arXiv subjects

Benjamin Schmidt

Publications and source records attributed to Benjamin Schmidt.

At least 19 recordsLinked to original sources

arXiv2026

Performance of a SuperCDMS HVeV Detector with Sub-eV Energy Resolution and Single Charge-sensitivity

We present a detailed characterization of a new generation of athermal-phonon single-charge sensitive Si HVeV detectors, the best of which achieved 589~meV~$\pm$~5~meV baseline resolution. Our sub-eV energy resolution enables precise measurements of single-photon events and reveal consistent energy losses of 0.92~eV~$\pm$~0.02~eV per charge excitation across two facilities. We demonstrate that the noise for these detectors is well described using a standard Transition Edge Sensor noise model. We also measure a nominal phonon collection efficiency of 45\%~$\pm$~3\%~(stat.)~$\pm$~6\%~(syst.) in the best performing device, establishing these detectors as the most efficient athermal phonon detectors to date, limited only by intrinsic limitations of quasiparticle generation.

physics.ins-det↗

arXiv2024

Performance of a Kinetic Inductance Phonon-Mediated Detector at the NEXUS Cryogenic Facility

Microcalorimeters that leverage microwave kinetic inductance detectors to read out phonon signals in the particle-absorbing target, referred to as kinetic inductance phonon-mediated (KIPM) detectors, offer an attractive detector architecture to probe dark matter (DM) down to the fermionic thermal relic mass limit. A prototype KIPM detector featuring a single aluminum resonator patterned onto a 1-gram silicon substrate was operated in the NEXUS low-background facility at Fermilab for characterization and evaluation of this detector architecture's efficacy for a dark matter search. An energy calibration was performed by exposing the bare substrate to a pulsed source of 470 nm photons, resulting in a baseline resolution on the energy absorbed by the phonon sensor of $2.1\pm0.2$ eV, a factor of two better than the current state-of-the-art, enabled by millisecond-scale quasiparticle lifetimes. However, due to the sub-percent phonon collection efficiency, the resolution on energy deposited in the substrate is limited to $\sigma_E=318 \pm 28$ eV. We further model the signal pulse shape as a function of device temperature to extract quasiparticle lifetimes, as well as the observed noise spectra, both of which impact the baseline resolution of the sensor.

physics.ins-det↗

arXiv2022

Variation of stability for moduli spaces of unordered points in the plane

We study compactifications of the moduli space of unordered points in the plane via variation of GIT quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given.

math.AG↗

arXiv2022

Characterizing unit spheres in Euclidean spaces via reach and volume

Let $M$ be a smooth, connected, compact submanifold of $\mathbb{R}^n$ without boundary and of dimension $k\geq 2$. Let $\mathbb{S}^k \subset \mathbb{R}^{k+1}\subset \mathbb{R}^n$ denote the $k$-dimesnional unit sphere. We show if $M$ has reach equal to one, then its volume satisfies $\text{vol}(M)\geq \text{vol}(\mathbb{S}^k)$ with equality holding only if $M$ is congruent to $\mathbb{S}^k$.

math.DG↗

arXiv2021

Sheaves of low rank in three dimensional projective space

We classify Chern characters of semistable sheaves up to rank four in three dimensional projective space. As a corollary we show that moduli spaces of semistable sheaves between rank zero and four with maximal third Chern character are smooth and irreducible.

math.AG↗

arXiv2021

On Fast Johnson-Lindenstrauss Embeddings of Compact Submanifolds of $\mathbb{R}^N$ with Boundary

Let $\mathcal{M}$ be a smooth $d$-dimensional submanifold of $\mathbb{R}^N$ with boundary that's equipped with the Euclidean (chordal) metric, and choose $m \leq N$. In this paper we consider the probability that a random matrix $A \in \mathbb{R}^{m \times N}$ will serve as a bi-Lipschitz function $A: \mathcal{M} \rightarrow \mathbb{R}^m$ with bi-Lipschitz constants close to one for three different types of distributions on the $m \times N$ matrices $A$, including two whose realizations are guaranteed to have fast matrix-vector multiplies. In doing so we generalize prior randomized metric space embedding results of this type for submanifolds of $\mathbb{R}^N$ by allowing for the presence of boundary while also retaining, and in some cases improving, prior lower bounds on the achievable embedding dimensions $m$ for which one can expect small distortion with high probability. In particular, motivated by recent modewise embedding constructions for tensor data, herein we present a new class of highly structured distributions on matrices which outperform prior structured matrix distributions for embedding sufficiently low-dimensional submanifolds of $\mathbb{R}^N$ (with $d \lesssim \sqrt{N}$) with respect to both achievable embedding dimension, and computationally efficient realizations. As a consequence we are able to present, for example, a general new class of Johnson-Lindenstrauss embedding matrices for $\mathcal{O}(\log^c N)$-dimensional submanifolds of $\mathbb{R}^N$ which enjoy $\mathcal{O}(N \log (\log N))$-time matrix vector multiplications.

