In this article, we prove a Legendrian Whitney trick which allows for the removal of intersections between codimension-two contact submanifolds and submanifolds, assuming such smooth cancellation is possible. This technique applied to show existence h-principle embeddings with prescribed structure.

The flag Whitney numbers (also referred to as the flag f -numbers) of a geometric lattice count the number of chains of the lattice with elements having specified ranks. We give a collection of inequalities which imply all the linear inequalities satisfied by the flag Whitney numbers of rank 3 geometric lattices. We further describe the smallest closed convex set containing the flag Whitney num...

A fundamental theorem of Whitney from 1933 asserts that 2-connected graphs $G$ and $H$ are 2-isomorphic, or equivalently, their cycle matroids isomorphic if only can be transformed into by a series operations called switches. In this paper we consider the quantitative question arising Whitney's theorem: Given two 2-isomorphic graphs, transform one another applying at most $k$ switches? This pro...

These introductory notes on Whitney towers in 4-manifolds, as developed collaboration with Jim Conant and Peter Teichner, are an expansion of three expository lectures given at the Winter Braids X conference February 2020 Pisa, Italy. Topics presented include local manipulations surfaces 4–space, fundamental definitions related to their associated trees, geometric Jacobi identities, classificat...

We introduce a natural partial order ≤ in structurally natural finite subsets the cobweb prefabs sets recently constructed by the present author. Whitney numbers of the second kind of the corresponding subposet which constitute Stirling-like numbers' triangular array-are then calculated and the explicit formula for them is provided. Next-in the second construction-we endow the set sums of prefa...