the approximating function is monotonic, the classical Khintchine–Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of . The famous Duffin–Schaeffer counterexample shows that the monotonicity assumption on is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n ≥ 3 (Schmidt) or when n = 1 and m ≥ 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine–Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

Conformal Dimension: Cantor Sets and Fuglede Modulus

Hakobyan, H. — Wed, 05 Aug 2009 09:48:05 PDT

In this paper, we give several conditions for a space to be minimal for conformal dimension. We show that there are sets of zero length and conformal dimension 1, thus answering a question of Bishop and Tyson. Another sufficient condition for minimality is given in terms of a modulus of a system of measures in the sense of Fuglede [7]. It implies in particular that if is minimal for conformal dimension and supports a measure such that for every > 0 there is a constant 0 < C < such that C^–1 r¹⁺ ≤ (E B(x, r)) ≤ C r^1–, then X x Y is minimal for conformal dimension for every compact Y.

Homological Resonances for Hamiltonian Diffeomorphisms and Reeb Flows

Ginzburg, V. L., Kerman, E. — Tue, 28 Jul 2009 06:45:17 PDT

We show that whenever a Hamiltonian diffeomorphism or a Reeb flow has a finite number of periodic orbits, the mean indices of these orbits must satisfy a resonance relation, provided that the ambient manifold meets some natural requirements. In the case of Reeb flows, this leads to simple expressions (purely in terms of the mean indices) for the mean Euler characteristics. These are invariants of the underlying contact structure which are capable of distinguishing some contact structures that are homotopic but not diffeomorphic.

The Metric Theory of p-Adic Approximation

Haynes, A. K. — Fri, 24 Jul 2009 07:33:01 PDT

Metric Diophantine approximation in its classical form is the study of how well almost all real numbers can be approximated by rationals. There is a long history of results which give partial answers to this problem, but still there are questions which remain unknown. The Duffin–Schaeffer conjecture is an attempt to answer all of these questions in full, and it has withstood more than 50 years of mathematical investigation. In this paper, we establish a strong connection between the Duffin–Schaeffer conjecture and its p-adic analog. Our main theorems are transfer principles which allow us to go back and forth between these two problems. We prove that if the variance method from probability theory can be used to solve the p-adic Duffin–Schaeffer conjecture for even one prime p, then almost the entire classical Duffin–Schaeffer conjecture would follow. Conversely, if the variance method can be used to prove the classical conjecture, then the p-adic conjecture is true for all primes. Furthermore, we are able to unconditionally and completely establish the higher dimensional analog of this conjecture in which we allow simultaneous approximation in any finite number and combination of real and p-adic fields, as long as the total number of fields involved is greater than one. Finally, by using a mass transference principle for Hausdorff measures, we are able to extend all of our results to their corresponding analogs with Haar measures replaced by the Hausdorff measures associated with arbitrary dimension functions.

Symplectic Forms and Cohomology Decomposition of almost Complex Four-Manifolds

Draghici, T., Li, T.-J., Zhang, W. — Wed, 22 Jul 2009 07:19:54 PDT

For any compact almost complex manifold (M, J), the last two authors [8] defined two subgroups H⁺_J(M), H^–_J(M) of the degree 2 real de Rham cohomology group . These are the sets of cohomology classes which can be represented by J-invariant, respectively, J-antiinvariant real 2-forms. In this paper, it is shown that in dimension 4 these subgroups induce a cohomology decomposition of . This is a specifically four-dimensional result, as it follows from a recent work of Fino and Tomassini [6]. Some estimates for the dimensions of these groups are also established when the almost complex structure is tamed by a symplectic form and an equivalent formulation for a question of Donaldson is given.

Combinatorial R-Matrices for Kirillov-Reshetikhin Crystals of Type D(1)n, B(1)n, A(2)2n-1

Okado, M., Sakamoto, R. — Mon, 20 Jul 2009 09:03:09 PDT

We calculate the image of the combinatorial R-matrix for any classical highest weight element in the tensor product of Kirillov–Reshetikhin crystals B^r,k B^1,l of type D⁽¹⁾_n, B⁽¹⁾_n, A⁽²⁾_2n–1. The notion of ±-diagrams is effectively used for the identification of classical highest weight elements in B^1,l B^r,k.