arXiv research

Calculates lower bounds for type III Reidemeister moves in link diagrams.

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.

The paper studies deformations of Kundt metrics using nil-Killing vector fields.

Kähler-Ricci flow singularity type is independent of initial metric.

In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.

This paper proves that all Type III holonomy metrics in G2* can be realized.

The local structure of the manifolds named in the title is described. Although curvature homogeneous, they are not, in general, locally homogeneous. Not all of them are Ricci-flat, which answers an existence question about type III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo (2006).

In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …

Study of special geometric structures on Lie groups.

Classification of 3-symmetric spaces with Ricci solitons.

Constructs a representation of the string 2-group on a von Neumann algebra.

Study shows how certain expanding spacetimes can collapse into flat or Kasner spacetimes.

Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton's compactness theorem that does not assume a l…

In this note, we answer affirmatively the question if a warped product of a compact manifold with a line as an ancient solution to the Ricci flow is trivial. We also consider the global behavior of the Type III warping product Ricci flow.

The paper studies conical structures and Perelman's functionals on Ricci flows.

The study classifies manifolds based on their geometric properties and invariants.

We discuss the Ricci flow on homogeneous 4-manifolds. After classifying these manifolds, we note that there are families of initial metrics such that we can diagonalize them and the Ricci flow preserves the diagonalization. We analyze the long time behavior of these families. We find that if a solution exists for all t…

It is shown that a self-dual neutral Einstein four-manifold of Petrov type III, admitting a two-dimensional null parallel distribution compatible with the orientation, cannot be compact or locally homogeneous, and its maximum possible degree of mobility is 3. Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo found a general c…

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently by Baird--Danielo and Lott: nongradient homogeneous expanding Ricci solitons on…

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as t goes to infinity. We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the square of the diamete…

Study solutions and singularities of G2-structures flows on specific manifolds.

Minimal generating sets of Reidemeister moves identified and classified.

Study shows convergence of immortal Ricci flows under certain conditions.

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-…

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $σ$-connections on finitely generated projective modules. This ma…

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar curvature and $\ep>0$ is a uniform constant, $M^3$ is compact. One of the key i…

The paper studies loxodromes on twisted surfaces in a specific 3D space.

We give a local expression for the {\it scalar curvature} of the noncommutative two torus $A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and Weyl factor. This is achieved by evaluating the value of the (analytic continuation of the) {\it spectral zeta functional} $ζ_a(s): …

We announce a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle $(X,\F)$ with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary foliation, that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Our theorem generalizes the cl…

This paper proves the realizability of all Type II holonomy groups in G2*.

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions ar…

This paper compares machine learning methods for recognizing lane change intentions from vehicle trajectories.

Credit estimation and bankruptcy prediction methods have been utilizing Altman's $z$ score method for the last several years. It is reported in many studies that $z$ score is sensitive to changes in accounting figures. Researches have proposed different variations to conventional $z$ score that can improve the predicti…

Study nearly-Kahler 6-manifolds with cohomogeneity-two action, proving existence and properties.

We study isometric actions on Riemannian symmetric spaces of noncompact type which are induced by reductive algebraic subgroups of the isometry group. We show that for such an action there exists a corresponding isometric action on a dual compact symmetric space, which reflects many properties of the original action. F…

Relational data representations have become an increasingly important topic due to the recent proliferation of network datasets (e.g., social, biological, information networks) and a corresponding increase in the application of statistical relational learning (SRL) algorithms to these domains. In this article, we exami…

Proves the Kundt conjecture in arbitrary dimensions, confirming its validity.

Paper solves the minimal generating set problem for singular Reidemeister moves.

Hessian metrics on a sphere with specific properties.

In this paper we prove a new matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow. The form of this new Li-Yau-Hamilton estimate is obtained by the interpolation consideration originated in \cite{Ch1}. This new inequality is shown to be connected with Perelman's entropy formula through a family of differential equalit…

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

We prove a Godbillon-Vey index formula for longitudinal Dirac operators on a foliated bundle with boundary; in particular, we define a Godbillon-Vey eta invariant on the boundary-foliation; this is a secondary invariant for longitudinal Dirac operators on type-III foliations. Moreover, employing the Godbillon-Vey index…