arXiv research

Affine maps reveal higher rank structures in certain spaces.

Study finite rank bundle over J-Holomorphic map moduli spaces with exponential decay.

Maps from rational homology solid tori yield rank inequalities in Heegaard Floer homology.

This work simplifies proximal mapping for low-rank norms.

Study proper holomorphic maps between symmetric domains, proving rigidity and isometric properties.

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.

Develops complex harmonic maps for Teichmüller theory, proving new theorems.

Study maps 4-manifolds with 1-handles, revealing infinite rank centers.

We show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most of the higher rank symmetric spaces. As a consequence, we prove that the locally finite simplicial volume of most $\mathbb Q$-rank $1$ locally symmetric spaces is positive, which has been open for many y…

New concept of relatively dominated representations for higher-rank groups.

If Pi: M -> B is an onto smooth maximal rank map between complete Riemannian manifolds M and B with bounded geometry, we prove sufficient conditions for M to be roughly isometric to the Riemannian product FxB, where F is a fiber of M.

We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is an affirmative solution to Brock-Farb's Rank Conjecture which as…

New methods classify H-like Lie algebras with rank 2 maps.

Let $K$ be a closed polydisc or ball in $\C^n$, and let $Y$ be a quasi projective algebraic manifold which is Zariski locally equivalent to $\C^p$, or a complement of an algebraic subvariety of codimension $\ge 2$ in such manifold. If $r$ is an integer satisfying $(n-r+1) (p-r+1)\geq 2$ then every holomorphic map from …

The paper explores non-amenability in infinite-type surfaces and graphs.

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integral homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f from Y' to Y between Seifert homology spheres yie…

Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.

The paper calculates ranks and bounds for Stiefel manifolds over different fields.

Maps preserving Carathéodory distance between symmetric domains are rigid.

A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite …

Injective map found between Poisson algebras.

Let Pi: M -> B be an onto maximal rank map or a Riemannian submersion between Riemannian manifolds M and B. Initially, we prove necessary and sufficient conditions for any fiber F to be roughly isometric to M. Then, we prove necessary and sufficient conditions for Pi to be a rough isometry. As a corollary M is roughly …

New examples show some convex-cocompact subgroups are separable.

This paper concerns a study of three families of non-compact type symmetric spaces of infinite dimension. Although they have infinite dimension they have finite rank. More precisely, we show they have finite telescopic dimension. We also show the existence of Furstenberg maps for some group actions on these spaces. Suc…

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

We construct a tangential map from a locally symmetric space of noncompact type to its dual compact type twin. By comparing the induced map in cohomology to a map defined by Matsushima, we conclude that in the equal rank case the map has a nonzero degree.

In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces and of CAT(0)--groups. We show that, for mapping class groups of surfaces, these functions exhibit phase transitions at the rank (as measured by thrice the genus plus the number of punctures minus 3). We also pr…

The paper shows bi-f-harmonic and f-biharmonic maps are equivalent and explores their properties.

Let $D, Ω_1, ..., Ω_m$ be irreducible bounded symmetric domains. We study local holomorphic maps from $D$ into $Ω_1 \times... Ω_m$ preserving the invariant $(p, p)$-forms induced from the normalized Bergman metrics up to conformal constants. We show that the local holomorphic maps extends to algebraic maps in the rank …

Analyzes the structure and rank of neural network Hessians.

We introduce a new framework for optimal transport using Schatten-p regularization to recover low-rank structures.

MARS automatically selects tensor decomposition ranks, improving performance in neural network tasks.

The paper explores meridional ranks of knotted surfaces and welded knots, proving equalities and relationships.

We prove geometric superrigidity for actions of cocompact lattices in semisimple Lie groups of higher rank on infinite dimensional Riemannian manifolds of nonpositive curvature and finite telescopic dimension.

Study closed manifolds with rank one ray structures, proving completeness or covering properties.

Proves regularity of harmonic maps into Euclidean buildings and applies to superrigidity of algebraic groups.

New perspective on CNNs using Hessian maps reveals their structure.

Constructs measurable equivariant maps for higher rank Lie groups.

Baker and Riley proved that a free group of rank 3 can be contained in a hyperbolic group as a subgroup for which the Cannon-Thurston map is not well-defined. By using their result, we show that the phenomenon occurs for not only a free group of rank 3 but also every non-elementary hyperbolic group. In fact it is shown…

Systems rank PubMed abstracts and sentences for RDoC criteria, achieving high mAP and MAA.

Improved GoF statistics using entropy-regularized optimal transport for multivariate rank.

The paper proves rigidity for cocycles from higher rank lattices to Out(FN).

We establish a "low rank property" for Sobolev mappings that pointwise solve a first order nonlinear system of PDEs, whose smooth solutions have the so-called "contact property". As a consequence, Sobolev mappings from an open set of the plane, taking values in the first Heisenberg group and that have almost everywhere…

The abstract discusses transversality for infinite dimensional manifolds.

For any positive integer $n$, we exhibit a cofinite subgroup $Γ_n$ of the mapping class group of a surface of genus at most two such that $Γ_n$ admits an epimorphism onto a free group of rank $n$. We conclude that $H^1(Γ_n;\Z)$ has rank at least $n$ and the dimension of the second bounded cohomology of each of these ma…

A rank-n tensor on a Lorentzian manifold V whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general properties in dimension N. We prove that rank-2 tensors which map the n…

Study shows rank of Heegaard Floer homology increases in spliced knot complements.

Study improves Heegaard Floer homology relations for knots in homology spheres.