Essential dimension of hyperkähler manifolds families is at most 1.
problem Essential dimension of complex manifold families over compact bases.
method Proved essential dimension not greater than 1 for hyperkähler manifolds families over compact simply connected bases.
result Essential dimension of hyperkähler manifolds families is at most 1.
Paper develops a new method for harmonic maps into symmetric spaces.
problem Tackles harmonic maps of finite uniton number into symmetric spaces.
method Uses loop group description and DPW theory to transform results.
result Establishes a 1-1 relation between harmonic maps and normalized potentials.
The paper proves compactness of warped product metrics on S²×S¹ with varying base metrics.
problem Proving compactness of warped product metrics on S²×S¹ with nonnegative scalar curvature.
method Using Gromov-Sormani MinA scalar curvature compactness conjecture, the paper proves a uniform diameter bound for the base surfaces, compactness of the base warping functions, and convergence of the metrics.
result The metrics converge to a limit metric with nonnegative scalar curvature in the distributional sense.
The paper generalizes spectral section concepts to non-compact spaces.
problem Generalizing spectral sections to non-compact base spaces.
method Generalization to arbitrary base spaces, applications to cobordism theorems, investigation of Riesz continuity.
result If a family of operators has a spectral section, it is Riesz continuous.
Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
problem Characterize compact quasi-Einstein metrics with constant scalar curvature.
method Connection to Sasakian geometry and circle bundles over Einstein metrics.
result Compact quasi-Einstein metrics with constant scalar curvature are locally homogeneous in 3D.
Formulae for Dixmier trace and noncommutative residue on compact manifolds.
problem Calculating traces and residues for pseudo-differential operators on compact manifolds.
method Using global symbols, Fourier analysis, representation theory, and pseudo-differential calculus.
result Formulae for Dixmier trace and noncommutative residue on compact manifolds.
Compact clustering in latent space improves semi-supervised learning.
problem Improving semi-supervised learning with unlabeled data.
method Dynamic graph creation over embeddings, label propagation, and Markov chain regularization.
result Compact clustering in latent space facilitates better separation and separation of labeled and unlabeled data.
The Kähler-Ricci flow and continuity method converge to compact metrics on Calabi-Yau fibrations.
problem Analyzing convergence of Kähler-Ricci flow and continuity method on Calabi-Yau fibrations.
method Uniform lower bound on Ricci curvature, Gromov-Hausdorff topology, metric completion of generalized Kähler-Einstein current.
result Convergence to compact metrics on Calabi-Yau fibrations.
Vanishing theorem for certain tensor fields on compact Hermitian manifolds.
problem Vanishing theorem for holomorphic tensor fields on compact Hermitian manifolds.
method Inspired by X. Yang and L. Ni-F. Zheng's ideas, the proof uses the definiteness of holomorphic sectional curvature.
result Spaces of certain holomorphic tensor fields are trivial under the definiteness of holomorphic sectional curvature.
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
Existence and convergence of ancient Ricci flow solutions on compact homogeneous spaces.
problem Existence and characterization of ancient solutions to the Ricci flow on compact homogeneous spaces.
method General existence theorem and Gromov-Hausdorff convergence under rescaling.
result Convergence of collapsed ancient solutions to Einstein metrics on torus fibrations.
Based on the representation theory and the study on the involutions of compact simple Lie groups, we show that F4 admits non-naturally reductive Einstein metrics.
The present research work proposes a new fast fixed-point averaging algorithm on the compact Stiefel manifold based on a mixed retraction/lifting pair. Numerical comparisons between fixed-point algorithms based on the proposed non-associated retraction/lifting map pair and two associated retraction/lifting pairs confir…
The paper classifies compact affine quaternionic curves and surfaces.
problem Classifying compact affine quaternionic curves and surfaces.
method Affine quaternionic manifolds, Kodaira Theorem, fundamental groups, Lie Groups.
result Only quaternionic tori and primary Hopf surface S^3 x S^1 are compact affine quaternionic curves.
