arXiv research

The $(α,β)$-Ricci-Yamabe flow exists on closed manifolds.

Proposes a method to prove closing of periodic orbits in dynamical systems.

We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed tran…

Study of closed trajectories in hyperbolic plane with specific curvature constraints.

Algorithms find second and third shortest non-trivial closed walks on surfaces.

New self-shrinkers found in higher dimensions.

We modify the Laplacian coflow of co-closed G2-structures - $\frac{d}{dt}ψ=Δψ$ where $ψ$ is the closed dual 4-form of a $G_{2}$-structure $\varphi$. The modified flow is now parabolic in the direction of closed forms upto diffeomorphisms. We then prove short time existence and uniqueness of solutions to the modified f…

Paper provides closed-form time derivatives for rigid body systems.

We prove Wadsley's theorem for foliations by closed non-lightlike geodesics. As an application we show that every pseudo-Riemannian and non-Riemannian 2-mainfold, all of whose time- or spacelike geodesics are closed, is diffeomorphic to $S^1\times \R$. Further we show that every pseudo-Riemannian 2-manifold with index …

Develops semi-closed form solutions for barrier and American options on time-dependent OU process.

Estimates lower bound for simplicial volume of certain manifolds.

We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared.

Study shows stability of travel time data reconstruction from closed subsets.

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ interse…

For each $n\geq 2$ we construct a new closed embedded mean curvature self-shrinking hypersurface in $\mathbb{R}^{2n}$. These self-shrinkers are diffeomorphic to $S^{n-1}\times S^{n-1}\times S^1$ and are $SO(n)\times SO(n)$ invariant. The method is inspired by constructions of Hsiang and these surfaces generalize self-s…

The paper examines Ricci flows with closed and smooth tangent flows, proving uniqueness and characterizing ancient flows.

In this note we construct a first example of a closed 3-form of $\tilde G_2$-type on $S^3\times S^4$. We prove that $S^3\times S^4$ does not admit a homogeneous 3-form of $\tilde G_2$-type. Thus our example is a first example of a closed 3-form of $\tilde G_2$-type on a compact 7-manifold which is not stably homogeneou…

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initi…

Every knot can be transformed into an unknot without cutting.

Compactifies maximal component of surface group representations into a closed ball.

The paper proves smoothness of mean curvature flow for generic initial data in 3D and 4D.

In this paper we consider the steepest descent $H^{-1}$-gradient flow of the length functional for immersed plane curves, known as the curve diffusion flow. It is known that under this flow there exist both initially immersed curves which develop at least one singularity in finite time and initially embedded curves whi…

In this paper, we give the generalization of the criterion for a 3-space curve to be closed given by [3] to an n-space curve in Minkowski space-time E_v^n. Furthermore, we apply this criterion for a curve lying on an oriented surface in the Minkowski space E_v^n as an application.

RPE detects anomalies robustly in time-series data.

We study the classification of closed, smooth, spin, $1$-connected $7$-manifolds whose integral cohomology ring is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$. We also prove that if the integral cohomology ring of a closed, smooth, spin, $1$-connected $7$-manifold is isomorphic to $H^*(\mathbb{C}P^2\times S^3)$ or $H^…

Author of this article created for the first time the method for finding solutions of the Minkowski problem for closed surfaces in Riemannian space.

The study reveals conditions for infinite closed geodesics on specific surfaces.

We study the evolution of closed inextensible planar curves under a second order flow that decreases the $p$-elastic energy. A short time existence result for $p \in (1,\infty)$ is obtained via a minimizing movements method. For $p = 2$, that is in the case of the classic elastic energy, long-time existence is retrieve…

To a closed braid in a solid torus we associate a trace graph in a thickened torus in such a way that closed braids are isotopic if and only if their trace graphs can be related by trihedral and tetraherdal moves. For closed braids with a fixed number of strands, we recognize trace graphs up to isotopy and trihedral mo…

In this paper we show the existence of a closed, embedded $λ$-hypersurfaces $Σ\subset \mathbb{R}^{2n}$. The hypersurface is diffeomorhic to $\mathbb{S}^{n-1} \times \mathbb{S}^{n-1} \times \mathbb{S}^1$ and exhibits $SO(n) \times SO(n)$ symmetry. Our approach uses a "shooting method" similar to the approach used by McG…

We obtain new closed-form pricing formulas for contingent claims when the asset follows a Dupire-type local volatility model. To obtain the formulas we use the Dyson-Taylor commutator method that we have recently developed in [5, 6, 8] for short-time asymptotic expansions of heat kernels, and obtain a family of general…

New examples of mean curvature flow converge to minimal surfaces with multiplicity 2.

The paper classifies closed Einstein manifolds with specific curvature properties.

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

We compute the value of the simplicial volume for closed, oriented Riemannian manifolds covered by $\mathbb{H}^{2}\times\mathbb{H}^{2}$ explicitly, thus in particular for products of closed hyperbolic surfaces. This gives the first exact value of a nonvanishing simplicial volume for a manifold of nonconstant curvature.

The paper constructs stable minimal hypersurfaces with specific singularities.

Closed Riemannian manifolds with positive mixed sectional curvature

The closure of a braid in a closed orientable surface $Σ$ is a link in $Σ\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close t…

The study examines the long-term behavior of mean curvature flows in closed 3-manifolds.

Proves stability of spacetime Penrose inequality for spherical symmetric initial data.

We prove that a closed immersed plane curve with total curvature $2πm$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2πm$ whose entropy is less than $m$ …

Ricci flow modelled on specific singularities on closed manifolds.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

We study curve shortening flows in two types of warped product manifolds. These manifolds are $S^1\times N$ with two types of warped metrics where $S^1$ is the unit circle in $R^2$ and $N$ is a closed Riemannian manifold. If the initial curve is a graph over $S^1$, then its curve shortening flow exists for all times an…

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically…

In this paper, we introduce the notion of motif closure and describe higher-order ranking and link prediction methods based on the notion of closing higher-order network motifs. The methods are fast and efficient for real-time ranking and link prediction-based applications such as web search, online advertising, and re…

We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We …