相关论文: The structural invariants of Goursat distributions
This is the second of a pair of papers devoted to the local invariants of Goursat distributions. The study of these distributions naturally leads to a tower of spaces over an arbitrary surface, called the monster tower, and thence to…
A Goursat structure on a manifold of dimension n is a rank two distribution D such that dim D(i)=i+2, for i=0,...,n-2, where D(i) denotes the derived flag of D, which is defined by D(0)=D and D(i+1)=D(i)+[D(i),D(i)]. Goursat structures…
In this note we summarize our results from earlier work with the Monster Tower (A Monster Tower Approach to Goursat Multi-Flags). In particular, we give an overview of the problem of classifying the orbits within a tower of fibrations with…
The monster tower is a tower of spaces over a specified base; each space in the tower is a parameter space for curvilinear data up to a specified order. We describe and analyze a natural stratification of these spaces.
We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the…
Our study of Goursat distributions originates new types of $k$-contact distributions and Lie systems with applications. In particular, families of generators for Goursat distributions on $\mathbb{R}^4, \mathbb{R}^5$ and $\mathbb{R}^6$ give…
We consider here the problem of classifying orbits of an action of the dif- feomorphism group of 3-space on a tower of fibrations with P2-fibers that generalize the Monster Tower due to Montgomery and Zhitomirskii. As a corollary we give…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient…
In earlier work, we introduced the `Monster tower', a tower of fibrations associated to planar curves. We constructed an algorithm for classifying its points with respect to the equivalence relation generated by the action of the contact…
We consider the Goursat's (n+1)-webs of codimension one of two kinds on an n-dimensional manifold. They are characterized by the specific closed form equations or by two special relations between components of the torsion tensor of the web.…
We give an overview of results on irregular complex surfaces of general type, discussing in particular the distribution of the numerical invariants self-intersection of a canonical divisor and holomorphic Euler characteristic for the…
Germs of Goursat distributions can be classified according to a geometric coding called an RVT code. Jean (1996) and Mormul (2004) have shown that this coding carries precisely the same data as the small growth vector. Montgomery and…
Using the monster/Semple tower construction, we study the structure of the Cartan prolongation of the family $x_1x_2 = t$ of plane curves with nodal central member.
In this paper we discuss open problems concerning L^2-invariants focusing on approximation by towers of finite coverings.
In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective…
Regular variation of distributional tails is known to be preserved by various linear transformations of some random structures. An inverse problem for regular variation aims at understanding whether the regular variation of a transformed…
The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…
We define scrollar invariants of tropical curves with a fixed divisor of rank 1. We examine the behavior of scrollar invariants under specialization, and provide an algorithm for computing these invariants for a much-studied family of…
This is an expository paper in which we define projective GIT quotients and introduce toric varieties from this perspective. It is intended primarily for readers who are learning either invariant theory or toric geometry for the first time.