Study of Anosov representations in pseudo-Riemannian hyperbolic spaces.
problem Understanding Anosov representations in higher-dimensional spaces.
method Examining representations into projective indefinite orthogonal groups and their action on H^{p,q-1}.
result Intimate connection between Anosov representations and convex cocompactness in this setting.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
Study on counting special-orthogonal Anosov orbits in geometric spaces.
problem Counting points in specific orbits of projective Anosov subgroups.
method Analysis of totally geodesic copies in Riemannian and pseudo-Riemannian spaces.
result Number of points in the orbit is finite and asymptotically exponential.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
String structures have played an important role in algebraic topology, via elliptic genera and elliptic cohomology, in differential geometry, via the study of higher geometric structures, and in physics, via partition functions. We extend the description of String structures from connected covers of the definite-signat…
We classify the effective and transitive actions of a Lie group G on an n-dimensional non-degenerate hyperboloid (also called real pseudo-hyperbolic space), under the assumption that G is a closed, connected Lie subgroup of SO0(n−r,r+1), the connected component of the indefinite special orthogonal group. Assumin…
Method produces Einstein metrics on Lie groups.
problem Finding non-Ricci-flat Einstein metrics on Lie groups.
method Systematic method to produce left-invariant, non-Ricci-flat Einstein metrics.
result Classifications of Einstein metrics in dimensions 8 and 9.
Let SO+(p,q) denote the identity connected component of the real orthogonal group with signature (p,q). We give a complete description of the spaces of continuous and generalized translation- and SO+(p,q)-invariant valuations, generalizing Hadwiger's classification of Euclidean isometry-invari…
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
problem Constructing a Hermitian TQFT from a non-semisimple category.
method Endowed a non-semisimple category of quantum sl(2) modules with a Hermitian structure and proved the resulting TQFT is Hermitian.
result Projective representations of the mapping class group in indefinite unitary matrices.
This paper classifies holonomy groups of K-contact sub-pseudo-Riemannian manifolds.
problem The problem of subspace degeneracy in indefinite signature metrics.
method Adapted for metrics of indefinite signature, bypassing subspace degeneracy.
result Horizontal holonomy group either coincides with the adapted holonomy group or acts as its normal subgroup of codimension one.
Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
problem Characterizing simply connected indefinite homogeneous spaces of finite volume.
method Analyzing Lie algebras with abelian solvable radical and symmetric bilinear form.
result Simply connected indefinite homogeneous spaces are compact and have specific Lie algebra structures.
Study symplectic and orthogonal groups over involutive algebras, realizing geometric models for symmetric spaces and applications to Higgs bundles.
problem Understanding symplectic and orthogonal groups over involutive algebras and their geometric properties.
method Explicitly describe complexified tangent spaces and their diffeomorphisms, providing geometric models for symmetric spaces.
result New geometric interpretations of Higgs bundle data and exact component counts for moduli spaces.
Study new Hopf real hypersurfaces in indefinite complex projective space.
problem Problems posed by H.~Anciaux and K.~Panagiotidou on non-degenerate real hypersurfaces.
method Changed point of view, constructed new families, obtained rigidity results.
result Classified η-umbilical real hypersurfaces and characterized Killing Reeb vector field. 3-Sasaki structures linked to projective geometry.
problem Understanding 3-Sasaki structures via projective geometry.
method Establishing a connection between 3-Sasaki structures and projective structures with specific holonomy reductions.
result 3-Sasaki structures are described as projective structures with a particular holonomy reduction to the unitary quaternionic group.
This paper studies ruled real hypersurfaces in indefinite complex projective space.
problem Characterizing and classifying ruled real hypersurfaces in indefinite complex projective space.
method Introduced and studied ruled real hypersurfaces with maximal holomorphic distribution integrable and leaves totally geodesic holomorphic hyperplanes. Detailed shape operator computation and method of construction by gluing totally geodesic hyperplanes along a curve.
result Classification of all minimal ruled real hypersurfaces in terms of three main families of curves.