We classify all compact simply connected homogeneous CR manifolds M of codimension one and with non-degenerate Levi form up to CR equivalence. The classification is based on our previous results and on a description of the maximal connected compact group G(M) of automorphisms of M. We characterize also the standa…
Paper answers Gromov's compactness question on noncompact manifolds.
problem Gromov's compactness question on positive scalar curvature metrics.
method Construct examples and prove vanishing of certain index theoretic invariants.
result Positive answer for a class of 1-tame manifolds.
Compact surfaces with constant mean curvature found in Killing submersions.
problem Finding compact surfaces with constant mean curvature in Killing submersions.
method Classification of Killing submersions, proving existence of minimal sections, solving Bernstein and Plateau problems.
result Compact stable surfaces with constant mean curvature are either entire minimal sections or tangent to the Killing direction.
In non-compact manifolds, geodesic flowers exist.
problem Existence of geodesic flowers in non-compact manifolds.
method Proving the existence of non-trivial geodesic flowers in complete non-compact manifolds with locally convex ends.
result Non-trivial geodesic flowers exist in every complete non-compact manifold with locally convex ends.
Study shows algebraic nature of manifold submetries on compact spaces.
problem Understanding manifold submetries on compact homogeneous spaces.
method Analyzes singular Riemannian foliations and manifold submetries on compact normal homogeneous spaces.
result Establishes a one-to-one correspondence between algebras of preserved functions and manifold submetries.
Novel approach ensures stability of compact schemes for variable PDEs.
problem Ensuring stability of compact schemes for variable coefficient PDEs.
method Difference equation approach to derive stability conditions.
result Derives sufficient condition for unconditional stability.
We investigate the finiteness structure of a complete non-compact n-dimensional Riemannian manifold M whose radial curvature at a base point of M is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than π. We show, as our main theorem, that all Buse…
Compact fibration metrics inherit positive holomorphic curvature.
problem Conditions for positive curvature in fibration metrics.
method Warped product metric approach, differing from Cheung's negative curvature case.
result Total space of fibration inherits positive holomorphic sectional curvature.
Kerckhoff and Storm conjectured that compact hyperbolic n-orbifolds with totally geodesic boundary are infinitesimally rigid when n>3. This paper verifies this conjecture for a specific example based on the 4-dimensional hyperbolic 120-cell.
Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebr…
Study string topology on symmetric spaces, showing non-triviality and nilpotency results.
problem Understanding the structure of string topology on symmetric spaces.
method Used cycles from Bott-Samelson and Ziller to study coproduct and product.
result Showed non-triviality and nilpotency of Chas-Sullivan product and coproduct for higher rank symmetric spaces.
Compact SE models are created with PP and PQ techniques, reducing size by 10.03%.
problem Balancing denoising performance and computational cost in SE models.
method Parameter pruning and quantization techniques integrated for compactness.
result 10.03% reduction in model size with minor performance losses.
Study shows typical scales for manifolds with lower Ricci bounds.
problem Understanding typical scales in manifolds with lower Ricci curvature bounds.
method Analysis of collapsing sequences of Riemannian manifolds with uniform lower Ricci curvature bounds.
result Rescaled manifolds subconverge to a product of a Euclidean and a compact space.
Paper develops data-driven compact models for diodes.
problem Manual and time-consuming compact model development.
method Machine Learning techniques for automation.
result Data-driven models accurately predict diode behavior.
In this paper we first use the result in [12] to remove the assumption of the L2 boundedness of Weyl curvature in the gap theorem in [9] and then obtain a gap theorem for a class of conformally compact Einstein manifolds with very large renormalized volume. We also uses the blow-up method to derive curvature est…
The paper proves a compactness theorem for spaces with Bakry-Emery Ricci tensor.
problem Compactness in spaces with Bakry-Emery Ricci tensor.
method Proving f-mean curvature comparison and defining a Myers-type compactness theorem. result Improves a result from Soylu by using a weaker condition on f′(t). GNMC reduces XCSF population size while preserving function approximation and policy accuracy.
problem Population bloat in XCSF compaction.
method Introduced GNMC, a novel compaction algorithm.
result GNMC reduces population size significantly without compromising function approximation or policy accuracy.
