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We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.
We study modules with 1-dimensional socle for preprojective algebras for type A quivers. In particular, we classify such modules, determine all homomorphisms between them, and then explain how they may be used to describe the components of Lusztig quiver varieties.
We introduce a moduli functor for varieties whose tropicalization realizes a given weighted fan and show that this functor is an algebraic space in general, and is represented by a scheme when the associated toric variety is quasiprojective. We study the geometry of these tropical realization spaces for the matroid fans studied by Ardila and Klivans, and show that the tropical realization space of...
In previous work we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. Here we present an extract of the theory containing...
This survey of methods surrounding lattice point methods for binomial ideals begins with a leisurely treatment of the geometric combinatorics of binomial primary decomposition. It then proceeds to three independent applications whose motivations come from outside of commutative algebra: hypergeometric systems, combinatorial game theory, and chemical dynamics. The exposition is aimed at students and...
In the 1980’s, work of Green and Lazarsfeld (Invent. Math., 83, 1 (1985), 73–90; Compositio Math., 67, 3 (1988), 301–314), helped to uncover the beautiful interplay between the geometry of the embedding of a curve and the syzygies of its defining equations. Similar results hold for the first secant variety of a curve, and there is a natural conjectural picture extending to higher secant varieties...
The tangent space to a jet manifold at a point has a module structure; this fact allows us to endow the Cartan subspace with a canonical bracket, a point-wise definition of its curvature. This bracket is related with the Spencer differential and an algebraic proof of the criterion on formal integrability given by Kruglikov and Lychagin is outlined. On the other hand, we define the characteristic...
This article reviews some recent theoretical results about the structure of Darboux integrable differential systems and their relationship with symmetry reduction of exterior differential systems. The symmetry reduction representation of Darboux integrable equations is then used to derive some new and unusual transformations.
We give an account on Otto’s geometrical heuristics for realizing, on a compact Riemannian manifold M, the L2 Wasserstein distance restricted to smooth positive probability measures, as a Riemannian distance. The Hilbertian metric discovered by Otto is obtained as the base metric of a Riemannian submersion with total space, the group of diffeomorphisms of M equipped with the Arnol’d metric,...
Many constructions in mathematics give unreasonably nice results. In particular much compatible structure tends to imply that these structures are very regular. Also many counterexamples have nice underlying structures. This paper is a first attempt to analyze these phenomena.
We consider quantizations and braidings of modules with grading by an abelian group. In particular we investigate modules with grading by Zn and the algebra of polynomials in n variables. We find quantizations of this algebra and quantization of its differential structures. Exploiting the fact that the electromagnetic field tensor can be described by a curvature, the quantizations...
A bridge between Lie symmetry groups for differential equations and Galois groups for algebraic equations is suggested. It is based on calculation of Lie symmetries for algebraic equations and their restriction of the roots of the equations under consideration. The approach is illustrated by several examples. An alternative representation of Lie symmetries, called the Galois representation, is provided...
Focal systems are a generalization to the case of Pfaffian system with characteristics of the classical notion of focal curves for first-order scalar partial diffeerntial equations. We show how focal systems can be used to prove local normal form results for second-order scalar hyperbolic equations in the plane. We also illustrate their use in integration methods for first-order equations.
We sketch out a new geometric framework to construct Hamiltonian operators for generic, non-evolutionary partial differential equations. Examples on how the formalism works are provided for the KdV equation, Camassa-Holm equation, and Kupershmidt’s deformation of a bi-Hamiltonian system.
In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the second order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.
In the current paper we present a survey of our results on classification of the Monge-Ampère equations and operators with two independent variables. We use Lychagin’s approach to such equations.
I discuss integrable systems and their solutions arising in the context of supersymmetric gauge theories and topological string models. For the simplest cases these are particular singular solutions to the dispersionless KdV and Toda systems, and they produce in most straightforward way the generating functions for the Gromov-Witten classes, including well-known intersection and Hurwitz numbers, in...
In this paper we review our results on the quantization of a rigid body. The fact that the configuration space is not simply connected yields two inequivalent quantizations. One of the quantizations allows us to recover classically double-valued wave functions as single valued sections of a non-trivial complex line bundle. This reopens the problem of a physical interpretation of these wave functions.
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