Proposes PSCs for UQ in deep nets without retraining.
problem Estimating uncertainty in deep nets with a single pass.
method Identifies sensitive, smooth intermediate layer, fits probabilistic model.
result PSCs achieve UQ and OOD detection performance matching existing methods.
Study of psc metrics via block diffeomorphisms and cubical sets.
problem Understanding the concordance of psc metrics.
method Constructing cubical sets and using block Dirac operators.
result Construction of a cubical Kan set and comparison map.
Proves positive mass theorem for hyperbolic manifolds with ends.
problem Establishing positive mass theorem for complex initial data sets.
method Used spectral PSC, Jang equation, and quantitative shielding theorem.
result Proved positive mass theorem for asymptotically hyperbolic manifolds.
Let (Y,g) be a compact Riemannian manifold of positive scalar curvature (psc). It is well-known, due to Schoen-Yau, that any closed stable minimal hypersurface of Y also admits a psc-metric. We establish an analogous result for stable minimal hypersurfaces with free boundary. Furthermore, we combine this result wit…
Two positive scalar curvature metrics g0, g1 on a manifold M are psc-isotopic if they are homotopic through metrics of positive scalar curvature. It is well known that if two metrics g0, g1 of positive scalar curvature on a closed compact manifold M are psc-isotopic, then they are psc-concordant: i.e., …
We extend the deep and important results of Lichnerowicz, Connes, and Gromov-Lawson which relate geometry and characteristic numbers to the existence and non-existence of metrics of positive scalar curvature (PSC). In particular, we show: that a spin foliation with Hausdorff homotopy groupoid of an enlargeable manifold…
Experimental determination of protein function is resource-consuming. As an alternative, computational prediction of protein function has received attention. In this context, protein structural classification (PSC) can help, by allowing for determining structural classes of currently unclassified proteins based on thei…
We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{ä}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles…
The paper stratifies positive scalar curvature metrics on manifolds.
problem Understanding the structure of positive scalar curvature metrics.
method Stratification of the space of PSC metrics using a continuous functional.
result Intermediate values of the smooth constant are identified for specific manifolds.
The paper constructs manifolds without smooth psc metrics but with L∞-metrics that are psc outside singular points.
problem Constructing manifolds without smooth positive scalar curvature metrics.
method Constructing manifolds with point singularities and L∞-metrics that are psc outside the singular set. result Examples of manifolds with point singularities that do not admit smooth psc metrics but do admit L∞-metrics that are psc outside the singular set. Proves effective positive mass theorem for AF manifolds and singular spaces.
problem Proves positive mass theorem for AF manifolds with singularities.
method Dimension reduction techniques, bypassing N. Smale's regularity theorem.
result Effective positive mass theorem for AF manifolds of dimension n≤8 with singularities. In several application domains, high-dimensional observations are collected and then analysed in search for naturally occurring data clusters which might provide further insights about the nature of the problem. In this paper we describe a new approach for partitioning such high-dimensional data. Our assumption is that…
The paper proves non-existence of positive scalar curvature on certain fiber bundles.
problem The existence of positive scalar curvature metrics on fiber bundles.
method Analyzing fiber bundles over specific manifolds with incompressible or homotopically nontrivial fibers.
result Non-existence of PSC metrics on certain fiber bundles under specific conditions.
Study shows nonnegative scalar curvature on certain manifolds with specific properties.
problem Proving the nonexistence of metrics with positive scalar curvature on specific manifolds.
method Utilizing Gromov's μ-bubbles, the study examines properties of universal covers and applies them to show nonexistence of metrics with positive scalar curvature.
result The research demonstrates that certain manifolds do not admit complete metrics of positive scalar curvature.
New method to decompose 4-manifolds with positive scalar curvature.
problem Understanding and decomposing 4-manifolds with positive scalar curvature.
method 0 and 1-surgeries on topologically PSC 4-orbifolds.
result Every closed, oriented, topologically PSC 4-manifold can be obtained from a specific type of 4-orbifold.
The study explores psc-metrics on non-spin manifolds with pin^\pm or spin^c structures.
problem Existence of psc-metrics on non-spin manifolds with specific structures.
method Analysis of Dirac operators and index theory on pin^\pm or spin^c manifolds.
result The index of a Dirac operator controls the existence of psc-metrics on certain manifolds.
The study examines curvature conditions on a cylinder and its boundary.
problem Conditions for positive scalar curvature on a cylinder and its boundary.
method Analyzes metrics with specific curvature properties on a compact cylinder.
result If a cylinder has a metric with positive scalar curvature and nonnegative mean curvature on the boundary, it can be modified to have a metric with positive scalar curvature on the boundary.
