We show that there is no positive loop inside the component of a fiber in the space of Legendrian embeddings in the contact manifold ST M, provided that the universal cover of M is Rn. We consider some related results in the space of one-jets of functions on a compact manifold. We give an application to the positive isotopies in homogeneous neighborhoods of surfaces in a tight contact 3-manifold.

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Working paper

A tree, embedded into plane, is a dessin d'enfant and its Belyi function is a polynomial -- Shabat polynomial. Zapponi form of this polynomial is unique, so we can correspond to an embedded tree the Julia set of its Shabat-Zapponi polynomial. In this purely experimental work we study relations between the form of a tree and properties (form, connectedness, Hausdorff dimension) of its Julia set.

Added: Sep 5, 2016

Working paper

It is well known that by dualizing the Bochner-Lichnerowicz-Weitzenb\"{o}ck formula, one obtains Poincar\'e-type inequalities on Riemannian manifolds equipped with a density, which satisfy the Bakry-\'Emery Curvature-Dimension condition (combining the Ricci curvature with the "curvature" of the density). When the manifold has a boundary, the Reilly formula and its generalizations may be used instead. By systematically dualizing this formula for various combinations of boundary conditions of the domain (convex, mean-convex) and the function (Neumann, Dirichlet), we obtain new Poincar\'e-type inequalities on the manifold and on its boundary. For instance, we may handle Neumann conditions on a mean-convex domain, and obtain generalizations to the weighted-manifold setting of a purely Euclidean inequality of Colesanti, yielding a Brunn-Minkowski concavity result for geodesic extensions of convex domains in the manifold setting. All other previously known Poincar\'e-type inequalities of Lichnerowicz, Brascamp-Lieb, Bobkov-Ledoux, Nguyen and Veysseire are recovered, in some cases improved, and generalized into a single unified formulation, and their appropriate versions in the presence of a boundary are obtained. Finally, a new geometric evolution equation is proposed which extends to the Riemannian setting the Minkowski addition operation of convex domains, a notion previously confined to the linear setting, and for which a novel Brunn-Minkowski inequality in the weighted-Riemannian setting is obtained. Our framework allows to encompass the entire class of Borell's convex measures, including heavy-tailed measures, and extends the latter class to weighted-manifolds having negative "dimension".

Added: Nov 17, 2013

Working paper

We study the variety of Poisson structures and compute Poisson cohomology for two families of Fano threefolds - smooth cubic threefolds and the del Pezzo quintic threefold. Along the way we reobtain by a different method earlier results of Loray, Pereira and Touzet in the special case we are considering.

Added: Dec 27, 2013

Working paper

Added: Nov 1, 2012

Working paper

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and \omega, where \omega corresponds to the full uniform reflection schema, whereas n<\omega corresponds to its restriction to arithmetical \Pi_{n+1}-formulas. This calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time.

Added: Nov 22, 2013

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Added: Oct 18, 2019

Working paper

The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.

Added: Apr 15, 2021

Working paper

Added: Feb 6, 2013

Working paper

We prove that a foliation (M,F) of codimension q on a n-dimensional pseudo-Riemannian manifold is pseudo-Riemannian if and only if any geodesic that is orthogonal at one point to a leaf is orthogonal to every leaf it intersects. We show that on the graph G=G(F) of a pseudo-Riemannian foliation there exists a unique pseudo-Riemannian metric such that canonical projections are pseudo-Riemannian submersions and the fibres of different projections are orthogonal at common points. Relatively this metric the induced foliation (G,F) on the graph is pseudo-Riemannian and the structure of the leaves of (G,F) is described. Special attention is given to the structure of graphs of transversally (geodesically) complete pseudo-Riemannian foliations which are totally geodesic pseudo-Riemannian ones.

Added: Nov 29, 2016

Working paper

A small perturbation of a quadratic polynomial with a non-repelling fixed point gives a polynomial with an attracting fixed point and a Jordan curve Julia set, on which the perturbed polynomial acts like angle doubling. However, there are cubic polynomials with a non-repelling fixed point, for which no perturbation results into a polynomial with Jordan curve Julia set. Motivated by the study of the closure of the Cubic Principal Hyperbolic Domain, we describe such polynomials in terms of their quadratic-like restrictions.

