xGEMs explain black-box models by exploring data manifolds.
problem Understanding the behavior of black-box classifiers.
method Train an unsupervised generative model to represent the data manifold and perturb data points to analyze model behavior.
result Detect and quantify bias in model learning and changes in model behavior over training.
Combines generalized and graded geometry to explore new structures.
problem Exploring new structures on generalized tangent bundles of graded manifolds.
method Introduces canonical brackets, Dirac structures, and generalized complex structures.
result Canonical bracket on a generalized tangent bundle of a graded manifold.
Enhances deep networks with manifold exploration for better performance and robustness.
problem Improving deep neural network performance and robustness to random transformations.
method Geometric transformations learned through line-search manifold exploration to misclassify images while maintaining manifold consistency.
result Significantly increased robustness and performance on challenging classification tasks.
Paper explores cohomology classes on non-compact almost Kähler manifolds.
problem Understanding cohomology classes induced by symplectic forms.
method Provides criteria for non-trivial classes in Lp cohomology. result Symplectic forms induce non-trivial classes in Lp cohomology. The study explores weakly p-Kähler hyperbolic manifolds.
problem Generalization and application of weakly p-Kähler hyperbolic manifolds. method Investigation of generalizations and applications.
result Exploration of weakly p-Kähler hyperbolic manifolds. Exploring Sasaki metrics in joined manifolds.
problem Existence of constant scalar curvature Sasaki metrics in joined manifolds.
method Investigating the Sasaki cone of joined manifolds and considering continuous families of extremal Sasaki twins.
result Conditions for the existence of constant scalar curvature Sasaki metrics in joined manifolds.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
Study on G2-type flag manifolds, focusing on invariant metrics and Ricci flow.
problem Characterizing and analyzing metrics on G2-type flag manifolds. method Investigation of invariant metrics, analysis of g.o. metrics, and Ricci flow techniques.
result Characterization of metrics invariant under maximal compact subgroups.
Study explores Laplacian coflow versions on Calabi-Yau 7-manifolds.
problem Analyzing G2-structures on Calabi-Yau 7-manifolds. method Reduced Ansätze for the Laplacian coflow on various Calabi-Yau 7-manifolds.
result Obtained a modified Kähler-Ricci flow.
Integral geometry explores curvature conjectures on manifolds.
problem Proving curvature conjectures for manifolds of positive and negative curvature.
method Formulating curvature integral geometrically and using linear programming.
result Existence of a positive curvature metric on product manifolds.
The study explores scalar curvatures on manifolds with boundary properties.
problem Understanding scalar curvatures on manifolds with boundary constraints.
method Presentation of problems and results related to scalar curvatures and mean curvatures of boundaries.
result Exploration of natural and artificial constructions in scalar curvature studies.
We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.
Survey of astheno-Kähler manifolds and their challenges.
problem Understanding astheno-Kähler manifolds and their existence.
method Review of classical findings and recent developments.
result Challenges in the existence of astheno-Kähler manifolds.
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
Study explores how scalar functionals evolve under Ricci flow.
problem Understanding the evolution of functionals involving scalar quantities under Ricci flow.
method Deriving explicit expressions for the time derivative of integrals of scalar functionals under extended Ricci flow.
result Explicit expressions for the time derivative of integrals involving scalar functionals under Ricci flow.
This is a collection of notes on embedding problems for 3-manifolds. The main question explored is `which 3-manifolds embed smoothly in the 4-sphere?' The terrain of exploration is the Burton/Martelli/Matveev/Petronio census of triangulated prime closed 3-manifolds built from 11 or less tetrahedra. There are 13766 mani…
Improves latent space structure for better data representation.
problem Limited ability of conventional priors to encode data manifold structure.
method Introduces an Encoded Prior Sliced Wasserstein AutoEncoder with iterative training and geodesic interpolation.
result Learned manifold encoding preserves topological and geometric properties of data.
