The paper defines standard triples and finds Einstein metrics on Lie groups.
problem Finding Einstein metrics on compact simple Lie groups.
method Introduced standard triples and proved existence of Einstein metrics.
result Every compact simple Lie group admits at least 2 non-naturally reductive Einstein metrics.
A special geometric space has a unique curved shape.
problem Finding a curved shape in a specific geometric space.
method Using a unique representation of a subgroup and quotienting, the authors show the existence of a curved shape.
result A biquotient space admits a quasi-positively curved metric.
A Riemannian manifold is called almost positively curved if the set of points for which all 2-planes have positive sectional curvature is open and dense. We find three new examples of almost positively curved manifolds: Sp(3)/Sp(1)2, and two circle quotients of Sp(3)/Sp(1)2. We also show the quasi-positively cu…
New model for rational tropical points using sp4-webs and measures.
problem Understanding rational tropical points of Fock-Goncharov moduli space.
method Introducing rational bounded sp4-laminations and defining tropical coordinate systems. result Established a bijection between rational tropical points and sp4-webs. New Einstein metrics found on quaternionic Stiefel manifolds.
problem Finding invariant Einstein metrics on quaternionic Stiefel manifolds.
method Viewing the manifold as a total space over generalized Wallach and flag manifolds, using isotropy representations and equivalent summands.
result Obtained new Einstein metrics on VpHn. The paper classifies manifolds acted upon by a specific Lie group.
problem Characterizing manifolds acted upon by a specific Lie group.
method Analyzing the structure of manifolds under isometric action of a Lie group.
result Characterization of the structure of manifolds under specific Lie group action.
Study quaternionic symplectic groups and their actions, proving rigidity and classifying actions.
problem Deformation rigidity of quaternionic symplectic group actions.
method Study a non-associative algebra and compute automorphism groups to classify actions.
result Uniqueness and classification of isometric actions of quaternionic symplectic groups.
This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces SU(3)/Tmax, Sp(3)/Sp(1)×Sp(1)×Sp(1), and F4/Spin(8). We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…
Classifies Lie subgroups of Sp(1,n).
problem Classifying Lie subgroups of Sp(1,n).
method Conjugacy classification of connected H-irreducible Lie subgroups.
result All connected H-irreducible Lie subgroups of Sp(1,n) classified.
We show there are precisely 15 inhomogeneous biquotients of the form Sp(3)//Sp(1)2 and show that at least 8 of them admit metrics of quasi-positive curvature.
In this paper, we examine the homotopy classes of positive loops in Sp(2) and Sp(4). We show that two positive loops are homotopic if and only if they are homotopic through positive loops.
The spaces of Sp(n)-, Sp(n)U(1)- and Sp(n)Sp(1)- invariant, translation invariant, continuous convex valuations on the quaternionic vector space H^n are studied. Combinatorial dimension formulas involving Young diagrams and Schur polynomials are proved.
Study of sp4-webs on surfaces, proving cluster algebra structure.
problem Understanding sp4-webs and their cluster algebra properties. method Introduced skein algebra and cluster structure, proved positivity.
result Proved Ssp4,ΣZq[∂−1] is a quantum cluster algebra. Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.
problem Understanding Sp(n)-instantons on hyperkahler manifolds with conical singularities.
method Relating Sp(n)-instantons to deformed instantons and studying their properties on hyperkahler manifolds.
result Sp(n)-instantons on hyperkahler manifolds correspond to tri-contact instantons on the 3-Sasakian link.
Researchers create explicit p-harmonic functions on specific symmetric spaces.
problem Constructing explicit p-harmonic functions on compact Riemannian symmetric spaces.
method Explicit construction of complex-valued p-harmonic functions on specific symmetric spaces and their duals.
result Explicit p-harmonic functions constructed on SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), SU(2n)/Sp(n) and their duals.
