New progress on frame flow ergodicity for nearly pinched manifolds.
problem Ergodicity of frame flow on negatively-curved manifolds.
method New ideas leading to ergodicity for nearly 0.25-pinched manifolds.
result Achieved progress towards Brin's conjecture.
The paper connects disentanglement to manifold charts and commutativity.
problem Discovering local charts of the data manifold for disentanglement.
method Interpreting disentanglement as local charts of the data manifold and studying commutativity.
result Commutativity is a central property in disentanglement, as shown in manifold, group theoretic, and probabilistic frameworks.
New method calculates volume-renormalized mass from Hamiltonian perspective.
problem Calculating volume-renormalized mass for asymptotically hyperbolic manifolds.
method Using Michel's mass invariants and a reduced Hamiltonian perspective, the volume-renormalized mass is deduced.
result The reduced Hamiltonian recovers the volume-renormalized mass and its variations.
Geometric approach improves functional outlier detection.
problem Detecting outliers in functional data sets.
method Developed a geometric perspective on functional manifold for outlier detection.
result Improved understanding and differentiation of outliers.
New group theory insights on knot surgery results.
problem Understanding non-simply connected 3-manifolds from Dehn surgery.
method Group theoretic analysis of Property P conjecture variations.
result New group theoretic perspectives on Dehn filling.
Analyzes properties of Hopf manifolds from analytic and metric perspectives.
problem Properties of Hopf manifolds
method Analytic and metric structure
result Reviews old and new properties of Hopf manifolds
Surveying recent progress on flows of G2-structures on 7-manifolds.
problem Preserving metrics while modifying G2-structures on 7-manifolds. method Heat flows and other approaches in terms of 3-forms, octonions, vector fields, and geometric structures. result Comparison of different perspectives on G2-structure flows. This is a survey article on symplectically aspherical manifolds. The paper contains a discussion on constructions of symplectically aspherical manifolds, their topological properties and the role of this class in symplectic topology. Research perspectives are discussed.
Study monopole h-invariants from a topological viewpoint.
problem Understanding the h-invariants of 3-manifolds.
method Using Lidman and Manolescu's description of monopole Floer homology.
result Prove several properties of the h-invariants.
The paper improves GNN generalization theory by considering graph manifolds.
problem Improper GNN generalization bounds ignoring graph structures.
method Taking a manifold perspective, the paper establishes GNN generalization theory.
result GNN generalization bounds decrease linearly with graph size and spectral continuity.
New geometric perspective for optimal learning on hexagonal structures.
problem Optimal learning process on hexagonal structures.
method Local trivial fibrations and Ceva's theorem.
result Learning can be defined on hexagonal structures.
Global and local blowups of manifolds are proven equivalent.
problem Equivalence of global and local blowups in differential topology.
method Proof of equivalence between global and local constructions of blowups.
result Global and local constructions of blowups are shown to be equivalent.
Hermitian symmetric manifolds are Hermitian manifolds which are homogeneous and such that every point has a symmetry preserving the Hermitian structure. The aim of these notes is to present an introduction to this important class of manifolds, trying to survey the several different perspectives from which Hermitian sym…
Explores new perspectives in transverse index theory for Lie group actions.
problem Transverse index theory for compact Lie group actions.
method Kasparov's work on transverse index theory, connections to Berline-Vergne and Paradan-Vergne.
result Potential connections and new insights in transverse index theory.
Extends Weyl geometry from conformal to Weyl manifolds using ambient metrics.
problem Generalizing ambient constructions to Weyl manifolds.
method Introduces Weyl-ambient metric and Weyl-Fefferman-Graham gauge; shows Weyl-ambient space induces Weyl geometry; defines Weyl-connection and Weyl structure.
result Weyl-ambient construction for Weyl manifolds provides a well-defined initial value problem.
New homogeneous special Lagrangian submanifolds discovered in nearly Kähler CP3.
problem Exploring special Lagrangian submanifolds in nearly Kähler CP3.
method Intrinsically and extrinsically using moving frame and moment-type maps.
result Classification of totally geodesic special Lagrangian submanifolds and homogeneity of special Lagrangians.
Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
We construct a map from the suspension G-spectrum ΣG∞M of a smooth compact G-manifold to the equivariant A-theory spectrum AG(M), and we show that its fiber is, on fixed points, a wedge of stable h-cobordism spectra. This map is constructed as a map of spectral Mackey functors, which is compatible …
QP perspective on Poisson-Lie T-duality topology changes.
problem Understanding Poisson-Lie T-duality through QP manifolds.
method QP manifolds and canonical transformations for symplectic reductions.
result Canonical transformations mediate Poisson-Lie T-duality.
Bayesian analysis shows unlabeled data improve graph-based semi-supervised learning.
problem Improving semi-supervised learning with limited labeled data.
method Bayesian nonparametric approach using unlabeled data for graph-based learning.
result Posterior contracts optimally around the truth with sufficient unlabeled data.
We use the information metric to investigate the moduli space of a U(1) instanton on (anti)self-dual manifolds, finding an AdS geometry similar to that for the moduli space of a Yang-Mills instanton on flat space. We discuss our results from the perspective of gauge/gravity duality.
Study harmonic mappings and submanifolds using Bochner technique.
problem Classical theorems in harmonic mappings and submanifolds.
method Generalized Bochner technique.
result New insights into classical theorems.
The problem of learning a manifold structure on a dataset is framed in terms of a generative model, to which we use ideas behind autoencoders (namely adversarial/Wasserstein autoencoders) to fit deep neural networks. From a machine learning perspective, the resulting structure, an atlas of a manifold, may be viewed as …
Notes on embedding criteria for smooth manifolds.
problem Conditions for embedding smooth manifolds into Euclidean space.
method Linking recent results to classical criteria and K-theory.
result Recent results connect to classical embedding criteria.