Extends optimal regularity and Uhlenbeck compactness to non-Riemannian manifolds.
problem Establishing optimal regularity and compactness for connections on vector bundles over non-Riemannian manifolds.
method Proofs based on RT-equations for connections with Lp curvature, extending to non-compact gauge groups. result Removes singularities at GR shock waves, ensuring existence of geodesics and coordinates.
Survey on rigorous construction of supersymmetric path integral.
problem Rigorous construction of supersymmetric path integral.
method Construction based on joint work with collaborators.
result Rigorous construction of supersymmetric path integral.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex Monge-Ampère, Hessian and inverse Hessian equations. As an application we solve a class of He…
DCAE learns compact latent representations for one-class novelty detection.
problem Learning compact latent representations for one-class novelty detection.
method DCAE learns compact and collapse-free latent representations through internal discriminative layers of GANs, reconstructing in-class data finely and exclusively.
result DCAE achieves state-of-the-art performance on novelty and adversarial example detection.
These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds, aimed at graduate students with some background in differential geometry.
Proves conditions for separating regions in homogeneous spaces without trivial topology.
problem Separating regions in homogeneous, locally compact spaces without trivial topology.
method Analyzes properties of closed subsets and their boundaries in Čech cohomology.
result Conditions for irreducible separation without trivial topology.
The paper proves a structure theorem for compact Kähler manifolds with semi-positive holomorphic sectional curvature.
problem Understanding the fundamental groups of compact Kähler manifolds with specific curvature properties.
method Analyzing foliations and topological properties to prove a locally trivial fibration structure.
result Compact Kähler manifolds with semi-positive holomorphic sectional curvature admit a locally trivial fibration structure.
In this work, we describe how to obtain the structure of an infinite-dimensional Lie group on the group of compactly carried bundle automorphisms Autc(P) for a locally convex prinicpal bundle P over a finite-dimensional smooth sigma-compact base M. This is a generalization of previous work by Wockel, where the base M w…
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Algorithm solves American options with regime-switching using multigrid and compact finite difference.
problem Pricing American put options with regime-switching.
method Multigrid iterative algorithm based on compact finite difference schemes and Hermite interpolation.
result The algorithm provides a fast and efficient tool for pricing American put options with regime-switching.
New method grows many trees in parallel and selects a compact subset for optimization.
problem Creating interpretable and compact tree ensembles.
method Growing a large pool of trees in parallel with loss optimization on a subset.
result Compact tree ensemble achieves lower loss and misclassification error.
In this article, we prove new rigidity results for compact Riemannian spin manifolds with boundary whose scalar curvature is bounded from below by a non-positive constant. In particular, we obtain generalizations of a result of Hang-Wang \cite{hangwang1} based on a conjecture of Schroeder and Strake \cite{schroeder}.
Defines geometric quantization for non-compact Hamiltonian torus manifolds using index theory.
problem Geometric quantization for non-compact Hamiltonian torus manifolds.
method Deformation of Dirac operator along group orbits, localization to lattice points.
result Geometric quantization is independent of the choice of polarization.
Extends optimal regularity and compactness to vector bundles over non-Riemannian manifolds.
problem Optimal regularity and compactness for connections on vector bundles.
method Derive RT-equations, establish existence theory, handle curvature up to L1. result Optimal regularity and compactness extended to vector bundles over non-Riemannian manifolds.
New relation found between embolic volume and Betti numbers.
problem Relating embolic volume to Betti numbers of compact manifolds.
method Proof based on Gromov's method in systolic geometry.
result Improved result compared to previous work by Durumeric.
Researchers extend Gamma index theorem to non-compact spacetimes.
problem Establishing an L2-Gamma index for non-compact spacetimes. method Rewriting L2-Gamma index in terms of spectral flow and connecting to geometric expressions. result Extends Bär and Strohmaier's work to non-compact Cauchy hypersurfaces.
Paper proposes angular loss for better face recognition and object classification.
problem Improving intra-class compactness and preventing overfitting in face recognition and object classification.
method Angular loss function to maximize angular gradient, reducing overfitting and requiring only one adjustable constant.
result Our method outperforms other methods in accuracy, discriminative information, and time-efficiency.