cs.IT↗

arXiv2021

Lower Bounds on the Low-Distortion Embedding Dimension of Submanifolds of $\mathbb{R}^n$

Let $\mathcal{M}$ be a smooth submanifold of $\mathbb{R}^n$ equipped with the Euclidean (chordal) metric. This note considers the smallest dimension $m$ for which there exists a bi-Lipschitz function $f: \mathcal{M} \mapsto \mathbb{R}^m$ with bi-Lipschitz constants close to one. The main result bounds the embedding dimension $m$ below in terms of the bi-Lipschitz constants of $f$ and the reach, volume, diameter, and dimension of $\mathcal{M}$. This new lower bound is applied to show that prior upper bounds by Eftekhari and Wakin (arXiv:1306.4748) on the minimal low-distortion embedding dimension of such manifolds using random matrices achieve near-optimal dependence on both reach and volume. This supports random linear maps as being nearly as efficient as the best possible nonlinear maps at reducing the ambient dimension for manifold data. In the process of proving our main result, we also prove similar results concerning the impossibility of achieving better nonlinear measurement maps with the Restricted Isometry Property (RIP) in compressive sensing applications.

math.NA↗

arXiv2020

The desingularization of the theta divisor of a cubic threefold as a moduli space

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaves on a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$ is smooth and of dimension four. Moreover, the Abel-Jacobi map to the intermediate Jacobian of $X$ maps it birationally onto the theta divisor $\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to the singular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of the Torelli theorem for cubic threefolds, which says that $X$ can be recovered from its Kuznetsov component $\operatorname{Ku}(X) \subset \mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of the description of the singularity of the theta divisor, and thus of the classical Torelli theorem for cubic threefolds, i.e., that $X$ can be recovered from its intermediate Jacobian.

math.AG↗

arXiv2020

The Autonomous Racing Software Stack of the KIT19d

Formula Student Driverless challenges engineering students to develop autonomous single-seater race cars in a quest to bring about more graduates who are well-prepared to solve the real world problems associated with autonomous driving. In this paper, we present the software stack of KA-RaceIng's entry to the 2019 competitions. We cover the essential modules of the system, including perception, localization, mapping, motion planning, and control. Furthermore, development methods are outlined and an overview of the system architecture is given. We conclude by presenting selected runtime measurements, data logs, and competition results to provide an insight into the performance of the final prototype.

cs.RO↗

arXiv2020

Analysis of process-induced damage in remote laser cut carbon fibre reinforced polymers

In this contribution a method is introduced that allows for a linkage between the process-induced structural damage and the fracture behaviour. Based on an anisotropic elastic material model, different modelling approaches for initial damage effects are introduced and compared. The approaches are applied to remote laser cut carbon fibre reinforced polymers in order to model various thermally induced damage effects like chemical decomposition, micro-cracks and delamination. The dimensions of this heat affected zone are calculated with 1D-heat conduction. In experiment and simulation milled and laser cut specimens with different process parameters are compared in order to quantify the impact of the cutting technology on the fracture behaviour. For this purpose open hole specimens were used.

physics.app-ph↗

arXiv2020

Stability and Applications

We give a brief overview of Bridgeland's theory of stability conditions, focusing on applications to algebraic geometry. We sketch the basic ideas in Bayer's proof of the Brill--Noether Theorem and in the authors' proof of a theorem by Gruson--Peskine and Harris on the genus of space curves. This note originated from the lecture of the first author at the conference From Algebraic Geometry to Vision and AI: A Symposium Celebrating the Mathematical Work of David Mumford, held at the Center of Mathematical Sciences and Applications, Harvard University, August 18-20, 2018.

