arXiv research

Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.

Study homotopy equivalence of spaces of gradient-like flows and Morse functions on surfaces.

Let $S$ be a set of critical points of a smooth real-valued function on a closed manifold $M$. Generalizing a well-known result of Lusternik--Schnirelmann, Reeken~[R] proved that $\cat S \geq \cat M$. Here we prove a generalization of Reeken"s inequality for gradient-like flows on compact spaces.

We derive general Novikov-Morse type inequalities in a Conley type framework for flows carrying cocycles, therefore generalizing our results in [FJ2] derived for integral cocycle. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distingu…

The topological classification of gradient like Morse-Smale vector fields and diffeomorphisms on 3-manifolds was obtained.

The complete invariant for gradient like Morse-Smale dynamical systems (vector fields and diffeomorphisms) on closed 4-manifolds are constructed. It is same as Kirby diagram in a case of polar vector field without fixed points of index 3.

In this paper we develop a Morse theory for the uniform energy. We use the one-sided directional derivative of the distance function to study the minimizing properties of variations through closed geodesics. This derivative is then used to define a one-sided directional derivative for the uniform energy which allows us…

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…

In our previous paper math.DG/0010008, we develop some new techniques in attacking the convergence problems for the Kähler Ricci flow. The one of main ideas is to find a set of new functionals on curvature tensors such that the Ricci flow is the gradient like flow of these functionals. We successfully find such functio…

We consider a Morse function $f$ and a Morse-Smale gradient-like vector field $X$ on a compact connected oriented 3-manifold $M$ such that $f$ has only one critical point of index 3. Based on Laudenbach's ideas, we will show that the flow of $X$ can be isotoped into one so that the trajectory spaces of the new flow pro…

Study on stability of free boundary Willmore problem using new gradient inequality.

We develop the formal analogue of the Morse theory for a pair of commuting gradient-like vector fields. The resulting algebraic formalism turns out to be very similar to the algebra of the infrared of Gaiotto, Moore and Witten (see [GMW], [KKS]): from a manifold M with the pair of gradient-like commuting vector fields,…

This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric c…

Topological complexity for closed 1-forms

Let $M$ be a closed connected manifold, $f$ be a Morse map from $M$ to a circle, $v$ be a gradient-like vector field satisfying the transversality condition. The Novikov construction associates to these data a chain complex $C_*=C_*(f,v)$. There is a chain homotopy equivalence between $C_*$ and completed simplicial cha…

This article deals with a continuous closed 1-form defined on a CW-complex. In particular, we show Lusternik-Schnirelmann type theory on continuous closed 1-forms which is related to gradient-like flows. M.Farber defined a continuous closed 1-form and a category with a respect to a cohomology class and constructed a Lu…

We study the phase retrieval problem, which solves quadratic system of equations, i.e., recovers a vector $\boldsymbol{x}\in \mathbb{R}^n$ from its magnitude measurements $y_i=|\langle \boldsymbol{a}_i, \boldsymbol{x}\rangle|, i=1,..., m$. We develop a gradient-like algorithm (referred to as RWF representing reshaped W…

Let f be a Morse map from a closed manifold to a circle. S.P.Novikov constructed an analog of the Morse complex for f. The Novikov complex is a chain complex defined over the ring of Laurent power series with integral coefficients and finite negative part. This complex depends on the choice of a gradient-like vector fi…

We will prove the equivariant version of Smale's transversality theorem: suppose that the compact Lie-group G acts on the compact differentiable manifold M on which an invariant Morse-function f and an invariant vector field X are given so that X is gradient-like with respect to f (i.e. X(f)<0 away from critical orbits…

Paper develops a gradient-like proposal for discrete distributions without requiring natural differentiability.

We notice that a generic nonsingular gradient field $v = \nabla f$ on a compact 3-fold $X$ with boundary canonically generates a simple spine $K(f, v)$ of $X$. We study the transformations of $K(f, v)$ that are induced by deformations of the data $(f, v)$. We link the Matveev complexity $c(X)$ of $X$ with counting the …

For a non-vanishing gradient-like vector field on a compact manifold $X^{n+1}$ with boundary, a discrete set of trajectories may be tangent to the boundary with reduced multiplicity $n$, which is the maximum possible. (Among them are trajectories that are tangent to $\partial X$ exactly $n$ times.) We prove a lower bou…

For a Morse map $f:M\to S^1$ Novikov [11] has introduced an analog of Morse complex, defined over the ring $\ZZZ[[t]][t^{-1}]$ of integer Laurent power series. Novikov conjectured, that generically the matrix entries of the differentials in this complex are of the form $\sum_ia_it^i$, where $a_i$ grow at most exponenti…

Zeta invariants study Morse forms on Riemannian manifolds, proving smoothness and convergence.

The paper studies 3-manifolds with specific Morse-Smale diffeomorphisms and finds they are homeomorphic to lens spaces.

This note deals with arbitrary Morse-Smale diffeomorphisms in dimension 3 and extends ideas from \cite{GrLaPo}, \cite{GrLaPo1}, where gradient-like case was considered. We introduce a kind of Morse-Lyapunov function, called dynamically ordered, which fits well dynamics of diffeomorphism. The paper is devoted to finding…

In this paper we study topological lower bounds on the number of zeros of closed 1-forms without Morse type assumptions. We prove that one may always find a representing closed 1-form having at most one zero. We introduce and study a generalization $cat(X,ξ)$ of the notion of Lusternik - Schnirelman category, depending…

Link concordance equals homotopy for high-dimensional spheres.

Discrete diffusion models improve text and image inference.

Large models are prevalent in modern machine learning scenarios, including deep learning, recommender systems, etc., which can have millions or even billions of parameters. Parallel algorithms have become an essential solution technique to many large-scale machine learning jobs. In this paper, we propose a model parall…

TopoNTK kernel captures higher-order interactions in simplicial complexes.

Deep learning improves nearshore bathymetry estimation from sparse data.

New flows introduced for symplectic geometry.

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

Proves uniqueness of geometric flow in various Riemannian manifolds.

Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.

Investigate scalar curvature under geometric flows

We consider four extended Ricci flow systems---that is, Ricci flow coupled with other geometric flows---and prove dynamical stability of certain classes of stationary solutions of these flows. The systems include Ricci flow coupled with harmonic map flow (studied abstractly and in the context of Ricci flow on warped pr…

Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.

i-flow uses normalizing flows for high-dimensional integration and sampling.

The article calculates the $\mathbb{F}$-convergence rate for Ricci flows with closed and smooth tangent flows.

Paper introduces Tensor Gauge Flow Models for better data encoding.

Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.

Ancient curve shortening flows have entropy and curvature bounds equivalent.

The study disproves rotating ancient flows in 4D space.

Simplifies residual flows to make flow-based modeling more practical.

New flow preserves almost Hermitian metrics for manifold study.

Existence of translating solutions shown for curve diffusion flow.