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.

This paper detects torsion elements in homology cylinder monoids.

Unified framework LPCD optimizes quantization of complex submodules.

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

Given a link $L$, the Blanchfield pairing $\operatorname{Bl}(L)$ is a pairing which is defined on the torsion submodule of the Alexander module of $L$. In some particular cases, namely if $L$ is a boundary link or if the Alexander module of $L$ is torsion, $\operatorname{Bl}(L)$ can be computed explicitly; however no f…

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

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

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…

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

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

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian…

A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general proble…

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.

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces $G/H$ with two irreducible submodules in…

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.

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…

Is cognition a collection of loosely connected functions tuned to different tasks, or can there be a general learning algorithm? If such an hypothetical general algorithm did exist, tuned to our world, could it adapt seamlessly to a world with different laws of nature? We consider the theory that predictive coding is s…

We show, finitely generated rational $\mathsf{VIC}_{\mathbb Q}$-modules and $\mathsf{SI}_{\mathbb Q}$-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the To…

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…

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 …

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…

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…

A complex vector space $V$ is a prehomogeneous $G$-module if $G$ acts rationally on $V$ with a Zariski-open orbit. The module is called etale if $\dim V=\dim G$. We study etale modules for reductive algebraic groups $G$ with one-dimensional center. For such $G$, even though every etale module is a regular prehomogeneou…

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…

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…

More and more neural network approaches have achieved considerable improvement upon submodules of speaker diarization system, including speaker change detection and segment-wise speaker embedding extraction. Still, in the clustering stage, traditional algorithms like probabilistic linear discriminant analysis (PLDA) ar…

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,…

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

Explicitly expresses torsion functions on lens spaces.

The paper compares two torsion invariants in complex vector bundles.

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

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

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 …

Inspired by the work of Boris Vertman on refined analytic torsion for manifolds with boundary, in this paper we extend the construction of the Cappell-Miller analytic torsion to manifolds with boundary. We also compare it with the refined analytic torsion on manifolds with boundary. As a byproduct of the gluing formula…