On the Kauffman skein modules

 Let k be a subring of the field of rational functions in α, s which contains α ^±1 ,s ^±1 . Let M be a compact oriented 3-manifold, and let K(M) denote the Kauffman skein module of M over k. Then K(M) is the k-module freely generated by isotopy classes of framed links in M modulo the Kauffman skein relations. In the case of , the field of rational functions in α, s, we give a basis for the Kauffman skein module of the solid torus and a basis for the relative Kauffman skein module of the solid torus with two points on the boundary. We then show that K(S ¹ × S ² is freely generated by the empty link, i.e., . Received: 20 October 2001 / Revised version: 20 March 2002     

The Tate-Voloch conjecture for Drinfeld modules

We study the v-adic distance from the torsion of a Drinfeld module to an affine variety.     

On the standard modules conjecture

Let G be a quasi-split p-adic group. Under the assumption that the local coefficients C _ψ defined with respect to ψ-generic tempered representations of standard Levi subgroups of G are regular in the negative Weyl chamber, we show that the standard module conjecture is true, which means that the Langlands quotient of a standard module is generic if and only if the standard module is irreducible.     

A remark on the monomial conjecture and Cohen-Macaulay canonical modules

In this paper, some sufficient conditions for rings and modules to satisfy the monomial conjecture are given. A characterization of Cohen-Macaulay canonical modules is presented.

     

On the uniform boundedness conjecture for Drinfeld modules

 In analogy with the famous theorems of Mazur and Merel on the torsion subgroups of elliptic curves, one can formulate similar conjectures for the torsion points of Drinfeld modules. We prove some partial results for rank 2 Drinfeld 𝔽 _q [T]-modules, for example the uniform boundedness of the 𝔭-primary torsion. Received: 22 May 2001; in final form 4 September 2002 / Published online: 1 April 2003     

The mixed Littlewood conjecture for pseudo-absolute values

In this paper we study the mixed Littlewood Conjecture with pseudo-absolute values. We show that if $p$ is a prime and $\mathcal D $ is a pseudo-absolute value sequence satisfying mild conditions then $$\begin{aligned} \inf _{n\in \mathbb N } n|n|_p|n|_\mathcal D \Vert n\alpha \Vert =0\quad \text{ for } \text{ all }\,\,\alpha \in \mathbb R . \end{aligned}$$ Our proof relies on a measure rigidity theorem due to Lindenstrauss and lower bounds for linear forms in logarithms due to Baker and Wüstholz. We also deduce the answer to the related metric question of how fast the infimum above tends to zero, for almost every $\alpha $ .     

The Gelfand-Kirillov conjecture and Gelfand-Tsetlin modules for finite W-algebras

We address two problems with the structure and representation theory of finite W-algebras associated with general linear Lie algebras. Finite W-algebras can be defined using either Kostant's Whittaker modules or a quantum Hamiltonian reduction. Our first main result is a proof of the Gelfand-Kirillov conjecture for the skew fields of fractions of finite W-algebras. The second main result is a parameterization of finite families of irreducible Gelfand-Tsetlin modules using Gelfand-Tsetlin subalgebra. As a corollary, we obtain a complete classification of generic irreducible Gelfand-Tsetlin modules for finite W-algebras.     

Graded Betti numbers of Cohen-Macaulay modules and the multiplicity conjecture

We give conjectures on the possible graded Betti numbers ofCohen–Macaulay modules up to multiplication by positiverational numbers. The idea is that the Betti diagrams shouldbe non-negative linear combinations of pure diagrams. The conjecturesare verified in the cases where the structure of resolutionsis known, that is: for modules of codimension two, for Gorensteinalgebras of codimension three and for complete intersections.The motivation for proposing the conjectures comes from theMultiplicity conjecture of Herzog, Huneke and Srinivasan.     

The Van der Waerden conjecture for mixed discriminants

We prove that the mixed discriminant of doubly stochastic n-tuples of semidefinite hermitian n×n matrices is bounded below by and that this bound is uniquely attained at the n-tuple . This result settles a conjecture posed by R. Bapat in 1989. We consider various generalizations and applications of this result.     

The dimension vectors of indecomposable modules of cluster-tilted algebras and the Fomin-Zelevinsky denominators conjecture

The main result of this paper is that any two non-isomorphic indecomposable modules of a cluster-tilted algebra of finite representation type have different dimension vectors. As an application to cluster algebras of Types A,D,E, we give a proof of the Fomin-Zelevinsky denominators conjecture for cluster variables, namely, different cluster variables have different denominators with respect to any given cluster.     

Jacobson's conjecture and modules over fully bounded Noetherian rings

The object of this paper is to investigate finitely generated modules and injective modules over fully bounded Noetherian rings. Our main results on f.g. modules (Theorems 3.1 and 3.4) provide an analog of the Jordan-Holder Theorem and seem to be new even for commutative Noetherian rings. They imply the validity of Jacobson's conjecture for fully bounded Noetherian rings. The main result on injectives (Theorem 5.3) describes f.g. submodules of an indecomposible injective and shows how it can be built from indecomposible injectives over certain Artinian rings.     

Some cases of a conjecture on L-functions of twisted Carlitz modules

We prove two polynomial identies which are particular cases of a conjecture arising in the theory of L-functions of twisted Carlitz modules. This conjecture is stated in [6 Grishkov, A., Logachev, D. (2016). Resultantal varieties related to zeroes of L-functions of Carlitz modules. Finite Fields Appl. 38:116–176. Available at: arxiv.org/pdf/1205.2900.pdf.[Crossref], [Web of Science ®] , [Google Scholar], p. 153, 9.3, 9.4] and [8 Logachev, D., Zobnin, A. (2017). L-functions of Carlitz Modules, Resultantal Varieties and Rooted Binary Trees, arXiv:1607.06147v3 [math.AG]. [Google Scholar], p. 5, 0.2.4].     

Periodic knots and the skein polynomial

This research was supported by NSERC-No. A4034 (Canada) and Polish scientific grant RP 1.10.     

The Yang-Mills measure in the Kauffman bracket skein module

For each closed, orientable surface , we construct a local, diffeomorphism invariant trace on the Kauffman bracket skein module . The trace is defined when |t| is neither 0 nor 1, and at certain roots of unity. At t = − 1, the trace is integration against the symplectic measure on the SU(2) character variety of the fundamental group of . Received: June 2, 2000