Some combinatorial conjectures for Jack polynomials

We present several combinatorial conjectures related to the expansion of Jack polynomials in terms of power sums.     

Some geometric conjectures in harmonic function theory

We prove some results on the geometry of the level sets of harmonic functions, particularly regarding their ‘oscillation’ and ‘pinching’ properties. These results allow us to tackle three recent conjectures due to De Carli and Hudson (Bull London Math Soc 42:83–95, 2010). Our approach hinges on a combination of local constructions, methods from differential topology and global extension arguments.     

Some problems and conjectures in number theory

In this article, we shall discuss some interesting, viable, meaningful, applicable and productive conjectures and methods to deal with some fundamental results in the theory of numbers.     

Some combinatorial conjectures for shifted Jack polynomials

We present several combinatorial conjectures related to Jack generalized binomial coefficients, or equivalently to shifted Jack polynomials. We prove these conjectures when the degree of these polynomials is 5.     

Some conjectures aboutL p norms of k-plane transforms

For 1≤k≤n-1, the k-plane transformT _k,n carries functionsf defined onR ⁿ to functionsT _k,n f defined on the set of affine k-planes inR ⁿ . It is known thatT _k,n mapsL ^p intoL ^q for certain values ofp andq. In this article we formulate conjectures for the exact values of the norm of theT _k,n , and state also a conjecture asserting that theL ^q norm ofT _k,n f changes monotonically whenf is replaced by its symmetric decreasing rearrangement. Conferenza tenuta il 6 marzo 1997 The first author was supported in part by NSF grants DMS-9206319 and DMS-9501293. The second author was supported in part by NSF grant DMS-9500840     

Some unsolvable conjectures in finite p-groups

We survey some unsolvable conjectures in finite p-groups and their research progress.     

A new invariant of stable equivalences of Morita type

It was proved in an earlier paper by the author that the Hochschild cohomology algebras of self-injective algebras are invariant under stable equivalences of Morita type. In this note we show that the orbit algebra of a self-injective algebra (considered as an --bimodule) is also invariant under stable equivalences of Morita type, where the orbit algebra is the algebra of all stable --bimodule morphisms from the non-negative Auslander-Reiten translations of to .

     

Some conjectures on modular representations of affine sl_2 and Virasoro algebra

We conjecture an explicit bound on the prime characteristic of a field, under which the Weyl modules of affine sl_2 and the minimal series modules of Virasoro algebra remain irreducible, and Goddard-Kent-Olive coset construction for affine sl_2 is valid.     

Category equivalences of clones

This note was inspired by McKenzie's recent paper [8] whose main result is a characterization of categorical equivalence (for short category equivalence) between varieties. Here we use this result to define category equivalent clones and to describe all clones category equivalent to maximal clones. If one is interested in maximal clones only up to category equivalence it is sufficient to consider very simple maximal clones, essentially the same as considered in Jablonski's early papers about maximal clones [6]. The results we have found encourage us in our feeling that category equivalence could be a useful tool in clone theory.Dedicated to the memory of Alan Day.Presented by J. Sichler.     

Stable equivalences of graded algebras

We extend the notion of stable equivalence to the class of locally finite graded algebras. For such an algebra Λ, we focus on the Krull–Schmidt category gr_Λ of finitely generated -graded Λ-modules with degree 0 maps, and the stable category obtained by factoring out those maps that factor through a graded projective module. We say that Λ and Γ are graded stably equivalent if there is an equivalence that commutes with the grading shift. Adapting arguments of Auslander and Reiten involving functor categories, we show that a graded stable equivalence α commutes with the syzygy operator (where defined) and preserves finitely presented modules. As a result, we see that if Λ is right noetherian (resp. right graded coherent), then so is any graded stably equivalent algebra. Furthermore, if Λ is right noetherian or k is artinian, we use almost split sequences to show that a graded stable equivalence preserves finite length modules. Of particular interest in the nonartinian case, we prove that any graded stable equivalence involving an algebra Λ with socΛ=0 must be a graded Morita equivalence.     

Coarse equivalences of Euclidean buildings

We prove the following rigidity results. Coarse equivalences between metrically complete Euclidean buildings preserve spherical buildings at infinity. If all irreducible factors have dimension at least two, then coarsely equivalent Euclidean buildings are isometric (up to scaling factors); if in addition none of the irreducible factors is a Euclidean cone, then the isometry is unique and has finite distance from the coarse equivalence.     

Adjoint functors and equivalences of subcategories

For any left R-module P with endomorphism ring S, the adjoint pair of functors P⊗_S− and Hom_R(P,−) induce an equivalence between the categories of P-static R-modules and P-adstatic S-modules. In particular, this setting subsumes the Morita theory of equivalences between module categories and the theory of tilting modules. In this paper we consider, more generally, any adjoint pair of covariant functors between complete and cocomplete Abelian categories and describe equivalences induced by them. Our results subsume the situations mentioned above but also equivalences between categories of comodules.     

Koszul duality and equivalences of categories

Let and be dual Koszul algebras. By Positselski a filtered algebra with gr is Koszul dual to a differential graded algebra . We relate the module categories of this dual pair by a Hom adjunction. This descends to give an equivalence of suitable quotient categories and generalizes work of Beilinson, Ginzburg, and Soergel.

     

The solution of Kato's conjectures

In this Note, we announce the solution of Kato's conjectures about the domain of square roots of differential elliptic operators in the Euclidean space or a Lipschitz domain.