Related papers: The small growth invariants of Goursat distributio…
This is the first 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…
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…
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…
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…
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…
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 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…
Let $\ell$ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell$-towers of multigraphs. In this context, growth patterns are realized by certain…
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 study positive definite kernels pulled back along a finite family of self-maps under a subinvariance inequality for the associated branching operator. Iteration produces an increasing kernel tower with defect kernels. Under diagonal…
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…
We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of $\mathbf{PGL}_n$ over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a…
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 prove a correspondence between Donaldson-Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov-Witten invariants of holomorphic symplectic cluster varieties. The proof relies on…
We consider the multi-time correlation and covariance structure of a random surface growth with a wall introduced in arXiv:0904.2607. It is shown that the correlation functions associated with the model along space-like paths have…
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.
It is well known that the compactifications of the canonical contact systems living on real jet spaces $J^{\,r}(1,1)$, $r \ge 2$, are locally universal Goursat distributions, $\Delta^r$, living on compact manifolds (called Goursat monsters)…
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…