We find G2-manifolds with specific asymptotic properties.
problem Existence and structure of G2-manifolds with ALC asymptotics.
method Robust Fredholm theory for ALC spaces, proving existence and rigidity results.
result Existence of a G2-analogue of the Atiyah-Hitchin metric and good moduli theory for ALC G2-holonomy metrics.
Constructs Spin(7) manifolds from self-dual Einstein 4-orbifolds.
problem Creating complete non-compact Spin(7) manifolds.
method Analytic construction using adiabatic limits of Spin(7) metrics on Seifert bundles over G2 orbifolds.
result Produces complete non-compact Spin(7) manifolds with arbitrary second Betti numbers and infinitely many families of ALC metrics.
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter…
Constructs non-abelian G2-instantons on ALC members of B7 family.
problem Constructing non-abelian G2-instantons on ALC members of B7 family.
method Using co-homogeneity one symmetries, classify and describe the solutions as perturbations of abelian instantons.
result Find a one-parameter family of instantons with polynomial decay.
The article constructs Spin(7) metrics with Aloff--Wallach spaces as orbits.
problem Creating Spin(7) metrics with specific geometric properties.
method Continuous 1-parameter families of non-compact Spin(7) metrics with chiralities, focusing on Aloff--Wallach spaces.
result Construction of Spin(7) metrics with Aloff--Wallach spaces as principal orbits, including geometric transitions.
New Spin(7) manifolds created from Calabi-Yau bundles.
problem Creating complete non-compact Spin(7) manifolds.
method Building on T2-bundles over AC Calabi-Yau manifolds. result First known examples of complete toric Spin(7) manifolds.
DALC customizes LSTM models for detectors in large-scale traffic networks.
problem Fine-grained traffic prediction for large-scale transportation networks.
method Formulated as a finite Markov decision process, introduced ALC algorithm for automatic customization, and developed DALC for distributed customization.
result DALC provides higher prediction accuracy than Apache Spark MLlib approaches.
New algorithm clusters time-series data faster and more efficiently.
problem Fast clustering of noisy, correlated time-series data.
method Agglomerative Likelihood Clustering (ALC) replaces genetic algorithm with recursive merging.
result ALC reduces compute time and resource usage for large datasets.
Constructs Einstein metrics on manifolds with specific orbits.
problem Finding Einstein metrics on manifolds with given orbits.
method Continuous families of metrics constructed using vector bundles and R4m+4. result Recovery of Spin(7) metrics A8 and B8. We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying…
The paper finds new Spin(7) metrics with specific orbits.
problem Existence of Spin(7) metrics with specified orbits. method Construction of three continuous families of non-compact Spin(7) metrics. result Existence of asymptotically conical and locally conical metrics.
We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by A_8, is complete and non-singular on R^8. The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S^4, and …
Paper improves preterm birth prediction using neural networks with noisy labels.
problem Predicting preterm birth from noisy EHR diagnosis codes.
method Developed ALC method to correct label noise in deep learning models.
result Improved prediction performance compared to baseline methods.
Defines (p,q) hermitian geometry and formulates it in generalised complex geometry.
problem Defines (p,q) hermitian geometry and formulates it in generalised complex geometry. method Formulates (p,q) hermitian geometry in generalised complex geometry. result Provides explicit formulae for the map to generalised geometry.
Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.
problem Characterize surfaces with constant ratio of principal curvatures in different geometries.
method Differential geometry, line geometry, Lie sphere geometry, ordinary differential equations, algebraic geometry.
result Characterized various types of surfaces like rotational, channel, ruled, helical, and translational.
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwe…
A car's motion shows flat parabolic geometry.
problem Understanding the geometry of a car's motion.
method Analyzing a car as a nonholonomic system.
result Shows a car's motion as a flat parabolic geometry.
Hessian geometry explains special geometries in N=2 theories.
problem Explaining special geometries in N=2 theories. method Formulating N=2 theories in terms of Hessian structures. result Hessian geometry relates special geometries of vector multiplets to hypermultiplet geometries.
Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.
problem Understanding non-lorentzian spacetimes.
method Classification and characterization of kinematical Lie algebras and their geometries.
result Characterization of Cartan geometries based on intrinsic torsion.
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
Surveying recent developments in Hitchin moduli space geometry.
problem Understanding the asymptotic geometry of Hitchin moduli space.
method Introduction to Hitchin moduli space and hyperkähler geometry.
result Recent developments in asymptotic geometry of Hitchin moduli space.
New definition of Born geometry connects to known geometries.
problem Defining and understanding Born geometries.
method Using Künneth structures and recursion operators.
result Born connection derived from Künneth connection for integrable geometries.
