arXiv research

Study ribbon homology concordances using link Floer homology.

New torsion patterns found in Khovanov homology of link diagrams.

This paper shows hyperbolic knots can have arbitrarily large torsion in knot Floer homology.

The article computes the Blanchfield pairing for any link.

This paper detects torsion elements in homology cylinder monoids.

Researchers compute the Kauffman bracket skein module of a specific 3-manifold.

Unified framework LPCD optimizes quantization of complex submodules.

Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.

We discuss the ribbon-move for 2-knots, which is a local move. Let $K$ and $K'$ be 2-knots. Then we have: Suppose that $K$ and $K'$ are ribbon-move equivalent. (1) Let ${\mathrm {Tor}} H_1(\widetilde X_K; {\Z})$ (resp. ${\mathrm {Tor}} H_1(\widetilde X_{K'}; {\Z})$) be the $\Z$-torsion submodule of the Alexander module…

The paper introduces lattice homology for integrally closed submodules and applies it to geometric invariants.

Solves Lempert's question on Nakano semi-positivity preservation.

Study etale modules for reductive groups with 1D center, finding constraints on their irreducible submodules.

Simplified conditions for GO metrics in homogeneous manifolds.

New algebraic structures called cyclic Lie-Rinehart algebras are defined.

Almost regular Poisson structures are characterized and their holonomy groupoids are shown to be Poisson groupoids.

A class of Z_2-graded Lie algebra and Lie superalgebra extensions of the pseudo-orthogonal algebra of a spacetime of arbitrary dimension and signature is investigated. They have the form g = g_0 + g_1, with g_0 = so(V) + W_0 and g_1 = W_1, where the algebra of generalized translations W = W_0 + W_1 is the maximal solva…

We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\fr…

New method for linear connections in ODEs with constraints.

Study classifies compact geodesic orbit spaces with two isotropy components.

Let ${\mathcal F}_λ(\mathbb{S}^n)$ be the space of tensor densities on $\mathbb{S}^n$ of degree $λ$. We consider this space as an induced module of the nonunitary spherical series of the group $\mathrm{SO}_0(n+1,1)$ and classify $(\mathrm{so}(n+1,1),\mathrm{SO}(n+1))$-sim$unitary submodules of${\mathcal F}_λ(\mathbb{S…

PredNet excels at natural videos but fails on artificial rules, suggesting a trade-off.

We give a sufficient condition under which vanishing property of Cochran-Orr-Teichner knot concordance obstructions splits under connected sum. The condition is described in terms of self-annihilating submodules with respect to higher-order Blanchfield linking forms. This extends results of Levine and the authors on di…

Geodesic orbit metrics on real flag manifolds identified.

The motivation of this work is to define cohomology classes in the space of knots that are both easy to find and to evaluate, by reducing the problem to simple linear algebra. We achieve this goal by defining a combinatorial graded cochain complex, such that the elements of an explicit submodule in the cohomology defin…

This thesis consists of two parts which share only a slight overlap. The first part is concerned with the study of ideals in the ring $C^\infty(M,R)$ of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated $C^\infty(M,R)$-module V. We define a topology on the space of all…

For each pseudo-Anosov map $φ$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,φ)$. $A(S,φ)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov maps. We will develop a few nice properties of $A(S,φ)$ and give a few examples to show …

Study of holonomy algebras and Kahler homogeneous structures in complex hyperbolic spaces.

Two finite Alexander quandles with the same number of elements are isomorphic iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a) where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific condition…

This paper studies posets associated with link diagrams and their algebraic properties.

In this paper, we consider the similarity and quasi-affinity problems for Hilbert modules in the Cowen-Douglas class associated with the complex geometric objects, the hermitian anti-holomorphic vector bundles and curvatures. Given a "simple" rank one Cowen-Douglas Hilbert module $\mathcal{M}$, we find necessary and su…

We consider the space of tensor densities on the n-dimensional sphere with degree lambda (or, equivalently, of conformal densities with degree lambda). This space is a module over the group of diffeomorphisms, and consequently over the Lie algebra of vector fields, on the sphere, and we first prove that as a module ove…

The paper proves representation stability for Torelli groups and their filtrations.

We present a complete classification and the construction of $\mathrm{Mp}(2n+2,\mathbb{R})$-equivariant differential operators acting on the principal series representations, associated to the contact projective geometry on $\mathbb{RP}^{2n+1}$ and induced from the irreducible $\mathrm{Mp}(2n,\mathbb{R})$-submodules of…

We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-ty…

We compute the equivariant bordism of free oriented $(\mathbb{Z}/p)^n$-manifolds as a module over $Ω_*^{SO}$, when $p$ is an odd prime. We show, among others, that this module is canonically isomorphic to a direct sum of suspensions of multiple tensor products of $Ω^{SO}_*(B \mathbb{Z}/p)$, and that it is generated by …

For homogeneous reductive spaces G/H with reductive complements decomposable into an orthogonal sum \mathfrak{m}=\mathfrak{m}_1 \oplus \mathfrak{m}_2 \oplus \mathfrak{m}_3 of three Ad(H)-invariant irreducible mutually inequivalent submodules we establish simple conditions under which an invariant metric f-structure (f,…

Study the monodromy and center-focus problems for rational maps defined by products of generic lines.

Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.

We consider the polynomial representation S(V*) of the rational Cherednik algebra H_c(W) associated to a finite Coxeter group W at constant parameter c. We show that for any degree d of W and nonnegative integer m the space S(V*) contains a single copy of the reflection representation V of W spanned by the homogeneous …

Explicitly expresses torsion functions on lens spaces.

Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.

The paper compares two torsion invariants in complex vector bundles.

Analytic torsion matches Ray-Singer for specific nilmanifolds.

New spectral torsion defined for rescaled Dirac operators.

The paper examines torsions in Minkowskian product of Finsler metrics.

Study of symmetric affine surfaces with non-vanishing torsion.

Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.