arXiv research

In this article, we give proofs on the Arnold Lagrangian intersection conjecture on the cotangent bundles, Arnold-Givental Lagrangian intersection conjecture and the Arnold fixed point conjecture.

Proves Arnold-Thom conjectures for motion by mean curvature.

Proves Arnold-Thom conjecture for surfaces' arrival times.

Proves Arnold conjecture for singular symplectic manifolds using novel techniques.

Proves Weinstein's and Arnold's conjectures using contact instantons.

The paper confirms Arnold's conjecture about hyperbolic polynomials.

Poincare function counts geometric moduli, solving Arnold's conjecture.

In [R2] and [RO] the Arnold conjecture for closed symplectic manifolds with trivial second homotopy group was proved. This proof used surgery and cobordism theory. Here we give a purely cohomological proof of this result.

The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.

We prove the Arnold conjecture for closed symplectic manifolds with $π_2(M)=0$ and $\cat M=\dim M$. Furthermore, we prove an analog of the Lusternik-Schnirelmann theorem for functions with ``generalized hyperbolicity'' property.

We prove that the Lusternik-Schnirelmann category $cat(M)$ of a closed symplectic manifold $(M, ω)$ equals the dimension $dim(M)$ provided that the symplectic cohomology class vanishes on the image of the Hurewicz homomorphism. This holds, in particular, when $π_2(M)=0$. The Arnold conjecture asserts that the number of…

The Arnold conjecture is proven for integers using Floer theory.

The Hessian Topology is a subject with interesting relations with some classical problems of analysis and geometry. In this article we prove a conjecture on this subject stated by V.I. Arnold concerning the number of connected components of hyperbolic homogeneous polynomials of degree $n$. The proof is constructive and…

Nearby pinwheels are isotopic, solving Arnold's conjecture.

The Kähler-Ricci flow smooths positive currents on Kähler manifolds.

We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold.

The abstract discusses conjectures about virtual Legendrian knots and their relation to causality.

We use virtual neighborhood technique to establish GW-invariants, Quantum cohomology, equivariant GW-invariants, equivariant quantum cohomology and Floer cohomology for general symplectic manifold. We also establish GW-invariants for a family of symplectic manifolds. As a consequence, we prove Arnold conjecture for non…

The paper studies co-Hamiltonian diffeomorphisms on compact cosymplectic manifolds.

New integral expression quantizes Arnold strangeness.

We present some results supporting the Iwase-Sakai conjecture about coincidence of the topological complexity $TC(X)$ and monoidal topological complexity $TC^M(X)$. Using these results we provide lower and upper bounds for the topological complexity of the wedge $X\vee Y$. We use these bounds to give a counterexample t…

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology theory for symplectomorphisms of surface and a calculation of Seidel's symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and asympltot…

Kolmogorov-Arnold Networks promise scalable performance in high dimensions.

New integer-valued functions for Legendrian knots.

Quotients $Y=X/conj$ of complex surfaces by anti-holomorphic involutions $conj\: X\to X$ tend to be completely decomposable when they are simply connected, i.e., split into connected sums, $n CP^2\#m\barCP2$, if $w_2(Y)\ne0$, or into $n(S^2\times S^2)$ if $w_2(Y)=0$. If $X$ is a double branched covering over $CP^2$, th…

Explain Arnold's proof of the Morse index theorem using Maslov index.

Revisits the connection between neural networks and the Kolmogorov-Arnold theorem.

We define the generalized connected sum for generic closed plane curves, generalizing the strange sum defined by Arnold, and completely describe how the Arnold invariants $J^{\pm}$ and $\mathit{St}$ behave under the generalized connected sums.

Arnold-Liouville systems cannot be bi-Hamiltonian generically.

Solves an Arnold trivium problem using calculus and topology.

Explains Arnold's J+ invariant for curves, using basic math.

A point q in a contact manifold is called a translated point for a contactomorphism φ, with respect to some fixed contact form, if φ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. In this article we discuss a version of the Arnold conjecture for translated points of contactomorphisms and…

A new Kolmogorov-Arnold network improves function approximation and optimization.

SVGP KAN integrates uncertainty quantification into Kolmogorov-Arnold networks.

Recently V. Arnold introduced Strangeness and $J^{\pm}$ invariants of generic immersions of an oriented circle to $\R^2$. Here these invariants are generalized to the case of generic immersions of an oriented circle to an arbitrary surface $F$. We explicitly describe all the invariants satisfying axioms, which naturall…

In this paper we will prove that for a compact, symplectic manifold $(M, ω)$ and for $ω$-compatible almost-complex structure J any properly perturbed J-holomorphic curve has a non-negative symplectic area. This non-negative property provides us with a new obstruction to the bubbling off phenomenon and thus allows us to…

The paper develops obstructions for embedding 2D complexes into 4D space.

Combines geometric hydrodynamics with magnetic systems to derive new equations and prove well-posedness.

In this paper we prove a lower bound for the least number of one-periodic solutions of nondegenerate locally Hamiltonian equations on compact symplectic manifolds in terms of the Betti numbers of the Novikov homology associated to the Calabi invariant of the locally Hamiltonian equations. Our result improves lower boun…

Study shows almost all Arnold stable solutions have no conjugate points.

Arnold discovered geodesics in fluid dynamics.

Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.

We apply Arnold's theory of generic smooth plane curves to Stark-Zeeman systems. This is a class of Hamiltonian dynamical systems that describes the dynamics of an electron in an external electric and magnetic field, and includes many systems from celestial mechanics. Based on Arnold's $J^+$-invariant, we introduce inv…

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, ω, τ)$ with nonempty and compact real part $L={\rm Fix}(τ)$. For given $Λ\in (0, +\infty]$ and $m\in\N\cup\{0\}$ we show the equivalence of the following two claims: (i) $\sharp(L\capφ^H_1(L))\ge m$ for any Hamiltonia…

Hamiltonian minimality (H-minimality) for Lagrangian submanifolds is a symplectic analogue of Riemannian minimality. A Lagrangian submanifold is called H-minimal if the variations of its volume along all Hamiltonian vector fields are zero. This notion was introduced in the work of Y.-G. Oh in connection with the celebr…

A new formula detects differences between counterexamples and standard embeddings of circles.

Kolmogorov-Arnold Networks achieve optimal convergence rates in nonparametric regression.

Floer homology theories solve complex dynamics problems.