Classifies matrices in the quaternionic hyperbolic unitary group.
problem Understanding the structure of matrices in the quaternionic hyperbolic unitary group.
method Used complex representation and characteristic polynomial to study matrices.
result Computed the characteristic polynomial and studied its sign.
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on SU(n) and showed applicability to SO(n) and Sp(n). Let E be a Real or Quaternionic Hermitian vector bundle over a Klein surface M. We study the action of the gauge group of E on the space of Galois-invariant unitary connections and we show that the closure of a semi-stable orbit contains a unique unitary orbit of projectively flat, Galois-invariant connections. We then…
The paper uses permutation representations to visualize group extensions and subgroups.
problem Visualizing and understanding group extensions and subgroups.
method Developing metaphoric rope-thread diagrams to represent semi-direct products and their constituents.
result Injective homomorphisms into semi-direct products are established.
Local classification of quaternion-Kähler metrics with rotating S1-symmetry.
problem Classifying quaternion-Kähler metrics with specific symmetries.
method Quaternionic Feix--Kaledin construction and explicit construction of holomorphic contact distributions.
result Quaternion-Kähler metrics with rotating S1-symmetry are determined by a Kähler metric and a line bundle. We study a problem of the geometric quantization for the quaternion projective space. First we explain a Kaehler structure on the punctured cotangent bundle of the quaternion projective space, whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and show that the canonical line bundle o…
Pairs of elements in quaternionic hyperbolic space have zero measure of being strongly doubly reversible.
problem Characterizing pairs of elements in quaternionic hyperbolic space that are strongly doubly reversible.
method Analyzing conjugacy conditions and using Haar measure.
result The set of strongly doubly reversible pairs has Haar measure zero in $\PSp(n,1) imes \PSp(n,1)$.
Given a discrete subgroup Γ of finite co-volume of PGL(2,R), we define and study parabolic vector bundles on the quotient Σ of the (extended) hyperbolic plane by Γ. If Γ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vecto…
A hypercomplex manifold M is a manifold equipped with three complex structures satisfying quaternionic relations. Such a manifold admits a canonical torsion-free connection preserving the quaternion action, called Obata connection. A quaternionic Hermitian metric is a Riemannian metric on which is invariant with resp…
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.
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
Contact group retracts to unitary subgroup.
problem Understanding contact structures on 3-sphere.
method Proving deformation retraction to unitary subgroup.
result Group of contactomorphisms retracts to U(2).
New structure on unitary group of Hilbert space.
problem No specific problem stated; constructing a new structure.
method Constructing a Banach Poisson-Lie group structure.
result Banach Poisson-Lie group structure on unitary group of Hilbert space.
We construct explicit left invariant quaternionic contact structures on Lie groups with zero and non-zero torsion, and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of quaternionic contact manifolds not locally quaternionic contact conformal to the quaternionic sphere. W…
Study counts and equidistributes rational points in quaternionic Heisenberg groups.
problem Counting and equidistribution of rational points in quaternionic Heisenberg groups.
method Arithmetic group actions on quaternionic hyperbolic spaces, Mertens counting formula, Neville equidistribution theorem.
result Proved Mertens counting formula and Neville equidistribution theorem for rational points over definite quaternion algebras.
Study calculates Kulkarni limit sets for quaternionic projective groups.
problem Computing Kulkarni limit sets for quaternionic projective groups.
method Natural action of quaternionic projective linear group on quaternionic projective space.
result Computed Kulkarni limit sets for cyclic subgroups.
Study Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
problem Deformation of spectrum and length spectrum on compact nilmanifolds.
method Ricci-Bourguignon flow on Heisenberg and quaternion Lie groups.
result Construct a solution of the Ricci-Bourguignon flow on Heisenberg and quaternion nilpotent Lie groups.
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.
UGConvs improve CNN accuracy with unitary transforms.
problem Improving CNN accuracy with richer representations.
method UGConvs combine group convolutions with unitary transforms.
result HadaNets achieve similar accuracy to circulant networks with lower complexity.
Six quaternionic lines with optimal angles found in 2D quaternion space.
problem Finding optimal configurations of quaternionic lines in 2D space.
method Simple presentation of lines as orbit of a reflection group, finding other optimal designs.
result Optimal spherical designs of 10, 15, and 20 lines in quaternion space.
Researchers describe unitary representations of mixed braid groups.
problem Understanding unitary representations of mixed braid groups.
method Explicitly describe unitary representations on cohomology of Abelian branched covers.
result Image of the representation is generated by complex reflections and related to the multivariate Burau representation.
The study finds conditions for quaternionic structures on symmetric spaces.
problem Conditions for quaternionic structures on symmetric spaces.
method Analysis of Lie group actions and representations.
result Symmetric spaces have invariant quaternionic structures under specific conditions.