Added: Oct 6, 2013

Working paper

We concider the Wiener algebra A(T^d) of absolutely convergent Fourier series on the d -torus. For phase functions \phi of a certain special form we obtain lower bounds for the A -norms of e^{i\lambda\phi} as \lambda tends to \infty.

Added: Nov 11, 2016

Working paper

Drinfeld zastava is a certain closure of the moduli space of maps from the projective line to the Kashiwara flag scheme of an affine Lie algebra g^. In case g is the symplectic Lie algebra spN, we introduce an affine, reduced, irreducible, normal quiver variety Z which maps to the zastava space isomorphically in characteristic 0. The natural Poisson structure on the zastava space Z can be described in terms of Hamiltonian reduction of a certain Poisson subvariety of the dual space of a (nonsemisimple) Lie algebra. The quantum Hamiltonian reduction of the corresponding quotient of its universal enveloping algebra produces a quantization Y of the coordinate ring of Z. The same quantization was obtained in the finite (as opposed to the affine) case generically in arXiv:math/0409031 . We prove that Y is a quotient of the affine Borel Yangian. The analogous results for g=slN were obtained in our previous work arXiv:1009.0676 .

Added: Dec 27, 2013

Working paper

We introduce and study noncommutative (or "quantized") versions of the algebras of holomorphic functions on the polydisk and on the ball in C^n. Specifically, for each nonzero complex number q we construct Fréchet algebras O_q(D^n) and O_q(B^n) such that for q=1 they are isomorphic to the algebras of holomorphic functions on the open polydisk D^n and on the open ball B^n, respectively. We show that O_q(D^n) and O_q(B^n) are not isomorphic provided that |q|=1 and n>1. This result can be interpreted as a q-analog of Poincaré's theorem, which asserts that D^n and B^n are not biholomorphically equivalent unless n=1. In contrast, O_q(D^n) and O_q(B^n) are shown to be isomorphic for |q| ≠ 1.. Next we prove that O_q(D^n) is isomorphic to a quotient of J. L. Taylor's "free polydisk algebra" (1972). This enables us to construct a Fréchet O(C^x)-algebra O_def(D^n) whose "fiber" over each q is isomorphic to O_q(D^n). Replacing the free polydisk algebra by G. Popescu's "free ball algebra" (2006), we obtain a Fréchet O(C^x)-algebra O_def(B^n) with fibers isomorphic to O_q(B^n). The algebras O_def(D^n) and O_def(B^n) yield continuous Fréchet algebra bundles over C^x which are strict deformation quantizations (in Rieffel's sense) of O(D^n) and O(B^n), respectively. Finally, we study relations between our deformations and formal deformations of O_q(D^n) and O_q(B^n).

Added: Aug 27, 2015

Working paper

Kazeev V., Oseledets I.,

et al. arxiv.org. math. Cornell University, 2020
Homogenization in terms of multiscale limits transforms a multiscale problem with n+1 asymptotically separated microscales posed on a physical domain D⊂ℝd into a one-scale problem posed on a product domain of dimension (n+1)d by introducing n so-called "fast variables". This procedure allows to convert n+1 scales in d physical dimensions into a single-scale structure in (n+1)d dimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with the recently developed quantized tensor-train finite-element method (QTT-FEM). The method is based on restricting computation to sequences of nested subspaces of low dimensions (which are called tensor ranks) within a vast but generic "virtual" (background) discretization space. In the course of computation, these subspaces are computed iteratively and data-adaptively at runtime, bypassing any "offline precomputation". For the purpose of theoretical analysis, such low-dimensional subspaces are constructed analytically to bound the tensor ranks vs. error τ>0. We consider a model linear elliptic multiscale problem in several physical dimensions and show, theoretically and experimentally, that both (i) the solution of the associated high-dimensional one-scale problem and (ii) the corresponding approximation to the solution of the multiscale problem admit efficient approximation by the QTT-FEM. These problems can therefore be numerically solved in a scale-robust fashion by standard (low-order) PDE discretizations combined with state-of-the-art general-purpose solvers for tensor-structured linear systems. We prove scale-robust exponential convergence, i.e., that QTT-FEM achieves accuracy τ with the number of effective degrees of freedom scaling polynomially in logτ.

