arXiv research

A Poisson realization of the simple real Lie algebra $\mathfrak {so}^*(4n)$ on the phase space of each $\mathrm {Sp}(1)$-Kepler problem is exhibited. As a consequence one obtains the Laplace-Runge-Lenz vector for each classical $\mathrm{Sp}(1)$-Kepler problem. The verification of these Poisson realizations is greatly s…

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…

We study geometric first order differential operators on quaternionic Kähler manifolds. Their principal symbols are related to the enveloping algebra and Casimir elements for $\Sp(1)\Sp(n)$. This observation leads to anti-symmetry of the principal symbols and Bochner-Weitzenböck formulas for operators. As an applicatio…

Geometric quantization on hyperKähler manifolds via brane quantization.

We classify hyperbolic monopoles with continuous symmetries and construct new examples.

Study of Riemannian geometry on quaternionic unit ball linked to Sp(1,1) group.

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle…

We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(…

We scan for massive type IIA SU(3)-structure compactifications of the type AdS4 x CP3 with internal symmetry group SO(4). This group acts on CP3 with cohomogeneity one, so that one would expect new non-homogeneous solutions. We find however that all such solutions enhance their symmetry group to Sp(2) and form, in fact…

For an almost product structure $J$ on a manifold $M$ of dimension $6$ with non-degenerate Nijenhuis tensor $N_J$, we show that the automorphism group $G=Aut(M,J)$ has dimension at most 14. In the case of equality $G$ is the exceptional Lie group $G_2^*$. The next possible symmetry dimension is proved to be equal to 10…

Characterizes group-equivariant neural networks for three groups.

New geometric quantisation scheme for hyper-Kähler manifolds.

ConvNP improves SP prediction with translation equivariance and coherent samples.

New tools prove smooth actions on exotic spheres.

In the paper "Einstein metrics on compact simple Lie groups attached to standard triples", the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except …

The purpose of this note is to define tri-moment maps for certain manifolds that carry closed non-degenerate 4-forms and an $Sp(1)^n$-action. Examples include quaternionic vector spaces and flag manifolds. We show how this map can be used ro reduce such manifolds to the ones with fewer symmetries. The images of such ma…

Suppose $φ_3:Sp(1)\rightarrow Sp(2)$ denotes the unique irreducible $4$-dimensional representation of $Sp(1) = SU(2)$ and consider the two subgroups $H_1, H_2\subseteq Sp(3)$ with $H_1 = \{\operatorname{diag}(φ_3(q_1), q_1): q_1 \in Sp(1)\}$ and $H_2 = \{\operatorname{diag}(φ_3(q_2),1):q_2\in Sp(1)\}$. We show that the…

We consider invariant Einstein metrics on the quaternionic Stiefel manifolds $V_p\mathbb{H} ^n$ of all orthonormal $p$-frames in $\mathbb{H}^n$. This manifold is diffeomorphic to the homogeneous space $\mathrm{Sp}(n) / \mathrm{Sp}(n-p)$ and its isotropy representation contains equivalent summands. We obtain new Einstei…

It is known that the hard Lefschetz action, together with Kähler identities for Kähler (resp. hyperkähler) manifolds, determines a $\mathfrak{su}(1,1)_{sup}$ (resp. $\mathfrak{sp}(1,1)_{sup}$) Lie superalgebra action on differential forms. In this paper, we explain the geometric origin of this action, and we also gener…

Let $M$ be an analytic complete finite volume pseudo-Riemannian manifold and $\widetilde{Sp}(n,\mathbb{R})\times\widetilde{Sp}(1,\mathbb{R})$ a connected semisimple Lie group such that its Lie algebra is $\mathfrak{sp}(n,\mathbb{R})\oplus\mathfrak{sp}(1,\mathbb{R})$. We characterize the structure of the manifold $M$ as…

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…

Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by studying a natural non-associative algebra $\mathfrak{m}$ comming from the affine struc…

New model for rational tropical points using $\mathfrak{sp}_4$-webs and measures.

Let $Z \to Y^{2n+1}$ be the bundle of Legendrian $n$-planes over a contact manifold $Y$. We consider a foliation of $Z$ by canonical lifts of Legendrian submanifolds, called \emph{Legendrian submanifold path geometry}, whose flat model is \[ Sp(n+1, R) \to RP^{2n+1}. \] The equivalence problem provides an $sp(n+1, R)$ …

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curv…

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 $\mathfrak{sp}_4$-webs on surfaces, proving cluster algebra structure.

Sp(n)-instantons linked to complex Lagrangian graphs via Fourier-Mukai transform.

Researchers create explicit p-harmonic functions on specific symmetric spaces.

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 $G/K:=Sp(1,1) /(Sp(1) \times Sp(1))$. We use the Szegö map to give expressions for the restric…

Maximal and Borel Anosov representations in $Sp(4,\mathbb{R})$ are proven to be Hitchin.

New complete minimal submanifolds found in specific Riemannian spaces.

We study a geometry associated with rank 3 distributions in dimension 8, whose symbol algebra is constant and has a simple Lie algebra sp(3,R) as Tanaka prolongation. We restrict our considerations to only those distributions that are defined in terms of a systems of ODEs of the form $\dot{z}_{ij}=\frac{\partial^2 f(\d…

The study examines geodesics and tight geodesics in surface curve complexes.

Study of webs in quantum type C, proving equivalence to quantum representations.

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 $\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds $\mathrm{Gr}_k(\mathbb{H}^2)$ 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 $\mathbb{H}^2$ w…

New method for constructing contact Lie systems on various spaces.

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.

Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.

DD-SP uses ML to improve SP for Lorenz 96 systems, outperforming LR and DD-P.

The SPS method constructs confidence regions for true parameters with optimal sample complexity.

Researchers found a family of Sp(2)-invariant solitons for Laplacian flow.

The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.