Study calculates the maximal dimension of totally geodesic submanifolds in symmetric spaces.
problem Understanding the maximal dimension of totally geodesic submanifolds in symmetric spaces.
method Defined and computed the splitting rank for symmetric spaces, and analyzed bounded cohomology.
result Explicitly computed the splitting rank for each irreducible symmetric space and showed the surjectivity of a comparison map.
We show that every p-fold strictly-cyclic branched covering of a b-bridge link in the 3-sphere admits a p-symmetric Heegaard splitting of genus g=(b-1)(p-1). This gives a complete converse to a result of Birman and Hilden, and gives an intrinsic characterization of p-symmetric Heegaard splittings as p-fold strictly-cyc…
We show that every p-fold strictly-cyclic branched covering of a b-bridge link in S3 admits a p-symmetric Heegaard splitting - in the sense of Birman and Hilden - of genus g=(b−1)(p−1). This gives a complete converse of one of the results of the two authors. Moreover, we introduce the concept of weakly p-symmetric…
Study characterizes totally geodesic submanifolds in quotient spaces.
problem Characterizing totally geodesic submanifolds in quotient spaces.
method Characterization through totally geodesic submanifolds and holomorphic tangent sequence splitting.
result Characterization of totally geodesic submanifolds in quotient spaces.
Study splitting submanifolds in specific homogeneous spaces.
problem Classify splitting submanifolds in rational homogeneous spaces of Picard number one.
method Use global holomorphic vector fields and projection maps to analyze submanifolds.
result Proves submanifolds in certain spaces are rational or Hermitian symmetric.
Study non-split supermanifolds from complex manifolds.
problem Classify non-split supermanifolds retracting to complex manifolds.
method Construct supermanifolds from d-closed (1,1)-forms on complex manifolds. result Complete classification of non-split supermanifolds for certain flag manifolds.
In this paper, I prove a splitting theorem for equifocal submanifolds with non-flat section in a simply connected symmetric space of compact type. Also, by using the splitting theorem, I prove that the sections of equifocal submanifolds with non-flat section in an irreducible simply connected symmetric space of compact…
New Einstein manifolds split into symmetric and compact parts.
problem Understanding Einstein manifolds with unimodular isometry groups.
method Theory of polar actions, Lie-theoretic arguments, and maximum principles.
result Negative Einstein manifolds split into symmetric and compact parts.
New method uses symmetric splitting for efficient HMC inference in large neural networks.
problem Efficient inference for Bayesian neural networks with large datasets.
method Introduces a symmetric integration scheme for Hamiltonian Monte Carlo (HMC) that does not rely on stochastic gradients.
result Symmetric splitting leads to more efficient HMC inference over large data sets.
Study of foliations on symmetric spaces and mean curvature flow results.
problem Understanding foliations on symmetric spaces with non-negative curvature.
method Proving foliations are isoparametric and applying mean curvature flow.
result Ancient solutions to mean curvature flow of regular leaves.
Unique submaximal symmetry found for certain parabolic geometries.
problem Determining the next realizable symmetry dimension in parabolic geometries.
method Analyzing submaximally symmetric structures of type (G,P) for specific Lie groups. result Local uniqueness of submaximally symmetric structures established.
Study PSCT manifolds splitting into well-understood factors.
problem Understanding Riemannian manifolds with specific torsion properties.
method Prove PSCT manifolds locally split into products of known factors.
result PSCT manifolds locally split into products of well-understood factors.
New algorithm samples Bayesian neural networks for improved calibration.
problem Improving calibration of Bayesian neural networks.
method Symmetric Minibatch Splitting-UBU (SMS-UBU) algorithm.
result SMS-UBU provides better calibration performance than standard methods.
Study maximally symmetric distribution of An-Nurowski surface rolling on a plane.
problem Maximally symmetric (2,3,5)-distribution of An-Nurowski surface rolling without slipping or twisting. method Calculated vector fields defining a split g2 Lie algebra and projected to an action of SL(3,R). result Obtained an action of SL(3,R) on the configuration space without a surface. Proves conjecture about foliations on curved spaces.
problem Completeness of dual foliations on curved spaces.
method Analyzes Riemannian foliations on nonnegatively curved symmetric spaces.
result Foliations split into trivial and single dual leaf foliations.
