Study families of Morse functions for manifolds with boundary.
problem Characterize degeneracies in 1-parameter families of Morse functions.
method List all possible degeneracies in generic 1-parameter families.
result Identified all degeneracies in generic 1-parameter families.
Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connect…
Geodesic orbit metrics on real flag manifolds identified.
problem Classifying real flag manifolds with geodesic orbit metrics.
method Investigated invariant metrics on real flag manifolds, focusing on those where geodesics are orbits of one-parameter subgroups.
result Non-trivial geodesic orbit metrics exist on real flag manifolds, unlike in the complex case.
The paper establishes a correspondence between special Kähler manifolds and their deformations.
problem Mapping between affine and projective special Kähler manifolds.
method Formulation of a correspondence and application to r-maps in string theory.
result One-parameter deformations of projective special Kähler manifolds correspond to perturbative α'-corrections.
Solves geodesics in homogeneous manifolds using one-parameter subgroups.
problem Finding geodesics in homogeneous manifolds with various symmetries.
method Explicitly solving the geodesic equation for a wide class of homogeneous manifolds, proving geodesic completeness, and showing geodesics as orbits of one-parameter subgroups.
result Geodesics in homogeneous manifolds are orbits of one-parameter subgroups, and metrics can be N-parameter deformations of a Riemannian metric with special symmetries.
Study explores warped geometries of tensor manifolds, finding non-geodesic connections for some parameters.
problem Investigate non-geodesic connections in warped Segre-Veronese manifolds.
method Investigate a one-parameter family of warped geometries, presenting closed expressions for maps and distance.
result Segre-Veronese manifolds are not geodesically connected in Euclidean geometry but can be for some warping parameters.
A flow defined by a nonsingular smooth vector field X on a closed manifold M is said to be parameter rigid if given any real valued smooth function f on M, there are a smooth funcion g and a constant c such that f=X(g)+c holds. We show that the parameter rigid flows on closed orientable 3-manifolds are sm…
Maximizing mutual information selects simple models from limited data.
problem Selecting simple models from finite and potentially noisy data.
method Prior choice that maximizes mutual information between parameters and predictions.
result The method selects a lower-dimensional effective theory by ignoring poorly constrained parameters.
Sparse reduced-rank regression selects variables and ranks via manifold optimization.
problem Traditional rank selection fails when true rank is high.
method Sparse regularization and manifold optimization for rank and variable selection.
result Accurate estimation of coefficient parameter with high true rank.
This paper extends VB to manifolds for more flexible parameter spaces.
problem Limited applicability of VB due to Euclidean parameter spaces.
method Developed manifold-based VB algorithms exploiting geometric and information geometry.
result Proven convergence and improved performance compared to existing methods.
Superized Kaehler manifolds with continuous parameters.
problem How to extend Lie algebra actions to Lie superalgebras.
method Continuous parameterization of Kaehler manifolds and hyper-Kaehler supermanifolds.
result Definitions of hyper-Kaehler supermanifolds with parameters.
We give a simple proof of the local version of a result of R. Bryant, stating that any 3-dimensional Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We refine the theorem proving that a certain class of one-parameter families of metrics on a 3-torus can be…
Given a geometrically finite hyperbolic cone-manifold, with the cone singularity sufficiently short, we construct a one parameter family of cone-manifolds decreasing the cone angle to zero. We also control the geometry of this one parameter family via the Schwarzian derivative of the projective boundary and the length …
On a main class of the almost contact manifolds with B-metric, it is described the family of the linear connections preserving the manifold's structures by 4 parameters. In this family there are determined the canonical-type connection and the connection with zero parameters.
We introduce a method for constructing skills capable of solving tasks drawn from a distribution of parameterized reinforcement learning problems. The method draws example tasks from a distribution of interest and uses the corresponding learned policies to estimate the topology of the lower-dimensional piecewise-smooth…
This Ph.D. thesis is devoted to the constructions of Lagrangian formulation on Finsler and Kawaguchi manifolds. While Finsler geometry is a natural extension of Riemannian geometry, Kawaguchi geometry is the extension of Finsler geometry to higher order derivatives and to k-dimensional parameter space. The latter exten…
New method for identifying autoregressive systems on manifolds.
problem Identifying autoregressive systems on Stiefel and Grassmann manifolds.
method Defining parameters as orthogonal group elements, averaging over observations, conjugate gradient descent on manifolds.
result System parameters can be estimated efficiently using the proposed algorithm.
Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on g(g+1)/2 complex parameters where g is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of…
We study the volume growth of hyperkaehler manifolds of type A∞ constructed by Anderson-Kronheimer-LeBrun and Goto. These are noncompact complete 4-dimensional hyperkaehler manifolds of infinite topological type. These manifolds have the same topology but the hyperkaehler metrics are depends on the choice of …
Constructing compact non-Kähler manifolds with and without the Hard Lefschetz Condition
problem Symplectic non-Kähler manifolds
method One-parameter family of symplectic forms on orbifold
result Symplectic manifolds with HLC and non-HLC structures
A parametric manifold is a manifold on which all tensor fields depend on an additional parameter, such as time, together with a parametric structure, namely a given (parametric) 1-form field. Such a manifold admits natural generalizations of Lie differentiation, exterior differentiation, and covariant differentiation, …
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space and anti-de Sitter 3-space is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula wh…
The paper classifies 3D spherical Sasakian manifolds using geometric and algebraic methods.
problem Classifying 3D spherical Sasakian manifolds with specific properties.
method Establishing correspondence between different sets of parameters and geometrically describing the moduli space.
result Determination of Sasakian automorphism groups and detection of homogeneous Sasakian manifolds.
The paper examines conditions for Einstein multiply warped products and estimates their parameters.
problem Existence and non-existence of non-trivial Einstein multiply warped products.
method Analyzes conditions for the existence or non-existence of Einstein multiply warped products, especially generalized Kasner type.
result Estimates the Einstein parameter that conditions the existence of such metrics.
Inversion-free natural gradient method for Riemannian manifolds.
problem Hindered by the need for Euclidean space, Fisher information matrix inversion, and computational cost.
method Intrinsic, inversion-free natural gradient method on Riemannian manifolds, using moving approximation of inverse FIM.
result Almost-sure convergence rates and sub-quadratic storage complexity for large-scale applications.
In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parame…
Constructing solutions to the heterotic G2 system on specific types of manifolds.
problem Finding solutions to the heterotic G2 system on certain types of manifolds. method Investigating a 1-parameter family of G2-connections on tangent bundles. result Obtaining several approximate and one new class of exact solutions on degenerate 3-(α,δ)-Sasaki manifolds. Constructs Lorentzian manifolds from Riemannian conformal structures.
problem Creating Lorentzian manifolds from Riemannian conformal structures.
method Starting from a Riemannian conformal structure, a family of Lorentzian manifolds is constructed using a metric in the conformal class and a 1-parameter family of tensor fields.
result Every Mobius structure on a Riemannian conformal structure arises from this construction.
Graphons connect graph structures to manifold properties.
problem Interpolating between graphs and manifolds.
method Graph-to-graphon and graphon-to-manifold convergence.
result Established monotonicity inequality linking combinatorial and geometric parameters.
Study shows consistency of shallow GCNNs on sampled point clouds under manifold assumption.
problem Consistency of shallow GCNNs on sampled point clouds under manifold assumption.
method Functional analysis perspective, weakly compact product of unit balls, Sobolev regularity, frequency cutoff.
result Proves Γ-convergence of regularized empirical risk minimization functionals and convergence of their global minimizers. Study on Lie groups and their geometric properties.
problem Classifying 4D Lie algebras and analyzing their geometric structures.
method Investigation of 4D Lie groups as manifolds, focusing on indecomposable real Lie algebras with two parameters and their almost hypercomplex structures with Hermitian-Norden metrics.
result Geometric characteristics of almost hypercomplex manifolds derived from 4D Lie groups.
The study reveals a persistent bias in the distribution of holonomy on compact hyperbolic 3-manifolds.
problem The distribution of holonomy on compact hyperbolic 3-manifolds is not uniformly distributed.
method An asymptotic count of closed geodesics by their length and holonomy, and analysis of spectral parameters.
result A normalized, smoothed bias count of holonomy is distributed according to a probability distribution, controlled by the number of zero spectral parameters.