New method reduces variance and bias in approximating indefinite kernels.
problem Approximating non-stationary indefinite kernels with low variance and bias.
method Generalized orthogonal random features (GORF)
result GORF achieves lower variance and approximation error compared to existing methods.
GOPO optimizes large models in Hilbert space, avoiding Kullback-Leibler's curvature.
problem Optimizing large language models with Kullback-Leibler divergence's curvature issues.
method GOPO uses Hilbert space L2(pi_k) with orthogonality constraints and a work-dissipation functional.
result GOPO achieves competitive generalization with stable gradient dynamics and entropy preservation.
Formula derived for discrete improper affine spheres.
problem Constructing discrete improper affine spheres.
method Loop group factorizations and Birkhoff decomposition.
result Representation formula for discrete indefinite affine spheres.
We study the relations between the quaternion H-type group and the boundary of the unit ball on two dimensional quaternionic space. The orthogonal projection of the space of square integrable functions defined on quaternion H-type group into its subspace of boundary values of q-holomorphic functions is consider. …
New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
Study shows non-spin 4-manifolds where smooth Nielsen realization fails.
problem Existence of non-spin 4-manifolds with non-realizable mapping class groups.
method Investigation of multi-twists, projective twists, and multi-reflections.
result Examples of non-spin 4-manifolds with non-realizable mapping class groups.
Projective geometry simplifies Sasaki-Einstein structures and their compactification.
problem Understanding and compactifying Sasaki-Einstein structures.
method Projective differential geometry and holonomy reductions.
result Characterization of Sasaki-Einstein structures and their compactification.
The paper creates a deformation retraction for homeomorphisms of the projective plane.
problem Deformation retraction of homeomorphisms of the projective plane.
method Equivariant strong deformation retraction from homeomorphism group to special orthogonal group.
result Induces a SO(3)-equivariant strong deformation retraction from projective plane homeomorphisms to SO(3).
A new double quasi-Poisson bracket on surface groups.
problem Constructing a new mathematical structure on surface groups.
method Proposing and proving a double quasi-Poisson bracket on group algebras.
result The double quasi-Poisson bracket is a noncommutative generalization of the Goldman bracket.
Paper derives local Plücker formulas for special orthogonal groups.
problem Deriving Plücker formulas for special orthogonal groups.
method Reduction to classical A_n case.
result Local Plücker formulas for special orthogonal groups derived.
Projective pre-compactness induces projective structure on boundary, relates to GR asymptotic forms.
problem Projective compactness for torsion-free linear connections on a manifold.
method Introduce and study a weakening of projective compactness for torsion-free linear connections on a manifold.
result Induces projective structure on the boundary and relates to asymptotic forms in GR.
We use reduced homogeneous coordinates to study Riemannian geometry of the octonionic (or Cayley) projective plane. Our method extends to the para-octonionic (or split octonionic) projective plane, the octonionic projective plane of indefinite signature, and the hyperbolic dual of the octonionic projective plane; we di…
This is the first in a series of papers devoted to an analogue of the metaplectic representation, namely, the minimal unitary representation of an indefinite orthogonal group; this representation corresponds to the minimal nilpotent coadjoint orbit in the philosophy of Kirillov-Kostant. We begin by applying methods fro…
Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.
problem Determining the minimal number of homogeneous geodesics in Finsler manifolds with indefinite Killing form.
method Analyzing examples of Lie groups with invariant Finsler metrics and presenting new examples.
result Homogeneous Finsler manifolds with indefinite Killing form admit at least four homogeneous geodesics.
The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.
New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.
problem Creating non-arithmetic lattices in projective orthogonal groups.
method Using anti-holomorphic involutions on complex arithmetic ball quotients, gluing fixed loci along geodesic subspaces.
result Explicit calculation of the volume of constructed non-arithmetic orbifolds.
Formula for Heisenberg group surface areas derived.
problem Deriving a formula for surface areas in Heisenberg groups.
method Analogy of Cauchy's surface area formula in Heisenberg groups.
result Formula for p-area of compact hypersurfaces in Heisenberg groups.