In this paper we study the analytic realisation of the discrete series representations for the group G=Sp(1,1) as a subspace of the space of square integrable sections in a homogeneous vector bundle over the symmetric space G/K:=Sp(1,1)/(Sp(1)×Sp(1)). We use the Szegö map to give expressions for the restric…
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. Poisson realization simplifies classical Sp(1)-Kepler problems.
problem Classical Sp(1)-Kepler problems and their symmetries. method Poisson realization of so∗(4n) on phase space. result Verification of Poisson realizations simplified via Weinstein's idea.
Two families of Sp(2,R) symmetric G2 structures found in 7D.
problem Identifying Sp(2,R) symmetric G2 structures in 7D. method Analyzing homogeneous spaces and root diagrams of sp(2,R). result Found two families of Sp(2,R) symmetric G2 structures with distinct properties. New complete minimal submanifolds found in specific Riemannian spaces.
problem Finding complete minimal submanifolds in non-compact Riemannian symmetric spaces.
method Constructing multidimensional families of submanifolds via complex-valued eigenfunctions.
result New families of complete minimal submanifolds in SL_n(R)/SO(n), Sp(n,R)/U(n), SO*(2n)/U(n), and SU*(2n)/Sp(n).
The study examines geodesics and tight geodesics in surface curve complexes.
problem Characterizing the spectrum of geodesics and tight geodesics in curve complexes.
method Analyzing the number of geodesics and tight geodesics of length d in curve complexes. result The spectrum of geodesics is a subset of the spectrum of tight geodesics, with equality for geodesics of length 2.
Study of webs in quantum type C, proving equivalence to quantum representations.
problem Equivalence of webs and quantum representations in type C.
method Defined a linear pivotal category diagrammatically, conjectured equivalence, and proved functorial results.
result Full, essentially surjective functor between web and quantum representation categories.
We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?
We describe the orbit space of the action of the group Sp(2)Sp(1) on the real Grassmann manifolds Grk(H2) in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H2 w…
We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough inve…
We consider two functions on Sp(g,R) with values in the cyclic group of order four {1,-1,i,-i}. One was defined by Lion and Vergne. The other is -i raised to the power given by an integer valued function defined by Masbaum and the author (initially on the mapping class group of a surface). We identify these functions w…
New compact minimal submanifolds found in Riemannian symmetric spaces.
problem Finding compact minimal submanifolds in Riemannian symmetric spaces.
method Constructing multi-dimensional families of compact minimal submanifolds via complex-valued eigenfunctions.
result New families of compact minimal submanifolds of codimension two in SU(n)/SO(n), Sp(n)/U(n), SO(2n)/U(n), and SU(2n)/Sp(n). DD-SP uses ML to improve SP for Lorenz 96 systems, outperforming LR and DD-P.
problem Improving computational efficiency in weather/climate modeling.
method Data-driven super-parameterization using recurrent neural networks.
result DD-SP is more accurate and cheaper than SP, especially with scale separation.
A Riemannian manifold (M,ρ) is called Einstein if the metric ρ satisfies the condition $\Ric (ρ)=c\cdot ρ$ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain…
The SPS method constructs confidence regions for true parameters with optimal sample complexity.
problem Constructing exact, non-asymptotic confidence regions for true system parameters.
method Sign-Perturbed Sums (SPS) method, generalized to various types of problems.
result High probability upper bounds for SPS confidence regions show optimal shrinkage rate.
Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.
problem Analyzing Sp(2)-invariant solitons for Laplacian flow on the 4-sphere.
method Used mathematical analysis and asymptotic cone determination.
result Identified a 1-parameter family of Sp(2)-invariant expanding solitons with specific asymptotic behavior.
The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in se…
New method improves GANs by estimating density ratios in feature space with SP loss.
problem Filtering out unrealistic images from GANs trained with suboptimal discriminators.
method Develops DRE-F-SP method based on Softplus loss for density ratio estimation in feature space, and proposes three subsampling methods.
result Empirically shows substantial improvement over existing methods on synthetic and CIFAR-10 datasets.