Study circumcenters in Finsler unitary groups with optimal convexity bounds.
problem Existence and convexity of circumcenters in Finsler unitary groups.
method Analysis of distance functions and p-Schatten norm on Lie algebra.
result Existence of circumcenters for sets with radius < π/2 in several metrics.
New unitary representations for mapping class groups without almost invariant vectors.
problem Understanding unitary representations of mapping class groups.
method Space of measured foliations and Teichmüller space.
result None of the representations has almost invariant vectors.
Study local control in a 7D quaternionic Heisenberg group.
problem Optimizing geodesics in a 7D quaternionic Heisenberg group.
method Matrix representation and analysis of sub-Riemannian structure symmetries.
result Impact of symmetries on geodesic optimality.
Resolves gap problem for quaternion-Hermitian structures.
problem Determine maximal and submaximal symmetry dimensions for quaternion-Hermitian structures.
method Classifies structures with specific symmetry dimensions and studies geometric properties of submaximally symmetric spaces.
result Identifies locally conformally quaternion-Kähler and quaternion-Kähler with torsion structures.
Geometric proof of contractibility of unitary group in strong topology.
problem Contractibility of unitary group in strong operator topology.
method Direct geometric proof and construction of special subspaces and operators.
result Direct geometric proof of contractibility theorem.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
Quaternionic hyperbolic groups stabilize complex subspaces if trace skew-field is commutative.
problem Characterizing discrete subgroups of quaternionic hyperbolic groups with commutative trace skew-fields.
method Analyzing the trace skew-field of discrete subgroups in Sp(n,1). result Quaternionic hyperbolic groups stabilize complex subspaces if their trace skew-field is commutative.
The paper studies metrics and geodesics on a quaternionic Heisenberg group.
problem Characterizing geodesics and distances on a quaternionic Heisenberg group.
method Defining and analyzing a sequence of Riemannian metrics, deriving formulas for mean curvature.
result Explicit description of Carnot-Carathéodory distance and spheres.
We answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven dimensional solvable Lie groups having an integra…
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
Study asymptotics of unitary matrix elements in quantum mechanics.
problem Asymptotic behavior of unitary matrix elements in quantum mechanics.
method Uses Berezin-Toeplitz quantization and symplectic geometry.
result Recover asymptotics of Wigner's d-matrix elements for spin representations.
Let Uc(H)=u:uisunitaryandu−1iscompact stand for the unitary Fredholm group. We prove the following convexity result. Denote by d∞ the rectifiable distance induced by the Finsler metric given by the operator norm in Uc(H). If u0,u1,u∈Uc(H) and the geodesic β joining u0 and u1 in $U…
Study describes moduli of quaternionic hyperbolic triples of points.
problem Tackles the congruence classes of triples of points in quaternionic hyperbolic space.
method Introduces invariants and defines quaternionic Goldman invariants for mixed configurations.
result Defines quaternionic analogues of Goldman invariants for mixed configurations.
Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.
problem Understanding the geometry induced by slice Riemannian metric.
method Developed Lie theoretic study, computed isometry group, compared with quaternionic Poincaré geometry.
result Isometry group of slice Riemannian metric is built from symmetries of Sp(1,1) group.
In this paper we give the characterization of Fuchsian groups acting on quaternionic hyperbolic 2-space.
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. …
Study of logarithms in SVD-closed subgroups of unitary group.
problem Understanding logarithms in SVD-closed subgroups of unitary groups.
method Analysis of generalized principal logarithms and minimizing geodesics.
result Set of generalized principal logarithms is a disjoint union of diffeomorphic subsets.
The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary p…
Study nonrigid dynamics of unitary groups on Lie groups via kinetic energy metrics.
problem Understanding the dynamics of unitary groups on Lie groups using kinetic energy metrics.
method Least action principle applied to geodesics of the kinetic energy metric on G. result Kinetic energy metric on G is not complete and not invariant. Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
With the usual definition of a super Hilbert space and a super unitary representation, it is easy to show that there are lots of super Lie groups for which the left-regular representation is not super unitary. I will argue that weakening the definition of a super Hilbert space (by allowing the super scalar product to b…
We consider the construction of the basic bundle gerbe on SU(n) introduced by Meinrenken and show that it extends to a range of groups with unitary actions on a Hilbert space including U(n), diagonal tori and the Banach Lie group of unitary operators differing from the identity by an element of a Schatten ideal. In all…
In this note, we study deformations of quaternionic hyperbolic lattices in larger quaternionic hyperbolic spaces and prove local rigidity results. On the other hand, surface groups are shown to be more flexible in quaternionic hyperbolic plane than in complex hyperbolic plane.
We construct new proper biharmonic functions defined on open and dense subsets of the special unitary group SU(2). Then we employ a duality principle to obtain new proper biharmonic functions from the non-compact 3-dimensional hyperbolic space H^3.