Paper explores prescribing Chern scalar curvatures on noncompact Hermitian manifolds.
problem Prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds.
method Generalizes Aviles-McOwen's existence results to higher-dimensional Hermitian manifolds.
result Existence results for Chern scalar curvatures on Hermitian manifolds.
The paper explores representations of graph manifolds to Seifert motion groups.
problem Existence of faithful representations of graph manifolds to Seifert motion groups.
method Discussion and proof of non-existence of certain representations.
result Graph manifolds can have virtually no faithful representations to the Seifert motion group.
The paper explores geometric decompositions for Ricci tensors and their applications.
problem Understanding Ricci tensors on compact Riemannian manifolds.
method Utilizes Berger-Ebin and York L2-orthogonal decompositions. result New insights into Ricci almost solitons and harmonic maps.
The study explores geometric properties of hyperbolic cohomology classes on Kähler manifolds.
problem Understanding the geometric effects of hyperbolic cohomology classes on Kähler manifolds.
method Introducing Kähler topologically hyperbolic manifolds and proving spectral gap theorems for positive holomorphic Hermitian vector bundles.
result Kähler topologically hyperbolic manifolds are not uniruled nor bimeromorphic to compact Kähler manifolds with trivial first real Chern class.
Improves exploration in reinforcement learning with diverse population.
problem Lack of diversity in reinforcement learning agents leads to limited exploration.
method Optimizes a population of agents using volume of behavioral manifold and online learning techniques.
result Improves exploration without sacrificing performance.
Researchers explore geometric dualities in statistical manifolds.
problem Understanding geometric dualities in statistical manifolds.
method Exploring the dualistic geometry of statistical manifolds, focusing on Hessian manifolds.
result Moduli space of univariate normal distributions corresponds to Siegel half-space and Siegel-Jacobi space.
Bing's house-like spines approximate all PL manifolds.
problem Understanding spines of PL manifolds.
method Approximation rigidity and h-principle.
result PL manifolds have spines similar to Bing's house.
Extends symphonic maps to bi-symphonic maps between Riemannian manifolds.
problem Defining and exploring new types of maps between Riemannian manifolds.
method Introduces bi-symphonic maps by analyzing the bi-energy functional.
result New types of maps (bi-symphonic) with associated bi-energy functional.
Arboricity of manifolds is explored, with specific results for 2D surfaces.
problem Understanding the arboricity of different types of manifolds.
method Analyzing discrete 2-spheres, other 2D surfaces, and d-manifolds of higher dimensions.
result Arboricity of 2D surfaces is 3 or 4, and for higher dimensions, it can be arbitrarily large.
This paper explores 4-planes in Spin(7) manifolds with symplectic structures.
problem Characterizing 4-planes in Spin(7) manifolds with symplectic structures.
method Detailed analysis of differential forms, including the Cayley 4-form, and exploration of mirror duality.
result Enhanced understanding of the interplay between Spin(7)-structures and symplectic geometry.
BelMan uses Bayesian methods to optimize decisions in multi-armed bandit problems.
problem Optimizing decisions in multi-armed bandit problems with varying rewards and beliefs.
method BelMan uses a geometric approach with information projection and reverse projection to balance exploration and exploitation.
result BelMan outperforms other algorithms in specific scenarios involving many arms and continuous rewards.
The mixed scalar curvature of a foliated Riemannian manifold, i.e., an averaged mixed sectional curvature, has been considered by several geometers. We explore the Yamabe type problem: to prescribe the constant mixed scalar curvature for a foliation by a conformal change of the metric in normal directions only. For a h…
Examines different approaches to Poisson structures in Banach spaces.
problem Exploring various definitions of Poisson structures in Banach spaces.
method Presenting and comparing different definitions of Poisson structures.
result Illustrates differences between existing definitions with examples.
