Robert M. Young
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View article: Vertical curves and vertical fibers in the Heisenberg group
Vertical curves and vertical fibers in the Heisenberg group Open
Let $\mathbb H$ denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in $\mathbb H$ and fibers of maps $\mathbb H \to \mathbb R^2$ from a metric perspective. We say that a set in $\mathbb H$ is a vertical …
View article: Ergodic maps and the cohomology of nilpotent Lie groups
Ergodic maps and the cohomology of nilpotent Lie groups Open
In this paper, we study how the cohomology of nilpotent groups is affected by Lipschitz maps. We show that, given a smooth Lipschitz map $f$ between two simply-connected nilpotent Lie groups $G$ and $H$, there is a map $ψ$ that induces an …
View article: Construction of fillings with prescribed Gaussian image and applications
Construction of fillings with prescribed Gaussian image and applications Open
We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the barycenter of the prescribed measure, considered as a…
View article: Undistorted fillings in subsets of metric spaces
Undistorted fillings in subsets of metric spaces Open
Lipschitz k-connectivity, Euclidean isoperimetric inequalities, and coning inequalities all measure the difficulty of filling a k-dimensional cycle in a space by a (k+1)-dimensional object. In many cases, such as Banach spaces and CAT(0) s…
View article: Constructing Hölder maps to Carnot groups
Constructing Hölder maps to Carnot groups Open
In this paper, we construct Hölder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group . Pansu and Gromov [Carnot-Carathéodory spaces seen from within, Birkhäuser, Basel, 1996] observed that any surfa…
View article: The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups
The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups Open
We show that the β-numbers of intrinsic Lipschitz graphs of Heisenberg groups ℍn {\mathbb{H}_{n}} are locally Carleson integrable when n≥2 {n\geq 2} . Our main bound uses a novel slicing argument to decompose intrinsic Lipschitz graphs int…
View article: Foliated corona decompositions
Foliated corona decompositions Open
We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this is…
View article: Filling functions of arithmetic groups
Filling functions of arithmetic groups Open
The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated …
View article: Dehn functions and Hölder extensions in asymptotic cones
Dehn functions and Hölder extensions in asymptotic cones Open
The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying…
View article: Vertical perimeter versus horizontal perimeter
Vertical perimeter versus horizontal perimeter Open
The discrete Heisenberg group $\mathbb{H}_{\mathbb{Z}}^{2k+1}$ is the group generated by $a_1,b_1,\ldots,a_k,b_k,c$, subject to the relations $[a_1,b_1]=\ldots=[a_k,b_k]=c$ and $[a_i,a_j]=[b_i,b_j]=[a_i,b_j]=[a_i,c]=[b_i,c]=1$ for every di…
View article: The integrality gap of the Goemans--Linial SDP relaxation for Sparsest Cut is at least a constant multiple of $\sqrt{\log n}$
The integrality gap of the Goemans--Linial SDP relaxation for Sparsest Cut is at least a constant multiple of $\sqrt{\log n}$ Open
We prove that the integrality gap of the Goemans--Linial semidefinite programming relaxation for the Sparsest Cut Problem is $Ω(\sqrt{\log n})$ on inputs with $n$ vertices.
View article: One-Step Fabrication of Microchannels with Integrated Three Dimensional Features by Hot Intrusion Embossing
One-Step Fabrication of Microchannels with Integrated Three Dimensional Features by Hot Intrusion Embossing Open
We build on the concept of hot intrusion embossing to develop a one-step fabrication method for thermoplastic microfluidic channels containing integrated three-dimensional features. This was accomplished with simple, rapid-to-fabricate imp…
View article: Erratum: Lipschitz connectivity and filling invariants in solvable groups and buildings
Erratum: Lipschitz connectivity and filling invariants in solvable groups and buildings Open
This note corrects some omissions in Section 2 of the paper “Lipschitz connectivity and filling invariants in solvable groups and buildings” [Geom. Topol. 18 (2014) 2375–2417].
View article: Examination of the temperature dependent electronic behavior of GeTe for switching applications
Examination of the temperature dependent electronic behavior of GeTe for switching applications Open
The DC and RF electronic behaviors of GeTe-based phase change material switches as a function of temperature, from 25 K to 375 K, have been examined. In its polycrystalline (ON) state, GeTe behaved as a degenerate p-type semiconductor, exh…