arXiv research

Study manifolds with positive intermediate Ricci curvature and large symmetry rank.

Let G be a rank two finite group, and let $\cH$ denote the family of rank one p-subgroups of G, at all primes where G has p-rank two. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X with isotropy in $\cH$, whose fixed sets are homotopy spheres. Ou…

We show that a rank two finite group G admits a finite G-CW-complex X homotopy equivalent to a sphere, with rank one prime power isotropy, if and only if G does not p'-involve Qd(p) for any odd prime p. This follows from a more general theorem which allows us to construct a finite G-CW-complex by gluing together a give…

Estimates covariance matrices for matrix-variate data via core covariance geometry.

Let $\mathcal{M}_{0}^n$ be the class of closed, simply-connected, non-negatively curved Riemannian manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if $M\in \mathcal{M}_{0}^n$, then $M$ is equivariantly diffeomorphic to the free linear quotient by a torus of a product of spheres…

Prove a global shadow lemma for Patterson-Sullivan measures associated with relatively Morse subgroups in higher-rank semisimple Lie groups.

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is var…

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existen…

Schwarz lemma extended to equality cases and curvature on manifolds.

A group action is called polar if there exists an immersed submanifold (a section) which intersects all orbits orthogonally. Such group actions have been studied extensively on symmetric spaces. We show how to construct a manifold admitting a polar group action by prescribing their isotropy groups along a fundamental d…

We consider numerical integrators of ODEs on homogeneous spaces (spheres, affine spaces, hyperbolic spaces). Homogeneous spaces are equipped with a built-in symmetry. A numerical integrator respects this symmetry if it is equivariant. One obtains homogeneous space integrators by combining a Lie group integrator with an…

The study examines the independence of GKM manifolds and symmetric spaces.

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kähler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.

We prove a rigidity theorem that shows that, under many circumstances, quasi-isometric embeddings of equal rank, higher rank symmetric spaces are close to isometric embeddings. We also produce some surprising examples of quasi-isometric embeddings of higher rank symmetric spaces. In particular, we produce embeddings of…

Explores local structure of morphisms and formal submanifolds in formal manifolds theory.

Collar lemma proven for certain surface group representations.

We prove that the Euler characteristic of an even-dimensional compact manifold with positive (nonnegative) sectional curvature is positive (nonnegative) provided that the manifold admits an isometric action of a compact Lie group $G$ with principal isotropy group $H$ and cohomogeneity $k$ such that $k - (\rank G - \ran…

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describi…

Let $A=(a^i_j)$ be an orthogonal matrix with no entries zero. Let $B=(b^i_j)$ be the matrix defined by $b^i_j=\frac 1{a^i_j}$. M. Kontsevich conjectured that the rank of $B$ is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statment can be understood as a generalization of …

We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups.

Develops sublinear Morse theory in symmetric spaces.

This paper surveys various methods for dimensionality reduction and nearest neighbor search.

The paper introduces isotropy as a regularizer to enhance portfolio stability.

For a compact homogeneous space $G/K$, we study the problem of existence of $G$-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of $G$. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, w…

In this paper, we show that small spherical soap bubbles in irreducible simply connected symmetric spaces of rank greater than one are constructed from the limits of a certain kind of modified mean curvature flows starting from small spheres in the Euclidean space of dimension equal to the rank of the symmetric space, …

We show that simply connected Riemannian homogeneous spaces of compact semisimple Lie groups with polar isotropy actions are symmetric, generalizing results of Fabio Podesta and the third named author. Without assuming compactness, we give a classification of Riemannian homogeneous spaces of semisimple Lie groups whose…

We prove several results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard's improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.

We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly with covolume. We obtain similar results for isotropy subgroups in lattices. Ge…

We show that if $G\times M \to M$ is a cohomogeneity one action of a compact connected Lie group $G$ on a compact connected manifold $M$ then $H^*_G(M)$ is a Cohen-Macaulay module over $H^*(BG)$. Moreover, this module is free if and only if the rank of at least one isotropy group is equal to the rank of $G$. We deduce …

Metric graphs have subgraphs with entropy at least λ.

Study isotropy groups for complex orthogonal and skew-symmetric matrices.

Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.

Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.

Study the smallest Laplace eigenvalue in special geometric spaces.

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric space…

Geodesic orbit metrics proven on specific homogeneous spaces.

We consider a homogeneous fibration $G/L \to G/K$, with symmetric fiber and base, where $G$ is a compact connected semisimple Lie group and $L$ has maximal rank in $G$. We suppose the base space $G/K$ is isotropy irreducible and the fiber $K/L$ is simply connected. We investigate the existence of $G$-invariant Einstein…

The paper reveals low-rank structure in neural network gradients, influenced by data and model parameters.

We classify all simply connected Riemannian manifolds whose isotropy groups act with cohomogeneity less than or equal to two.

The study connects matroids with torus representations and positive curvature.

Given a singular foliation, we attach an "essential isotropy" group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with t…

The space of $G$-invariant metrics on a homogeneous space $G/H$ is in one-to-one correspondence with the set of inner products on the tangent space $\fr{m}\cong T_{\it o}(G/H)$, which are invariant under the isotropy representation. When all the isotropy summands are inequivalent to each other, then the metric is calle…

Studying the isotropy orbits of compact symmetric spaces Reiswich introduced a family of explicit polynomials in one variable in order to describe the unique minimal isotropy orbit of compact symmetric spaces with Dynkin diagram of type $D_m$. Based on this geometric interpretation he conjectured that these polynomials…

Estimates true Sharpe ratio of selected assets with various methods.

Proves a 1930s Hopf conjecture about positive curvature manifolds.

Paper develops a theory for Patterson-Sullivan measures in higher rank symmetric spaces.

The concept of an objective spatial direction in special relativity is investigated and theories assuming light-speed isotropy while accepting the existence of a privileged spatial direction are classified. A natural generalization of the proper time principle is introduced which makes it possible to devise experimenta…