arXiv research

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

Paper shows how to represent Milnor's triple linking number using chord diagrams and doodle invariants.

Paper develops invariants for spherical curves using chord diagrams.

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 Arnol'd's chord conjecture for conormal bundles.

Proves Arnold-Thom conjecture for surfaces' arrival times.

Proves Arnold conjecture for singular symplectic manifolds using novel techniques.

The paper confirms Arnold's conjecture about hyperbolic polynomials.

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.

Vassiliev invariants can be studied by studying the spaces of chord diagrams associated with singular knots. To these chord diagrams are associated the intersection graphs of the chords. We extend results of Chmutov, Duzhin and Lando to show that these graphs determine the chord diagram if the graph has at most one loo…

We prove conjectures of Rene Thom and Vladimir Arnold for C^2 solutions to the degenerate elliptic equation that is the level set equation for motion by mean curvature. We believe these results are the first instances of a general principle: Solutions of many degenerate equations behave as if they are analytic, even wh…

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.

We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em triangular diagrams} and {\em connected hypergraphs} in the same way as the calcul…

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

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for…

It can be conjectured that the colored Jones function of a knot can be computed in terms of counting paths on the graph of a planar projection of a knot. On the combinatorial level, the colored Jones function can be replaced by its weight system. We give two curious formulas for the weight system of a colored Jones fun…

The Magnus expansion is a universal finite type invariant of pure braids with values in the space of horizontal chord diagrams. The Conway polynomial composed with the short circuit map from braids to knots gives rise to a series of finite type invariants of pure braids and thus factors through the Magnus map. We descr…

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 Poincare function is a compact form of counting moduli in local geometric problems. We discuss its property in relation to V.Arnold's conjecture, and derive this conjecture in the case when the pseudogroup acts algebraically and transitively on the base. Then we survey the known counting results for differential in…

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…

New methods prove non-squeezing in locally conformal symplectic geometry.

We construct the TQFT on symplectic cohomology and wrapped Floer cohomology, possibly twisted by a local system of coefficients, and prove that the TQFT respects Viterbo restriction maps and the canonical maps from ordinary cohomology. We also construct the module structure of wrapped Floer cohomology over symplectic c…

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

The paper shows that knot projections without triple chords can be simplified.

Paper explores relationships between triple chords and a specific homotopy relation in knot theory.

Formula for $\mathfrak{sl}_2$ weight system on complete bipartite graphs.

New integral expression quantizes Arnold strangeness.

Solves $L_p$-Gaussian chord Minkowski problem using Gauss curvature flow.

A chord diagram consists of a circle, called the backbone, with line segments, called chords, whose endpoints are attached to distinct points on the circle. The genus of a chord diagram is the genus of the orientable surface obtained by thickening the backbone to an annulus and attaching bands to the inner boundary cir…

We consider framed chord diagrams, i.e. chord diagrams with chords of two types. It is well known that chord diagrams modulo 4T-relations admit Hopf algebra structure, where the multiplication is given by any connected sum with respect to the orientation. But in the case of framed chord diagrams a natural way to define…

A chord index homomorphism for knots in thickened surfaces is constructed.

The paper finds at least N orthogonal Finsler geodesic chords in a disk-like manifold.

Paper solves a generalized chord Minkowski problem using Gauss curvature flows.

In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones pol…

Enhances Vassiliev knot invariants using chord diagrams.

Study counts sub-chord diagrams to classify spherical curves.

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…

The paper disproves the properness conjecture for higher-dimensional minimal hypersurfaces.

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…

In view of the result of Kontsevich, now often called ``the fundamental theorem of Vassiliev theory'', identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the ``4T-relation'' (and in the unframed case, th…

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.