Generalized Lame operators

We introduce a class of multidimensional Schr\"odinger operators with elliptic potential which generalize the classical Lam\'e operator to higher dimensions. One natural example is the Calogero--Moser operator, others are related to the root systems and their deformations. We conjecture that these operators are algebraically integrable, which is a proper generalization of the finite-gap property of the Lam\'e operator. Using earlier results of Braverman, Etingof and Gaitsgory, we prove this under additional assumption of the usual, Liouville integrability. In particular, this proves the Chalykh--Veselov conjecture for the elliptic Calogero--Moser problem for all root systems. We also establish algebraic integrability in all known two-dimensional cases. A general procedure for calculating the Bloch eigenfunctions is explained. It is worked out in detail for two specific examples: one is related to B_2 case, another one is a certain deformation of the A_2 case. In these two cases we also obtain similar results for the discrete versions of these problems, related to the difference operators of Macdonald--Ruijsenaars type.Comment: 38 pages, latex; in the new version a reference was adde

An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring $R$, i.e., a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over $R$. For a finitely generated maximal ideal $\mathfrak{m}$ in a commutative ring $R$ we show how solving (in)homogeneous linear systems over $R_{\mathfrak{m}}$ can be reduced to solving associated systems over $R$. Hence, the computability of $R$ implies that of $R_{\mathfrak{m}}$. As a corollary we obtain the computability of the category of finitely presented $R_{\mathfrak{m}}$-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a by-product, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.Comment: Fixed a typo in the proof of Lemma 4.3 spotted by Sebastian Posu

On Klein's Icosahedral Solution of the Quintic

We present an exposition of the icosahedral solution of the quintic equation first described in Klein's classic work "Lectures on the icosahedron and the solution of equations of the fifth degree". Although we are heavily influenced by Klein we follow a slightly different approach which enables us to arrive at the solution more directly.Comment: 29 pages, 5 figure

Galois differential algebras and categorical discretization of dynamical systems

A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. The integrable maps obtained share with their continuous counterparts a large class of solutions and, in the linear case, the Picard-Vessiot group.Comment: 19 pages (examples added

Moduli of products of stable varieties

We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties, (b) this map is always finite \'etale, and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension 1. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.Comment: 26 pages, suggestions and comments are welcome

Interactions between non-commutative algebraic geometry and skew PBW extensions

Abstract. We study some relations and interactions between non-commutative algebraic geometry and the skew PBW extensions. For this we will introduce a new class of noncommutative rings, the semi-graded rings, and for them we will prove a generalization of the famous Serre-Artin-Zhang-Verevkin theorem. Semi-graded rings extend not only skew PBW extension but also graded rings.Estudiamos algunas relaciones entre geometría algebraica no conmutativa y las extensiones PBW torcidas. Para esto, introducimos una nueva clase de anillos, los anillos semi-graduados, y para ellos demostraremos una versión generalizada del famoso teorema de Serre-Artin-Zhang-Verevkin. Los anillos semi-graduados no solo generalizan las extensiones PBW torcidas sino también los anillos graduados.Maestrí