Related papers: The Kontsevich tetrahedral flow revisited

In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I

In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of…

In this paper, we obtain the existence criteria for a geometic flow on noncompact affine Riemannian manifolds. Our results can be regarded as a real version of Lee-Tam [19]. As an application, we prove that a complete noncompact Hessian…

A new proof is given of the fact that the particle trajectories of the ideal incompressible fluid are analytic curves, though the solutions of the Euler equations may have a finite regularity. This is a consequence of a general fact that…

As an absolute invariant of smooth conjugacy, the multiplier group described the types of space-time symmetries that the flow has, and for a quasiperiodic flow on the $n$-torus, is the determining factor of the structure of its generalized…

We present the noncanonical Hamiltonian structure of the relativistic Euler equations for a perfect fluid in Minkowski spacetime. By identifying the system's noncanonical Poisson bracket and Hamiltonian, we show that relativistic fluid…

In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness…

This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal…

Quasisymmetry (QS) is a hidden symmetry of the magnetic field strength, B, that effectively confines charged particles in a three-dimensional toroidal plasma equilibrium. Here, we show that QS has a deep connection to the underlying…

We study the noncommutative Poincar\'e duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincar\'e…

We investigate the properties of the combinatorial Ricci flow for surfaces, both forward and backward -- existence, uniqueness and singularities formation. We show that the positive results that exist for the smooth Ricci flow also hold for…

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M x N of a quaternionic-Kaehler manifold M of negative scalar curvature and a very special real manifold N of dimension n >=0. Such gradient…

The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call {\it twisted polygons}. In this note we describe our recent work on the pentagram map, in which we find a…

The classical Bott-Samelson theorem states that if on a Riemannian manifold all geodesics issuing from a certain point return to this point, then the universal cover of the manifold has the cohomology ring of a compact rank one symmetric…

A polygon is called rational if the angle between each pair of sides is a rational multiple of $\pi.$ The main theorem we will prove is Theorem 1: For rational polygons, periodic points of the billiard flow are dense in the phase space of…

We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in $\mathbb{T}\times [0,1]$ when the initial perturbation is in…

We give an Eynard-Orantin type topological recursion formula for the canonical Euclidean volume of the combinatorial moduli space of pointed smooth algebraic curves. The recursion comes from the edge removal operation on the space of ribbon…

For a transitive Anosov flow $\Phi$ on 3-dimensional closed manifold $M$ , we realize its Teichm\"uller space in the sense of smooth orbit-equivalence classes as a product of two function spaces. As an application, we show the…

The paper is devoted to the study of topological properties, structure and classification of Morse flows with fixed points on the boundary of three-dimensional manifolds. We construct a complete topological invariant of a Morse flow,…

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is…