The paper extends group constructions to coset geometries, creating new ways to combine geometries.
problem Combining and gluing incidence geometries in a general framework.
method Extending classical group-theoretic constructions to coset geometries.
result Provides a general framework for combining or gluing incidence geometries.
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
New symmetries found in Riemann-Cartan geometries.
problem Investigating symmetries in geometries with curvature and torsion.
method Mathematical tools to determine symmetries and subclasses of geometries.
result Determined all static and stationary spherically symmetric Riemann-Cartan geometries and subclasses with specific symmetries.
Survey explores interactions between convex and complex geometry.
problem Understanding intersections between convex and complex geometry.
method Survey and review of existing literature.
result Demonstrates fascinating interactions between convex and complex geometry.
Lecture notes on geodesics in differential geometry.
problem Understanding geodesics in differential geometry.
method Expository lecture notes with exercises.
result Explains the geometry of geodesics.
Lecture notes on Finslerian geometry.
problem No specific problem stated; covers Finslerian geometry.
method Lecture notes.
result No specific key result mentioned.
Spin(7) geometry linked to multisymplectic geometry.
problem Understanding Spin(7) structures through multisymplectic geometry.
method Utilized Spin(7) identities to prove non-degeneracy of Cayley four-form in multisymplectic context.
result Spin(7) geometry is a special case of multisymplectic geometry.
The extension functors between categories of Cartan geometries can be used to define different categories of Cartan geometries with additional morphisms. The Cartan geometries modeled on skeletons can be used for the description of such categories of Cartan geometries and therefore we develop the theory of Cartan geome…
A geometric transition is a continuous path of geometric structures that changes type, meaning that the model geometry, i.e. the homogeneous space on which the structures are modeled, abruptly changes. In order to rigorously study transitions, one must define a notion of geometric limit at the level of homogeneous spac…
Introduces logarithmic Cartan geometry on complex manifolds with singularities.
problem Holomorphic Cartan geometry with singularities.
method Definition and study of logarithmic Cartan geometry on complex manifolds with polar part supported on a normal crossing divisor.
result Push-forward of a Cartan geometry constructed using a finite Galois ramified covering is a logarithmic Cartan geometry.
Paper introduces a generalized Bures-Wasserstein geometry for SPD matrices.
problem Understanding the geometry of SPD matrices for machine learning.
method Proposes a generalized Bures-Wasserstein geometry parameterized by a symmetric positive definite matrix.
result The GBW geometry outperforms the BW geometry in machine learning applications.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Maps strong Kähler with torsion to Generalised Complex Geometry.
problem Representing strong Kähler with torsion in Generalised Complex Geometry.
method Analogy to Gualtieri map for (2,2) models. result Representation of strong Kähler with torsion in Generalised Complex Geometry.
Surveying probabilistic real algebraic geometry.
problem Classical problems in real algebraic geometry.
method Probabilistic perspective on classical topics.
result Modern approach to Hilbert's Sixteenth Problem.
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
The study sets limits on the complexity of Klein geometries.
problem Understanding the complexity of Klein geometries.
method Simple upper and lower bounds for the order of Klein geometries.
result Established upper and lower bounds for the order of Klein geometries.
Ray-marching method visualizes 8 Thurston geometries in real-time.
problem Accurately rendering and visualizing Thurston geometries in real-time.
method Ray-marching algorithms with theoretical framework for non-Euclidean geometries.
result Accurate interactive real-time views of Thurston geometries achieved.
The paper extends Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
problem Extending Ruh-Vilms theorem to hypersurfaces in Weitzenböck geometry.
method Extending the Ruh-Vilms theorem to Weitzenböck geometry, considering the flat geometry with torsion.
result The Laplacian of the Gauss map for hypersurfaces in Weitzenböck geometry is related to the mean curvature vector field.
New algebraic-geometry method for Ribaucour transformations.
problem Classical differential geometry problems.
method Algebraic-geometry approach to constructing orthogonal nets.
result Obtains smooth orthogonal nets as Ribaucour transformations.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Two open problems in Sasaki geometry are discussed.
problem Open problems in Sasaki geometry.
method Discussion and description of open problems.
result Two open problems in Sasaki geometry are identified.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
The study of special metrics in various parabolic geometries.
problem Finding special metrics in parabolic geometries.
method Investigating invariant conditions for special metrics in different parabolic geometries.
result Characterization of special metrics in hypersurface CR and contact Legendrean cases.
We classify the 5-dimensional homogeneous geometries in the sense of Thurston. The present paper (part 3 of 3) classifies those in which the linear isotropy representation is nontrivial but reducible. Most of the resulting geometries are products. Some interesting examples include a countably infinite family of inequiv…
We introduce the notion of manifolds of amalgamation geometry and its generalization, split geometry. We show that the limit set of any surface group of split geometry is locally connected, by constructing a natural Cannon-Thurston map.
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.