Integrally splits L-spectra of integers into simpler components.
problem Understanding the homotopy type of L-spectra of integers.
method Using Anderson duality and splitting into simpler spectra.
result Splits L-spectra of integers into simpler components.
New submersion proves complex-valued harmonic map existence.
problem Existence of non-constant harmonic morphisms.
method Constructing harmonic Riemannian submersions from symmetric spaces.
result Existence of non-constant, globally defined complex-valued harmonic morphism.
Efficient multisections found for odd-dimensional tori.
problem Finding optimal multisections of odd-dimensional tori.
method Constructing multisections with specific properties (genus n, symmetry) for odd-dimensional tori. result Optimal multisections of genus n found for odd-dimensional tori. A new method solves SymNMF problems faster and more efficiently.
problem Symmetric nonnegative matrix factorization (SymNMF) for data analytics.
method Nonconvex variable splitting method.
result The method converges to KKT points and has a global sublinear convergence rate.
New criteria for splitting definite 4-manifolds with cyclic groups.
problem Definite 4-manifolds with infinite cyclic fundamental groups.
method Two new criteria extending previous results, equivalent to algebraic representation conditions.
result Equivalence to algebraic representation conditions and production of new forms.
Cohomogeneity-one actions on symmetric spaces of mixed type
problem Classifying cohomogeneity-one actions on symmetric spaces
method Using a new family of diagonal cohomogeneity-one actions
result Reducing the classification problem to symmetric spaces of a single type
Local equivalence found between maximally symmetric rolling and flat Cartan distributions.
problem Establishing local equivalence between maximally symmetric rolling and flat Cartan distributions.
method Using complex parametrisation of su(2), a change of coordinates maps the maximally symmetric rolling (2,3,5)-distribution to the flat Cartan distribution. result Local equivalence between maximally symmetric rolling and flat Cartan distributions established.
Affine maps reveal higher rank structures in certain spaces.
problem Characterizing spaces with higher rank structures.
method Using Hadamard spaces with geometric group actions and affine maps.
result Affine maps not dilations indicate higher rank structures.
New algebraic structures for Lie 2-algebroids and their connections.
problem Characterizing and understanding Lie 2-algebroids and their structures.
method Construction of homotopy Poisson algebra and introduction of Dirac structures.
result One-to-one correspondence between Manin triples and Lie 2-bialgebroids.
We prove properties of linking forms on rational homology spheres.
problem Properties of linking forms on rational homology spheres.
method Use of Heegaard splittings and properties of Q/Z-valued linking forms. result Linking forms on rational homology spheres are symmetric or anti-symmetric.
Optimizes data splitting for shorter conformal prediction intervals.
problem Minimizing prediction interval length while maintaining coverage.
method Theoretical framework for optimal data splitting in split conformal prediction.
result Analytical characterizations of length-optimal split ratios in various settings.
New minimal surface theory disproves a conjecture in symmetric spaces.
problem Proving the existence of unstable minimal maps in symmetric spaces.
method Using Hitchin representations and equivariant maps, with a new index bound.
result Disproves the Labourie conjecture for PSL(n,R) with n≥4. In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to des…
We prove that given a Hitchin representation in a real split rank 2 group G0, there exists a unique equivariant minimal surface in the corresponding symmetric space. As a corollary, we obtain a parametrization of the Hitchin components by a Hermitian bundle over Teichmüller space. The proof goes through intr…
A family of one-vertex triangulations of 3-manifolds, layered-triangulations, is defined. Layered-triangulations are first described for handlebodies and then extended to all 3-manifolds via Heegaard splittings. A complete and detailed analysis of layered-triangulations is given in the cases of the solid torus and lens…
New ICA method uses third-order moment for better performance.
problem ICA methods often assume kurtosis, which may not always fit data.
method ICA based on Split Generalized Gaussian distribution (SGGD).
result Method works better for heavy-tailed and non-symmetric data.
New formulas for p-capacitary potentials in convex domains.
problem Analyzing geometric properties of p-capacitary potentials.
method Monotonicity formulas derived from p-Laplace equation solutions.
result New characterizations of rotationally symmetric solutions and domains.