Paper links set derivatives to its orthogonal projections.
problem Understanding the relationship between set derivatives and projections.
method Derives equations from topological link between Minkowski functional partial derivatives and set boundary.
result System of equations for orthogonal projections derived.
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
Study convex cocompact subgroups in real projective geometry.
problem Characterize discrete subgroups acting on real projective space.
method Define and characterize convex cocompactness, extend results from orthogonal groups.
result Equivalence of different convex cocompactness conditions for word hyperbolic groups.
We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new, algebraic geometric approach to the classification of orthogonal separable coordinate…
Complex and quaternionic projective spaces lack local orthogonal coordinates.
problem Lack of local orthogonal coordinates in complex and quaternionic projective spaces.
method Analysis of Riemannian manifolds and canonical metrics.
result Complex and quaternionic projective spaces do not have local systems of orthogonal coordinates.
New method learns complete orthogonal dictionary from samples with theoretical guarantees and efficiency.
problem Learning a complete orthogonal dictionary from sparsely generated signals.
method Maximizes the \(\ell^4\)-norm over the orthogonal group, using a novel algorithm based on matching, stretching, and projection (MSP).
result The MSP algorithm provably converges locally at a superlinear (cubic) rate and is significantly more efficient than existing methods.
Researchers study the conformal geometry of bivariate Gaussian manifolds.
problem Exploring the conformal structure of Fisher-Rao metric on statistical manifolds.
method Determined invariants of the conformal structure of the Fisher-Rao metric on the bivariate Gaussian manifold.
result The conformal holonomy group is SO0(1,6) for generic random variables, but SO0(1,4) for independent ones. NS-RGS improves orthogonal group synchronization with faster convergence.
problem Orthogonal group synchronization from pairwise measurements.
method Newton-Schulz iteration for Riemannian gradient optimization.
result NS-RGS achieves linear convergence and near-optimal accuracy.
Smooth projections on manifolds split into simpler parts.
problem Decomposing operators on Riemannian manifolds.
method Constructing a sum of smooth orthogonal projections.
result Extends decomposition on real line to manifolds.
Orthogonal projections improve learning accuracy in clinical image segmentation and music classification.
problem Improving accuracy in learning tasks with high-dimensional data.
method Investigation and application of orthogonal projections to balance variance and pairwise distances in dimension reduction. Extension to deep learning with augmented target loss functions.
result Augmented target loss functions increase accuracy in clinical image segmentation and music classification.
Geometric argument proves projection theorems in hyperbolic space.
problem Proving projection theorems for hyperbolic space.
method Geometric argument for orthogonal projections.
result Characterization of purely unrectifiable sets in hyperbolic space.
New method constructs Ricci-flat metrics on Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Combinatorial method to construct indefinite Ricci-flat metrics.
result Infinite families of Ricci-flat nilmanifolds constructed.
Paper addresses group synchronization with incomplete measurements and proves linear convergence of GPM.
problem Orthogonal group synchronization with incomplete measurements and additive noise.
method Generalized power method (GPM) with local error bound analysis.
result Linear convergence of GPM to a global maximizer under general additive noise model.
New methods find Ricci-flat metrics on specific Lie groups.
problem Finding Ricci-flat metrics on Lie groups.
method Two constructions based on gradings and filtrations.
result Every nilpotent Lie algebra of dimensions up to 9 admits an indefinite Ricci-flat metric.
We solve the equivalence problem for the orthogonally separable webs on the three-sphere under the action of the isometry group. This continues a classical project initiated by Olevsky in which he solved the corresponding canonical forms problem. The solution to the equivalence problem together with the results by Olev…
Principal component analysis (PCA) is an unsupervised method for learning low-dimensional features with orthogonal projections. Multilinear PCA methods extend PCA to deal with multidimensional data (tensors) directly via tensor-to-tensor projection or tensor-to-vector projection (TVP). However, under the TVP setting, i…