arXiv subjects

Robert Young

Publications and source records attributed to Robert Young.

At least 19 recordsLinked to original sources

arXiv2026

Area of H\"older curves and coarea formula on the Heisenberg group

We prove the coarea formula for Lipschitz maps from the subriemannian $n$th Heisenberg group $\mathbb H_n$ to $\mathbb R^{2n}$. Our result is new even when $n=1$ and provides the simplest vector-valued instance of the coarea formula in subriemannian geometry. This answers a question left open in the works of Magnani, Kozhevnikov, Magnani--Stepanov--Trevisan, and Julia--Nicolussi Golo--Vittone. The main difficulty of the proof is that a fiber of a $C^1_{\mathrm{H}}$ map $f: \mathbb H_n\to \mathbb R^{2n}$ is typically an unrectifiable curve. Its measure depends on the symplectic area of its projection to $\mathbb R^{2n}$. A bound on this area would imply the coarea formula, but examples of Kozhevnikov show that this area can be infinite or undefined. To overcome this, we introduce an integral that we use to define both the symplectic area of $\frac{1}{2}$--H\"older curves in $\mathbb R^{2n}$ and the symplectic area of projections of vertical curves in $\mathbb H_n$. Then, we give a geometric condition for this integral to converge. This yields, in addition, new results on the existence of the signed area of $\tfrac12$--H\"older planar curves that may be of independent interest. Finally, we use $\beta$--number estimates from the F\"assler--Orponen Dorronsoro Theorem to show that this geometric condition holds for almost every fiber.

math.MG↗

arXiv2026

$L_1$-distortion of Earth Mover Distances and Transportation Cost Spaces on High Dimensional Grids

We prove that the distortion of any embedding into $L_1$ of the transportation cost space or earth mover distance over a $d$-dimensional grid $\{1,\dots m\}^d$ is $\Omega(\log N)$, where $N$ is the number of vertices and the implicit constant is universal (in particular, independent of dimension). This lower bound matches the universal upper bound $O(\log N)$ holding for any $N$-point metric space. Our proof relies on a new Sobolev inequality for real-valued functions on the grid, based on random measures supported on dyadic cubes.

math.FA↗

arXiv2024

Vertical curves and vertical fibers in the Heisenberg group

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 curve if it satisfies a cone condition with respect to a homogeneous cone with axis $\langle Z \rangle$, the center of $\mathbb{H}$. This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally biH\"older equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions $\beta$ from the unit ball $B$ in $\mathbb{H}$ to $\mathbb{R}^2$. We show that, in contrast to the situation in Euclidean space, there are maps such that $\beta$ is arbitrarily close to the projection $\pi$ from $\mathbb{H}$ to the horizontal plane, but the average $H^2$ measure of a fiber of $\beta$ in $B$ is arbitrarily small.

math.MG↗

arXiv2024

Ergodic maps and the cohomology of nilpotent Lie groups

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 $\psi$ that induces an ergodic measure on the space of functions from $G$ to $H$. We call such maps ergodic maps. We show that when $\psi$ is an ergodic map, the pullback $\psi^*\omega$ of a differential form $\omega$ admits a well-defined amenable average $\overline{\psi^{*}}\omega$, and $\overline{\psi^*}$ is a homomorphism of cohomology algebras. In the case that $f$ is a quasi-isometry, the ergodic map $\psi$ is also a quasi-isometry, and $\overline{\psi^*}$ is an isomorphism. This lets us generalize and provide a simplified, self-contained proof of the theorem due to Shalom, Sauer, and Gotfredsen--Kyed that quasi-isometric nilpotent groups have isomorphic cohomology algebras.

math.GR↗

arXiv2024

Construction of fillings with prescribed Gaussian image and applications

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 measure on $\bigwedge^d \mathbb{R}^n$, is $0$) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov [Geom. funct. anal., 2004]; our work gives an explicit construction, which is also flexible to generalizations. For instance, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove the surprising fact that, for anisotropic integrands, polyconvexity is equivalent to quasiconvexity of the associated $Q$-integrands (that is, ellipticity for Lipschitz multigraphs) and to show that strict polyconvexity is necessary for the atomic condition to hold.