Enhances Hantzsche's theorem for 3-manifolds in 4D.
problem Embedding 3-manifolds in 4-dimensional space.
method Using Heegaard diagrams and a property called doubly unlinked (DU).
result Enhanced Hantzsche's embedding obstruction.
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the existing optimization-based approach, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear…
Decomposes smooth manifolds into algebraic submanifolds.
problem Understanding the structure of smooth manifolds induced by continuous selections.
method Generic continuous selection of smooth functions provides stratification of the manifold.
result Stratification leads to local topological structure with nondegenerate critical points.
We identify and study a class of hyperbolic 3-manifolds (which we call Macfarlane manifolds) whose quaternion algebras admit a geometric interpretation analogous to Hamilton's classical model for Euclidean rotations. We characterize these manifolds arithmetically, and show that infinitely many commensurability classes …
Establishes a link between heat diffusion and manifold distances in data.
problem No theoretical link between diffusion-based manifold learning and geodesic distances.
method Formulates heat geodesic embeddings based on Riemannian geometry.
result Method outperforms state-of-the-art in preserving manifold distances and cluster structure.
We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. We classify these structures by their intrinsic torsion and review the literature in terms of this scheme. Moreover, we determine necessary and sufficient conditions for the existence of metric connections with vectorial, totally…
New perspective on Heegaard splittings using square complexes and combinatorial measurements.
problem Measuring obstructions to Heegaard splittings in 3-manifolds.
method Square complexes and Guirardel's core, augmented Heegaard diagrams.
result Augmented Heegaard diagrams provide a new way to describe Heegaard splittings with desirable properties.
We give an up-to-date perspective with a general overview of the theory of causal properties, the derived causal structures, their classification and applications, and the definition and construction of causal boundaries and of causal symmetries, mostly for Lorentzian manifolds but also in more abstract settings.
Paper examines global Covid-19 data complexity and finds low intrinsic dimensions.
problem Understanding the complexity of Covid-19 data across countries.
method Used a Bayesian mixture model (Hidalgo) to estimate intrinsic dimensionality.
result Covid-19 data projects onto two low-dimensional manifolds without significant loss of information.
Survey and clarify manifold-supported data in deep generative models.
problem Understanding why some DGMs succeed or fail at low-dimensional data.
method Formal analysis and new model connections.
result DGMs on autoencoder representations minimize Wasserstein distance.
Diffeological submanifolds are a new type of submanifold in manifold theory.
problem Defining and understanding different types of submanifolds in manifold theory.
method Introducing diffeological submanifolds and comparing them with other types of submanifolds.
result A diffeological submanifold can be included in a manifold without being an immersion.
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
Motivated by the cosmic censorship conjecture in mathematical relativity, we establish the precise mass lower bound for an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and minimal surface boundary, in terms of angular momentum and charge. In particular this result does not require the res…
Unified theory for adaptive image convolutions using metric perspectives.
problem Fixed kernels in convolutions limit adaptability in image processing.
method Metric perspective on images as 2D manifolds with local distances, proposing metric convolutions.
result Metric convolutions provide better generalisation and competitive performance.
These notes summarize and expand on a mini-course given at CIRM in February 2018 as part of Winter Braids VIII. We somewhat obsessively develop the slogan `Trisections are to 4-manifolds as Heegaard splittings are to 3-manifolds', focusing on and clarifying the distinction between three ways of thinking of things: the …
A new stock index model simplifies high-dimensional stock data.
problem Reflecting the overall stock market activity in high-dimensional data.
method Manifold learning and feature detection on discrete Laplace-Beltrami operator.
result The MF index series approximates the stock market better and has lower risk.
The paper analyzes diffusion condensation for data geometry and topology.
problem Understanding the geometry and topology of high-dimensional data.
method Time-inhomogeneous diffusion process with geometric, spectral, and topological analysis.
result The condensation process defines intrinsic condensation homology and ambient persistent homology.
Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numeri…
We give a different perspective on the (by now) classic Basmajian identity, and point out some related results, both in the setting of hyperbolic manifolds, and in the polyhedral setting \emph{without} any group acting. In the new version we give more geometric and combinatorial applications of the main ideas.
Lecture notes on curves in complex projective plane from a topological viewpoint.
problem Understanding curves in complex projective plane from a topological perspective.
method Topological analysis of curves in complex projective plane.
result Curves in complex projective plane have unique topological properties.
We show that any compact half-conformally flat manifold of negative type, with bounded L2 energy, sufficiently small scalar curvature, and a non-collapsing assumption, has all betti numbers bounded. We show that this result is optimal from an analytic perspective by demonstrating singularity models that are 2-ended,…
The paper connects complex normalizing flows to Kähler-Ricci flows using geometric and statistical perspectives.
problem Understanding the relationship between complex normalizing flows and Kähler-Ricci flows.
method Develops connections between complex normalizing flows and Kähler-Ricci flows by relating the log determinant to Ricci curvature and using a Bayesian perspective.
result Reconciles the complex normalizing flow and Kähler-Ricci flow, showing they are related under certain conditions.
Machine learning models predict which ideas will be innovated based on subjective perspectives.
problem Predicting high-impact innovation based on subjective perspectives and interpersonal innovation opportunities.
method Quantifying subjective perspectives and their interaction based on innovator positions within a geometric space of concepts.
result Subjective perspectives predict which ideas individuals and groups will creatively attend to and successfully combine in the future.
The authors introduce Morse foliated open books for studying contact manifolds.
problem Studying contact manifolds with boundary.
method Introducing Morse foliated open books and extending the right-veering concept.
result Right-veering plays a similar role in detecting overtwistedness in foliated open books.