The paper defines trans-para-Sasakian manifolds and explores their geometric properties.
problem Exploring the geometry of trans-para-Sasakian manifolds.
method Definition and study of curvature properties.
result Conditions for η−Einstein and Einstein manifolds. This paper explores Lorentzian manifolds with specific connections and their symmetries.
problem Characterizing Lorentzian manifolds with concircularly semi-symmetric metric connections.
method Investigates the properties of Lorentzian manifolds equipped with a concircularly semi-symmetric metric connection under specific conditions.
result Derives necessary and sufficient conditions for the manifold to be Einstein and proves that a perfect fluid space-time with a semi-symmetric metric P-connection is Ricci pseudo-symmetric manifold of constant type. In this paper we explore coarse properties of cusp-decomposable manifolds first defined by Nguyên Phan. We describe the large scale geometry of the universal cover of a cusp-decomposable manifold and of quasi-isometries between two such universal covers. This description will provide us the tools to prove quasi-isometr…
We explore the relation among volume, curvature and properness of a m-dimensional isometric immersion in a Riemannian manifold. We show that, when the Lp-norm of the mean curvature vector is bounded for some m≤p≤∞, and the ambient manifold is a Riemannian manifold with bounded geometry, properness …
The study explores conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
problem Conditions for a manifold to be a leaf of a complete foliation on the open unit ball.
method Analyzes conditions for manifolds to be leaves of complete foliations.
result Provides answers to the question of when a manifold can be a leaf of a complete foliation on the open unit ball.
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature (2,2). This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.
New uniformity definition on noncompact manifolds without metrics.
problem No Riemannian metric for uniformity on noncompact manifolds.
method Definition of uniformity without metrics, equivalent to bounded geometry.
result Equivalent definition of uniformity on noncompact manifolds.
The paper explores properties of special manifolds related to Yamabe solitons.
problem Characterizing and studying (LCS)n-manifolds with Yamabe solitons. method Analyzing curvature conditions and constructing specific manifolds.
result Characterizations and constructions of (LCS)n-manifolds with Yamabe solitons. This notes explores angle structures on ideally triangulated compact 3-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic 3-manifold with totally geodesic boundary has an ideal…
The paper explores algebraic and geometric structures on parallelizable manifolds.
problem Understanding algebraic and geometric structures on parallelizable manifolds.
method Definition of fundamental vector fields and their flows, leading to a product and loop structure.
result Induces a local loop structure and generalizes Lie algebra structure on the vector space.
New method finds large counterexamples by selectively exploring triangulations.
problem Finding small counterexamples in 3-manifold triangulations. method Selective enumeration of triangulations using heuristics.
result Found counterexamples to three conjectures about vertex triangulations.
In this paper, we explore the virtual technique that is very useful in studying moduli problem from differential geometric point of view. We introduce a class of new objects "virtual manifolds/orbifolds", on which we develop the integration theory. In particular, the virtual localization formula is obtained.
The paper explores inequalities on weighted Riemannian manifolds with boundary.
problem Developing inequalities on weighted Riemannian manifolds with boundary.
method Using a Reilly type integral formula associated with the φ-Laplacian.
result Provided inequalities of Brascamp-Lieb type and Colesanti type.
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 p-biharmonic and bi-p-harmonic map definitions.
problem No specific problem stated; extends definitions.
method Extends definitions of p-biharmonic and bi-p-harmonic maps. result Properties of extended maps explored.
Paper explores using 0-surgery to find exotic 4-manifolds.
problem Distinguishing smooth structures on 4-manifolds.
method Uses 0-surgery homeomorphisms to generate exotic pairs.
result Found 5 topologically slice knots potentially leading to exotic spheres.
The paper classifies self-replicating 3D shapes using algebraic models.
problem Understanding self-replicating 3D shapes.
method Using idempotents in the (2+1)-cobordism category to classify 3-manifolds.
result A classification theorem for self-replicating 3-manifolds.
Simple model explains manifold structure in high-dimensional data.
problem Understanding manifold structure in high-dimensional data.
method Latent Metric Model with latent variables, correlation, and stationarity.
result Establishes statistical explanation for manifold hypothesis.