math.AP↗

arXiv2023

The strong geometric lemma in the Heisenberg group

We prove that in the first Heisenberg group, unlike Euclidean spaces and higher dimensional Heisenberg groups, the best possible exponent for the strong geometric lemma for intrinsic Lipschitz graphs is $4$ instead of $2$. Combined with earlier work from arXiv:2004.11447 and arXiv:2207.03013, our result completes the proof of the strong geometric lemma in Heisenberg groups. One key tool in our proof, and possibly of independent interest, is a suitable refinement of the foliated coronizations which first appeared in arXiv:2004.12522.

math.MG↗

arXiv2022

The Riesz tranform on intrinsic Lipschitz graphs in the Heisenberg group

We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On the way, we use our construction to show that the strong geometric lemma fails in $\mathbb{H}$ for all exponents in $[2,4)$. Our results are in stark contrast to two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in $\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for $m\in (0,n)$.

math.MG↗

arXiv2022

Establishing the Capabilities of the Murchison Widefield Array as a Passive Radar for the Surveillance of Space

This paper describes the use of the Murchison Widefield Array, a low-frequency radio telescope at a radio-quiet Western Australian site, as a radar receiver forming part of a continent-spanning multistatic radar network for the surveillance of space. This paper details the system geometry employed, the orbit-specific radar signal processing, and the orbit determination algorithms necessary to ensure resident space objects are detected, tracked, and propagated. Finally, the paper includes the results processed after a short collection campaign utilising several FM radio transmitters across the country, up to a maximum baseline distance of over 2500 km. The results demonstrate the Murchison Widefield Array is able to provide widefield and persistent coverage of objects in low Earth orbit.

eess.SP↗

arXiv2022

Direct evaluation of progression or regression of disease burden in brain metastatic disease with Deep Neuroevolution

Purpose: A core component of advancing cancer treatment research is assessing response to therapy. Doing so by hand, for example as per RECIST or RANO criteria, is tedious, time-consuming, and can miss important tumor response information; most notably, they exclude non-target lesions. We wish to assess change in a holistic fashion that includes all lesions, obtaining simple, informative, and automated assessments of tumor progression or regression. Due to often low patient enrolments in clinical trials, we wish to make response assessments with small training sets. Deep neuroevolution (DNE) can produce radiology artificial intelligence (AI) that performs well on small training sets. Here we use DNE for function approximation that predicts progression versus regression of metastatic brain disease. Methods: We analyzed 50 pairs of MRI contrast-enhanced images as our training set. Half of these pairs, separated in time, qualified as disease progression, while the other 25 images constituted regression. We trained the parameters of a relatively small CNN via mutations that consisted of random CNN weight adjustments and mutation fitness. We then incorporated the best mutations into the next generations CNN, repeating this process for approximately 50,000 generations. We applied the CNNs to our training set, as well as a separate testing set with the same class balance of 25 progression and 25 regression images. Results: DNE achieved monotonic convergence to 100% training set accuracy. DNE also converged monotonically to 100% testing set accuracy. Conclusion: DNE can accurately classify brain-metastatic disease progression versus regression. Future work will extend the input from 2D image slices to full 3D volumes, and include the category of no change. We believe that an approach such as our could ultimately provide a useful adjunct to RANO/RECIST assessment.