No 3-manifolds have triangulations of bounded treewidth.
problem Understanding the limitations of triangulations in 3-manifolds.
method Analyzing dual graphs and connections between topology and graph width parameters.
result Explicit connections between 3-manifold topology and dual graph width parameters.
In this paper we study slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field. We prove that for non-geodesic such curves there exists a unique Frenet frame for which the original parameter is distinguished. Moreover, we obtain …
The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral repre…
Generalizes Aubin's result for Yamabe-type problem on smooth metric measure spaces.
problem Solving Yamabe-type problem on smooth metric measure spaces.
method Generalization of Aubin's result for nonlocally conformally flat manifolds with dimension ≥ 6 and parameter m close to nonnegative integers.
result Generalization of Aubin's result for Yamabe-type problem.
New bounds on treewidth for 3-manifolds, linking topology and computational tractability.
problem Understanding the relationship between the topology of 3-manifolds and their treewidth.
method Analyzing the dual graphs of triangulations of 3-manifolds to determine treewidth bounds.
result Closed, orientable 3-manifolds have treewidth at most 4g(M)-2, where g(M) is the Heegaard genus.
This paper proposes a new method to adapt ROMs for new parameter settings.
problem ROMs lack robustness when applied to new parameter settings.
method Regression trees on Grassmann Manifold to learn the mapping between parameters and POD bases.
result The proposed method is capable of establishing the mapping between parameters and POD bases, thus adapting ROMs for new parameters.
Algorithm calculates quantum invariants of 3-manifolds with polynomial time complexity.
problem Computing quantum invariants from Tambara-Yamagami categories is #P-hard.
method Fixed-parameter tractable algorithm with first Betti number as parameter.
result Existence of FPT algorithm for Tambara-Yamagami invariants.
Survey of stated skein modules/algebras of 3-manifolds/surfaces.
problem Understanding stated skein modules/algebras of 3-manifolds/surfaces.
method Discussion of splitting homomorphism, general structures, Frobenius homomorphism, center, dimension, representation theory.
result Skein algebra of non-closed marked surface at any root of 1 is a maximal order.
Method finds approximate Ricci-flat metrics on Calabi-Yau manifolds.
problem Finding analytic Kähler potentials for Calabi-Yau manifolds.
method Numerically calculating Ricci-flat Kähler potentials via machine learning and fitting to Donaldson's Ansatz.
result Simple analytic expressions for approximately Ricci-flat Kähler potentials are found, including explicit dependence on complex structure parameter.
Paper finds exotic diffeomorphisms on specific 4-manifolds.
problem Understanding exotic diffeomorphisms on 4-manifolds with b2+=2. method Comparing winding numbers of parameter families.
result 2\mathbb{C}\mathbb{P}^2 \# 10 (-\mathbb{C}\mathbb{P}^2) admits exotic diffeomorphisms.
We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius co…
New families of non-singular geodesic orbit nilmanifolds discovered.
problem Classifying non-singular geodesic orbit nilmanifolds.
method Complete classification through analysis of nilmanifolds.
result New families of non-singular GO nilmanifolds with dimensions 14 and 15.
Due to the growing ubiquity of unlabeled data, learning with unlabeled data is attracting increasing attention in machine learning. In this paper, we propose a novel semi-supervised kernel learning method which can seamlessly combine manifold structure of unlabeled data and Regularized Least-Squares (RLS) to learn a ne…
Study of Dirac-like operators on spin manifolds with large mass parameters.
problem Understanding spectra of Dirac-like operators with piecewise constant mass terms.
method Analysis of asymptotic regimes to derive effective operators.
result Extension of MIT Bag operator concept to spin geometry.
There are found exact values of (Matveev) complexity for the 2-parameter family of hyperbolic 3-manifolds with boundary constructed by Paoluzzi and Zimmermann. Moreover, ε-invariants for these manifolds are calculated.
Efficiently optimizes CNN and RNN parameters on Stiefel manifold.
problem Computational expense in optimizing orthonormal matrices on Stiefel manifold.
method Cayley transform for efficient retraction and vector transport on Stiefel manifold.
result Cayley SGD and ADAM achieve faster convergence and less training time.