This paper analyzes the sample complexity of SPS method for scalar linear regression.
problem Analyzing the sample complexity of the Sign-Perturbed Sums (SPS) identification method.
method The paper provides high probability upper bounds for the sizes of SPS confidence intervals under different sets of assumptions.
result The sizes of SPS confidence intervals shrink at a geometric rate around the true parameter, if observation noises are subgaussian.
The expectation value of Wilson loop operators in three-dimensional SO(N) Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method. With the same procedure the skein relation for Sp(N) are also obtained. Jones polyn…
The paper contains a characterization of compact groups $G\subseteq\GL(V)$, where V is a finite dimensional real vector space, which have the following property \SP{}: the family of convex hulls of G-orbits is a semigroup with respect to the Minkowski addition. If G is finite, then \SP{} holds if and only if G …
Construct model Higgs bundles in exceptional components of Sp(4, R) character variety.
problem Constructing model Higgs bundles in exceptional components of the Sp(4, R) character variety.
method Using the gluing construction for Higgs bundles over connected sums of Riemann surfaces and solutions to the Sp(4, R)-Hitchin equations.
result Provide model Higgs bundles in all 2g-3 exceptional components of the maximal Sp(4, R)-Higgs bundle moduli space.
Characterizes equigeodesics on specific homogeneous spaces.
problem Understanding geodesics on homogeneous spaces.
method Analyzing curves on Stiefel manifolds, generalized Wallach spaces, and spheres.
result Characterizations of algebraic equigeodesics on specific spaces.
The paper finds new invariant Einstein metrics on specific flag manifolds.
problem Finding Einstein metrics on generalized flag manifolds.
method Explicit construction and algebraic reduction of the Einstein equation.
result New invariant Einstein metrics on Sp(n) and SO(2n) flag manifolds. Classifies conjugation orbits of hyperbolic elements in various groups and surfaces.
problem Classifying conjugation orbits of hyperbolic elements in different groups.
method Using quaternionic hyperbolic planes and isometry groups, the classification is determined by real parameters.
result Conjugation orbits of `geometric' representations can be determined by a system of parameters.
We discuss twistor-like interpretation of the Sp(8) invariant formulation of 4d massless fields in ten dimensional Lagrangian Grassmannian Sp(8)/P which is the generalized space-time in this framework. The correspondence space C is SpH(8)/PH where SpH(8) is the semidirect product of Sp(8) with Heis…
We find the precise number of non-Kähler Sp(n)-invariant Einstein metrics on the generalized flag manifold M=Sp(n)/(U(p)×U(n−p)) with n≥3 and 1≤p≤n−1. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.
New method constructs complex-valued r-harmonic functions on Riemannian manifolds.
problem Constructing complex-valued r-harmonic functions on Riemannian manifolds.
method Introducing a new method for constructing complex-valued r-harmonic functions on Riemannian manifolds and applying it to specific semisimple Lie groups.
result The method successfully constructs complex-valued r-harmonic functions on various Riemannian manifolds, including specific Lie groups.
Theory connects Torelli subgroup homology to Sp(2g,ℤ)-modules.
problem Homology of Torelli subgroup of mapping class group.
method Equivariant group presentations and homology theory.
result Second homology group of Torelli subgroup is finitely generated as Sp(2g,ℤ)-module.
Introduces symplectic groups over noncommutative algebras and their geometric actions.
problem Understanding symplectic groups over noncommutative algebras.
method Introducing symplectic groups Sp2(A,σ) over noncommutative algebras and constructing geometric spaces. result New insights into structure theory of classical Lie groups and construction of symmetric spaces.
The paper defines subgroups of camomile type and studies singular braids and links.
problem Understanding subgroups of camomile type in singular braid groups.
method Presentations, defining relations, and group theory.
result The center of SGn is a direct factor in SPn but not in SPn. We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sp…