cs.NE↗

arXiv2021

Undistorted fillings in subsets of metric spaces

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

math.MG↗

arXiv2021

Constructing Hölder maps to Carnot groups

In this paper, we construct Hölder maps to Carnot groups equipped with a Carnot metric, especially the first Heisenberg group $\mathbb{H}$. Pansu and Gromov observed that any surface embedded in $\mathbb{H}$ has Hausdorff dimension at least 3, so there is no $α$-Hölder embedding of a surface into $\mathbb{H}$ when $α>\frac{2}{3}$. Züst improved this result to show that when $α>\frac{2}{3}$, any $α$-Hölder map from a simply-connected Riemannian manifold to $\mathbb{H}$ factors through a metric tree. In the present paper, we show that Züst's result is sharp by constructing $(\frac{2}{3}-ε)$-Hölder maps from $D^2$ and $D^3$ to $\mathbb{H}$ that do not factor through a tree. We use these to show that if $0<α< \frac{2}{3}$, then the set of $α$-Hölder maps from a compact metric space to $\mathbb{H}$ is dense in the set of continuous maps and to construct proper degree-1 maps from $\mathbb{R}^3$ to $\mathbb{H}$ with Hölder exponents arbitrarily close to $\frac{2}{3}$.

math.MG↗

arXiv2021

Area-minimizing ruled graphs and the Bernstein problem in the Heisenberg group

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab $\{-1<x<1\}$. We show that our condition is necessary by introducing a family of deformations of graphical strips based on varying a vertical curve. We show that it is sufficient by showing that strips satisfying the condition have monotone epigraphs. We use this condition to show that any area-minimizing ruled entire intrinsic graph in the Heisenberg group is a vertical plane and to find a boundary curve that admits uncountably many fillings by area-minimizing surfaces.

math.CA↗

arXiv2021

Foliated corona decompositions

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 isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.

math.MG↗

arXiv2020

Harmonic intrinsic graphs in the Heisenberg group

Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a calibration condition for an intrinsic Lipschitz graph to be energy-minimizing, construct energy-minimizing graphs with a variety of singularities, and prove a first variation formula for the energy of intrinsic Lipschitz graphs and piecewise smooth intrinsic graphs.

math.CA↗

arXiv2020

The strong geometric lemma for intrinsic Lipschitz graphs in Heisenberg groups

We show that the $β$--numbers of intrinsic Lipschitz graphs of Heisenberg groups $\mathbb{H}_n$ are locally Carleson integrable when $n \geq 2$. Our technique relies on a recent Dorronsoro inequality \cite{FO} as well as a novel slicing argument. A key ingredient in our proof is a Euclidean inequality bounding the $β$--number of a function on a cube of $\mathbb{R}^n$ using the $β$--number of the restriction of the function to codimension--1 slices of the cube.

math.MG↗

arXiv2020

Orbit Determination Before Detect: Orbital Parameter Matched Filtering for Uncued Detection

This paper presents a novel algorithm to incorporate orbital parameters into radar ambiguity function expressions by extending the standard ambiguity function to match Keplerian two-body orbits. A coherent orbital matched-filter will maximise a radar's sensitivity to objects in orbit, as well as provide rapid initial orbit determination from a single detection. This paper then shows how uncued detection searches can be practically achieved by incorporating radar parameters into the orbital search-space, especially for circular orbits. Simulated results are compared to results obtained from ephemeris data, showing that the orbital path determined by the proposed method, and the associated radar parameters that would be observed, match those derived from the ephemeris data.

eess.SP↗

arXiv2019

Improved Techniques for the Surveillance of the Near Earth Space Environment with the Murchison Widefield Array

In this paper we demonstrate improved techniques to extend coherent processing intervals for passive radar processing, with the Murchison Widefield Array. Specifically, we apply a two stage linear range and Doppler migration compensation by utilising Keystone Formatting and a recent dechirping method. These methods are used to further demonstrate the potential for the surveillance of space with the Murchison Widefield Array using passive radar, by detecting objects orders of magnitude smaller than previous work. This paper also demonstrates how the linear Doppler migration methods can be extended to higher order compensation to further increase potential processing intervals.

eess.SP↗

arXiv2019

Carnot rectifiability of sub-Riemannian manifolds with constant tangent

We show that if $M$ is a sub-Riemannian manifold and $N$ is a Carnot group such that the nilpotentization of $M$ at almost every point is isomorphic to $N$, then there are subsets of $N$ of positive measure that embed into $M$ by bilipschitz maps. Furthermore, $M$ is countably $N$--rectifiable, i.e., all of $M$ except for a null set can be covered by countably many such maps.

math.MG↗