Study automorphisms and real structures on a special super-Grassmannian.
problem Investigate automorphisms and real structures on a Π-symmetric super-Grassmannian. method Investigate automorphisms and real structures on a Π-symmetric super-Grassmannian ΠGrn,k using Galois cohomology. result Classify real structures on ΠGrn,k and compute corresponding supermanifolds of real points. We simplify symmetric NMF by transforming it into a nonsymmetric problem, enabling faster and more efficient solutions.
problem Efficiently solving symmetric nonnegative matrix factorization (NMF).
method Transforming symmetric NMF into a nonsymmetric problem, applying fast alternating algorithms, and rigorously proving convergence.
result Fast algorithms for symmetric NMF can converge to a critical point at least at a sublinear rate.
The study examines conditions for weak nearly cosymplectic manifolds to split into products.
problem Understanding the curvature and topology of weak nearly cosymplectic manifolds.
method Analyzes the conditions for splitting and characterizes specific manifolds.
result Conditions for weak nearly cosymplectic manifolds to become Riemannian products are identified.
Local equivalence shown between specific distributions and flat Cartan distribution.
problem Establishing local equivalence between specific distributions and flat Cartan distribution.
method Change of coordinates mapping specific distributions to flat Cartan distribution.
result Local equivalence between maximally symmetric (2,3,5)-distributions and flat Cartan distribution. Characterizes circles in self-dual symmetric R-spaces using geometric properties.
problem Defines and characterizes special curves (circles) in self-dual symmetric R-spaces.
method Characterizes elements of the transformation group G and describes circles in Riemannian geometric terms.
result Describes circles in terms of maximal compact subgroups and geodesics.
The paper studies geometries with parallel skew-symmetric torsion and their submersions.
problem Understanding geometries with parallel skew-symmetric torsion.
method Analyzing metric connections and submersions.
result Complete local classification of geometries with parallel skew-symmetric torsion in principal bundle cases.
Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields on M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…
Let G be a complex Lie group and ΛG denote the group of maps from the unit circle S1 into G, of a suitable class. A differentiable map F from a manifold M into ΛG, is said to be of \emph{connection order (ab)} if the Fourier expansion in the loop parameter λ of the S1-family …
The study examines subgroups of braid groups related to symmetric groups.
problem Characterizing subgroups of braid groups that are extensions of symmetric groups.
method Analyzing normal subgroups of braid groups and their quotient structures.
result There are exactly 8 commensurability classes of such subgroups for n≥4. New G2 symmetry found in 8D distribution with 6D square.
problem Discovering new G2 symmetry in geometric distributions. method Analyzing rank 3 distribution on 8D manifold with growth vector (3,6,8).
result Maximally symmetric rank 3 distribution with 6D square.
Study Lie algebras with specific bilinear forms, finding exceptions.
problem Characterizing Lie algebras with symmetric, invariant, and nondegenerate bilinear forms.
method Analyzing structural properties and exceptions of Lie algebras.
result Rare exceptions to the properties of Lie algebras with these bilinear forms.
Uniform systole bounds for arithmetic orbifolds and number fields.
problem Bounding systole lengths in arithmetic orbifolds.
method Geometric methods and Mahler measure.
result Uniform lower bounds for systole lengths.
New classification of conformal structures with maximal G2 symmetry.
problem Classifying conformal structures with maximal G2 symmetry. method Complete local classification of homogeneous 4D split-conformal structures.
result Established a complete local classification of conformal structures with maximal G2 symmetry. Suppose X/Gamma is an arithmetic locally symmetric space of noncompact type (with the natural metric induced by the Killing form of the isometry group of X), and let p be a point on the visual boundary of X. It was shown by T.Hattori that if each horoball based at p intersects every Gamma-orbit in X, then p is not on t…
The paper extends Einstein condition to 4-manifolds using Hodge splittings.
problem Extending Einstein condition to 4-manifolds.
method Variational characterization and Hodge splitting approach.
result Admissible (g,h) pairs are critical points of a conformally invariant functional. In this paper we obtain a splitting theorem for the symmetric diffusion operator Δφ=Δ−⟨∇φ,∇⟩ and a non-constant C3 function f in a complete Riemannian manifold M, under the assumptions that the Ricci curvature associated with Δφ satisfies Ricφ(∇f,∇f)≥0, that $|…