A parametric approach to correspondence analysis

We compare correspondence analysis (CA) and the alternative approach using Hellinger distance (HD), for representing categorical data in a contingency table. As both methods may be appropriate, we introduce a parameter and define a generalized version of correspondence analysis (GCA) which contains CA and HD as particular cases. Comparison with alternative approaches are performed. We propose a coefficient which globally measures the similarity between CA and GCA, which can be decomposed into several components, one component for each principal dimension, indicating the contribution of the dimensions on the difference between both representations. Two criteria for choosing the best value of the parameter are proposed.     

The Local gabriel correspondence

The first part of the present work introduces a new notion of ring of quotients by any filter of one sided ideals. The construction is a generalization of the one by Gabriel which works for idempotent filters. The second part is devoted to an extension of crossed action of Hopf algebras to such a ring of quotients. The last part comprises an application of this machinery to the question about crossed products being semiprime Goldie.     

On Jacquet-Langlands isogeny over function fields

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.     

Local theta correspondence for small unitary groups

In this paper we give an explicit parameterization of the local theta correspondence of supercuspidal representations for the reductive dual pairs , , , and of unitary groups over a nonarchimedean local field of odd residue characteristic.

     

A geometric Jacquet-Langlands correspondence for <Emphasis Type="Italic">U</Emphasis>(2) Shimura varieties

Let G be a unitary group over ℚ, associated to a CM-field F with totally real part F ⁺, with signature (1, 1) at all the archimedean places of F ⁺. Under certain hypotheses on F ⁺, we show that Jacquet-Langlands correspondences between certain automorphic representations of G and representations of a group G′ isomorphic to G except at infinity can be realized in the cohomology of Shimura varieties attached to G and G′.     

Local Langlands correspondence for classical groups and affine Hecke algebras

Mathematische Zeitschrift - Using the results of Colette Moeglin on the representations of p-adic classical groups (based on methods of James Arthur) and its relation with representations of affine...     

Local theta correspondence and minimalK-types of positive depth

In this paper we prove that two irreducible admissible representations of positive depth paired by the theta correspondence over ap-adic field have unrefined minimalK-types paired by the orbit correspondence. An application of our main result is that a positive depth character of a unitary group occurs as late as possible in the theta correspondence.     

Local minimizers of the Ginzburg-Landau functional with prescribed degrees

We consider, in a smooth bounded multiply connected domain D⊂R², the Ginzburg-Landau energy subject to prescribed degree conditions on each component of ∂D. In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: Eε(u) has, in domains D with 2,3,… holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control.     

Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence

The Richardson variety X _α ^γ in the Grassmannian is defined to be the intersection of the Schubert variety X ^γ and opposite Schubert variety X _α . We give an explicit Gröbner basis for the ideal of the tangent cone at any T-fixed point of X _α ^γ , thus generalizing a result of Kodiyalam-Raghavan (J. Algebra 270(1):28–54, 2003) and Kreiman-Lakshmibai (Algebra, Arithmetic and Geometry with Applications, 2004). Our proof is based on a generalization of the Robinson-Schensted-Knuth (RSK) correspondence, which we call the bounded RSK (BRSK). We use the Gröbner basis result to deduce a formula which computes the multiplicity of X _α ^γ at any T-fixed point by counting families of nonintersecting lattice paths, thus generalizing a result first proved by Krattenthaler (Sém. Lothar. Comb. 45:B45c, 2000/2001; J. Algebr. Comb. 22:273–288, 2005).     

Correspondance de Jacquet-Langlands pour les corps locaux de caractéristique non nulle

In this article we prove the Jacquet-Langlands local correspondence in non-zero characteristic. Let F be a local field of non-zero charactersitic and G′ an inner form of GL_n(F); then, following [17], we prove relations between the representation theory of G′ and the representation theory of an inner form of GL_n(L), where L is a local field of zero characteristic close to F. The proof of the Jacquet-Langlands correspondence between G′ and GL_n(F) is done using the above results and ideas from the proof by Deligne, Kazhdan and Vignéras [10] of the zero characteristic case. We also get the following, already known in zero characteristic: orthogonality relations for G′, inequality involving conductor and level for representations of G′ and finiteness for automorphic cuspidal representations with fixed component at almost every place for an inner form of GL_n over a global field of non-zero characteristic.     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Quasi-minimal enumeration degrees and minimal Turing degrees

We show that there exists a set A such that A has quasi-minimal enumeration degree, and there are uncountably many sets B such that A is enumeration reducible to B and B has minimal Turing degree. Answering a related question raised by Solon, we also show that there exists a nontotal enumeration degree which is not e-hyperimmune.During the preparation of this paper, Slaman was partially supported by the HCM European Program no. ERBCHRXCT930415 (while he was visiting the University of Leeds), by NSF Grant DMS-9500878 and was a CNR Visiting Professor at the University of Siena.The preparation of this paper was partially supported by the HCM European Program no. ERBCHRXCT930415 and by MURST 60%.     

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander-Reiten theory [O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22-50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195-222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749-770].     

McKay correspondence and orbifold equivalence

We prove that a pair of singularities related by a transformation arising from the McKay correspondence are orbifold equivalent. From this we deduce a McKay type category equivalence for the matrix factorization categories.