We prove that many spaces of positive scalar curvature metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside R+(Sd) if d≥6. To achieve that goal, we study the cobordism category of manifolds with positive scalar cu…
Study spectral flow for Callias operators to prove obstructions to positive scalar curvature.
problem Prove obstructions to positive scalar curvature on manifolds.
method Use spectral flow and odd K-cowaist to derive obstructions.
result Infinite odd K-cowaist is an obstruction to the existence of PSC metrics.
Proves Gromov's conjecture for a specific type of groups.
problem Gromov's conjecture for right-angled Artin groups.
method Analyzes universal covering spaces of manifolds with specific fundamental groups.
result Confirms Gromov's conjecture for right-angled Artin groups.
Study on metrics with positive scalar curvature on manifolds with singularities.
problem Understanding metrics with positive scalar curvature on manifolds with singularities.
method Proving homotopy invariance of the space of metrics with positive scalar curvature on manifolds with fibred singularities.
result Proved that the space of metrics with positive scalar curvature is homotopy invariant under certain surgeries.
The paper studies invariant metrics with positive scalar curvature on 3-manifolds.
problem Classifying G-invariant 3-manifolds with positive scalar curvature. method Analyzes the space of G-invariant Riemannian metrics with positive scalar curvature on closed 3-manifolds. result The space of G-invariant PSC metrics is either empty or contractible. PSC classifier improves HDLSS classification on class-imbalanced data.
problem Classification on high-dimension low-sample-size data with class imbalance.
method Population Structure-learned Classifier (PSC) maximizing inter-class and intra-class scatter matrices.
result PSC outperforms state-of-the-art methods on IHDLSS.
We study the space of positive scalar curvature (psc) metrics on a 4-manifold, and give examples of simply connected manifolds for which it is disconnected. These examples imply that concordance of psc metrics does not imply isotopy of such metrics. This is demonstrated using a modification of the 1-parameter Seiberg-W…
We construct complete Riemannian metrics to show that the total space of tangent bundles of orientable closed surfaces (except torus) admits complete uniformly PSC-metrics. It gives a partial positive answer to one of Gromov's question.
Study a relative aspherical conjecture and prove 3-manifold obstruction to positive scalar curvature.
problem Obstructing the existence of positive scalar curvature in higher dimensions.
method Introduced a relative aspherical condition and a new geometric quantity called spherical width.
result Proved results on how 3-manifolds obstruct the existence of positive scalar curvature.
Study optimizes HTL-free PSCs with MWCNTs, improving efficiency and stability.
problem Optimizing efficiency and degradation in HTL-free perovskite solar cells.
method Machine learning-driven framework integrating experimental validation and numerical simulations.
result Achieved RMSEs of 0.0179 and 0.0117 for efficiency and degradation, respectively.
In the first part of this paper, we consider the problem of fill-in of nonnegative scalar curvature (NNSC) metrics for a triple of Bartnik data (Σ,γ,H). We prove that given a metric γ on Sn−1 (3≤n≤7), (Sn−1,γ,H) admits no fill-in of NNSC metrics provided the prescribed mean cur…
The paper defines invariants for positive scalar curvature metrics on manifolds with boundary.
problem Invariants of bordism classes of positive scalar curvature metrics on manifolds with boundary.
method Definition of relative L2-ρ-invariant for Dirac operators on odd-dimensional spin manifolds with boundary. result Existence of infinitely many bordism classes of positive scalar curvature metrics on certain manifolds.
Introduces privilege scores to measure and interpret protected attribute-related privilege in machine learning models.
problem Lack of explicit formulation of non-neutrality in fairness-aware machine learning methods.
method Privilege scores (PS) and privilege score contributions (PSCs) to measure and interpret protected attribute-related privilege.
result Demonstrates the broad applicability of PS and PSCs in gender and racial privilege in mortgage and college admissions applications.
Study on positive scalar curvature on complex spaces with specific conditions.
problem Existence and sufficiency of positive scalar curvature metrics on stratified spaces.
method Index-theoretic conditions and cobordism analysis.
result Sufficient conditions for positive scalar curvature on certain stratified spaces.
Study detects non-trivial elements in diffeomorphism groups via trivalent graphs.
problem Detecting non-trivial elements in homotopy groups of diffeomorphism spaces.
method Using Kontsevich classes and trivalent graphs, we lift elements from one moduli space to another.
result Non-trivial elements in π∗(BDiff∂(Dd))⊗Q are lifted to π∗(BDiff⊔(DdimesI))⊗Q and π∗(M∂psc(Dd)h0)⊗Q. Constructs infinitely many examples of large manifolds with circle bundles of positive scalar curvature.
problem Existence of circle bundles over large manifolds with positive scalar curvature metrics.
method Symplectic geometry techniques.
result Infinitely many examples of macroscopically large manifolds with circle bundles of positive scalar curvature.