math.AG↗

arXiv2019

Geometric structures and the Laplace spectrum, part II

We continue our exploration of the extent to which the spectrum encodes the local geometry of a locally homogeneous three-manifold and find that if $(M,g)$ and $(N,h)$ are a pair of locally homogeneous, locally non-isometric isospectral three-manifolds, where $M$ is an elliptic three-manifold, then $(1)$ $N$ is also an elliptic three-manifold, $(2)$ $M$ and $N$ have fundamental groups of different orders, $(3)$ $(M,g)$ and $(N,h)$ both have non-degenerate Ricci tensors and $(4)$ the metrics $g$ and $h$ are sufficiently far from a metric of constant sectional curvature. We are unaware of any such isospectral pair and such a pair could not arise via the classical Sunada method. As part of the proof, we provide an explicit description of the isometry group of a compact simple Lie group equipped with a left-invariant metric---improving upon the results of Ochiai-Takahashi and Onishchik---which we use to classify the locally homogeneous metrics on an elliptic three-manifold $\Gamma \backslash S^3$ and we determine that any collection of isospectral locally homogeneous metrics on an elliptic three-manifold consists of at most two isometry classes that are necessarily locally isometric. In particular, the left-invariant metrics on $\operatorname{SO}(3)$ (respectively, $S^3$) can be mutually distinguished via their spectra. The previous statement has the following interpretation in terms of physical chemistry: the moments of inertia of a molecule can be recovered from its rotational spectrum.

math.DG↗

arXiv2019

Preserve one, preserve all

Isometries of metric spaces $(X,d)$ preserve all level sets of $d$. We formulate and prove cases of a conjecture asserting if $X$ is a complete Riemannian manifold, then a function $f:X \rightarrow X$ preserving at least one level set $d^{-1}(r)$, with $r>0$ small enough, is an isometry.

math.DG↗

arXiv2019

Geometric structures and the Laplace spectrum

Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we explore the extent to which compact locally homogeneous three-manifolds are characterized up to local isometry by their spectra. We observe that there are eight `metrically maximal' three-dimensional geometries on which all compact locally homogeneous three-manifolds are modeled and we demonstrate that for five of these geometries the associated compact locally homogeneous three-manifolds are determined up to local isometry by their spectra within the universe of locally homogeneous three-manifolds. Specifically, we show that among compact locally homogeneous three-manifolds, a Riemannian three-manifold is determined up to local isometry if its universal Riemannian cover is isometric to (1) a symmetric space, (2) $\mathbb{R}^2 \rtimes \mathbb{R}$ endowed with a left-invariant metric, (3) $\operatorname{Nil}$ endowed with a left-invariant metric, or (4) $S^3$ endowed with a left-invariant metric sufficiently close to a metric of constant sectional curvature. We then deduce that three-dimensional Riemannian nilmanifolds and locally symmetric spaces with universal Riemannian cover $\mathbb{S}^2 \times \mathbb{E}$ are uniquely characterized by their spectra among compact locally homogeneous three-manifolds. Finally, within the collection of closed manifolds covered by $\operatorname{Sol}$ equipped with a left-invariant metric, local geometry is `audible.'

math.DG↗

arXiv2019

Diameters of Ball Intersections

We prove the diameter of the intersection of two closed convex balls in a Riemannian manifold eventually decreases continuously as the centers of the balls move apart.

math.DG↗

arXiv2019

Characterizing Round Spheres Using Half-Geodesics

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.

math.DG↗

arXiv2018

Discriminants of stable rank two sheaves on some general type surfaces

We prove sharp bounds on the discriminants of stable rank two sheaves on surfaces in three-dimensional projective space. The key technical ingredient is to study them as torsion sheaves in projective space via tilt stability in the derived category. We then proceed to describe the surface itself as a moduli space of rank two vector bundles on it. Lastly, we give a proof of the Bogomolov inequality for semistable rank two sheaves on integral surfaces in three-dimensional projective space in all characteristics.

math.AG↗

arXiv2018

Rank two sheaves with maximal third Chern character in three-dimensional projective space

We give a complete classification of semistable rank two sheaves on three-dimensional projective space with maximal third Chern character. This implies an explicit description of their moduli spaces. As an open subset they contain rank two reflexive sheaves with maximal number of singularities. These spaces are irreducible, and apart from a single special case, they are also smooth. This extends a result by Okonek and Spindler to all missing cases and gives a new proof of their result. The key technical ingredient is variation of stability in the derived category.

math.AG↗