MAGIC uncovers disease heterogeneity across brain scales.
problem Understanding distinct subtypes of brain diseases at different spatial scales.
method Multi-scale Heterogeneity Analysis and Clustering (MAGIC) using semi-supervised clustering.
result Two main subtypes of AD identified with distinct atrophy patterns.
The paper sets new bounds on metrics with positive scalar curvature.
problem Classifying metrics with positive scalar curvature.
method Lower bounds on the rank of the group of psc metrics over M.
result Lower bounds on the rank of psc metrics up to bordism.
In a recent paper, the authors proved that no spin foliation on a compact enlargeable manifold with Hausdorff homotopy graph admits a metric of positive scalar curvature on its leaves. This result extends groundbreaking results of Lichnerowicz, Gromov and Lawson, and Connes on the non-existence of metrics of positive s…
The doubling conjecture for positive scalar curvature is proven under certain conditions.
problem Determining when a manifold with a specific boundary condition admits positive scalar curvature.
method Surgery techniques for positive scalar and mean curvature, and existence of area-minimizing hypersurfaces.
result The doubling conjecture holds true for manifolds with certain split conditions on fundamental groups.
The paper solves a problem related to scalar curvature and boundary metrics.
problem Proving the extensibility of boundary metrics to positive scalar curvature metrics.
method Introducing a fill-in invariant and proving relationships with positive mass theorems.
result The positive mass theorem for asymptotically hyperbolic manifolds implies the same for asymptotically flat manifolds.
The paper classifies certain 4D and 5D manifolds with positive scalar curvature.
problem Classifying manifolds with positive scalar curvature.
method Combining advances in positive scalar curvature with a novel argument.
result Closed manifolds with positive scalar curvature are homotopy equivalent to spheres or connected sums.
The study explores positive scalar curvature metrics on non-orientable manifolds and their covers.
problem Existence of positive scalar curvature metrics on non-orientable manifolds and their covers.
method Extends Schoen-Yau inductive descent approach to non-orientable manifolds.
result Examples of non-orientable manifolds with positive scalar curvature metrics on their orientation double covers but not on homotopy equivalent manifolds.
A scalable deep learning framework accelerates training of large neural networks for solving 3D Poisson equations.
problem Training large-scale neural networks for solving complex PDEs efficiently.
method Combines multigrid techniques with distributed deep learning to accelerate training.
result Solves 3D Poisson equations up to 512x512x512 resolution efficiently.
The paper constructs infinite rank summands in diffeomorphism groups via Seiberg-Witten theory.
problem Constructing infinite rank summands in diffeomorphism groups.
method Using Seiberg-Witten theory, the paper constructs spherical families and computes invariants.
result Infinite rank summands in homotopy and homology groups of diffeomorphism groups.
Proves generic existence of spectral networks for many cases.
problem Existence of spectral networks for a broad range of spectral data.
method Generic existence proof for spectral networks.
result Proves existence for a large class of spectral data.
Study spectral distances on compact RCD spaces.
problem Understanding spectral convergence in RCD spaces.
method Established relationships between different spectral convergences and constructed a spectral approximation map.
result Found canonical spectral approximation map for RCD spaces.
Paper defines spectral triple and computes functional for nonminimal de Rham-Hodge operator.
problem Computing spectral functions for nonminimal de Rham-Hodge operators.
method Definitions and computations of spectral triple and functional.
result Computed spectral Einstein functional for even-dimensional compact manifolds.
New spectral torsion defined for rescaled Dirac operators.
problem Defining spectral torsion for rescaled Dirac operators.
method Using three vector fields and noncommutative residue.
result Computed spectral torsion for one form rescaled Dirac operators.
The paper explores rigidity theorems for spectral curvature bounds in 3-manifolds.
problem Classical rigidity results in scalar curvature geometry are extended to the spectral setting.
method Warped μ-bubble method is systematically employed to classify stable weighted minimal hypersurfaces and establish band width estimates. result Classification theorems and band width estimates for spectral Ricci and scalar curvatures are proven.
Introduces new spectral triples for parabolic geometry.
problem Anisotropies and varying orders in parabolic geometry.
method Tangled spectral triples incorporating directional Dirac operators.
result Higher order spectral triples for hypoelliptic complexes and nilpotent group algebras.