Added: Oct 20, 2020

Working paper

Ogievetsky O.,

arxiv.org. math. Cornell University, 2019. No. arXiv:1910.08551 .
A notion of quantum matrix (QM-) algebra generalizes and unifies two famous families of algebras from the theory of quantum groups: the RTT-algebras and the reflection equation (RE-) algebras. These algebras being generated by the components of a `quantum' matrix M possess certain properties which resemble structure theorems of the ordinary matrix theory. It turns out that such structure results are naturally derived in a more general framework of the QM-algebras. In this work we consider a family of Birman-Murakami-Wenzl (BMW) type QM-algebras. These algebras are defined with the use of R-matrix representations of the BMW algebras. Particular series of such algebras include orthogonal and symplectic types RTT- and RE- algebras, as well as their super-partners. For a family of BMW type QM-algebras, we investigate the structure of their `characteristic subalgebras' --- the subalgebras where the coefficients of characteristic polynomials take values. We define three sets of generating elements of the characteristic subalgebra and derive recursive Newton and Wronski relations between them. We also define an associative ⋆-product for the matrix M of generators of the QM-algebra which is a proper generalization of the classical matrix multiplication. We determine the set of all matrix `descendants' of the quantum matrix M, and prove the ⋆-commutativity of this set in the BMW type.

Added: Oct 25, 2019

Working paper

The quantum period of a variety X is a generating function for certain Gromov-Witten invariants of X which plays an important role in mirror symmetry. In this paper we compute the quantum periods of all 3-dimensional Fano manifolds. In particular we show that 3-dimensional Fano manifolds with very ample anticanonical bundle have mirrors given by a collection of Laurent polynomials called Minkowski polynomials. This was conjectured in joint work with Golyshev. It suggests a new approach to the classification of Fano manifolds: by proving an appropriate mirror theorem and then classifying Fano mirrors. Our methods are likely to be of independent interest. We rework the Mori-Mukai classification of 3-dimensional Fano manifolds, showing that each of them can be expressed as the zero locus of a section of a homogeneous vector bundle over a GIT quotient V/G, where G is a product of groups of the form GL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses the Fano 3-fold as a toric complete intersection; in the remaining cases, it expresses the Fano 3-fold as a tautological subvariety of a Grassmannian, partial flag manifold, or projective bundle thereon. We then compute the quantum periods using the Quantum Lefschetz Hyperplane Theorem of Coates-Givental and the Abelian/non-Abelian correspondence of Bertram-Ciocan-Fontanine-Kim-Sabbah.

Added: May 27, 2013

Working paper

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.

Added: Jun 20, 2014

Working paper

http://arxiv.org/abs/1311.0309

Added: Nov 14, 2013

Working paper

In this paper we study quotients of rational conic bundles over arbitrary fields of characteristic zero by finite groups of automorphisms. We construct smooth minimal models for such quotients and show that there are many examples of birationally non-trivial quotients.

Added: Jan 24, 2014

Working paper

In this paper we study quotients of del Pezzo surface of degree $4$ and more
over arbitrary field $\Bbbk$ of characteristic zero by finite groups of
automorphisms. We show that if del Pezzo surface contains a point defined over
the ground field and degree is five or greater then the quotient is always
$\Bbbk$-rational. If degree is equal to four then the quotient can be
non-$\Bbbk$-rational only if the order of group is $1$, $2$ or $4$.

Added: Jan 24, 2014

Working paper

Davydov Y.,

arxiv.org. math. Cornell University, 2016. No. 1609.07066.
We consider the moving particle process in Rd which is defined in the following way. There are two independent sequences (Tk) and (dk) of random variables. The variables Tk are non negative and form an increasing sequence, while variables dk form an i.i.d sequence with common distribution concentrated on the unit sphere. The values dk are interpreted as the directions, and Tk as the moments of change of directions. A particle starts from zero and moves in the direction d1 up to the moment T1 . It then changes direction to d2 and moves on within the time interval T2 minus T1 , etc. The speed is constant at all sites. The position of the particle at time t is denoted by X(t). We suppose that the points (Tk) form a non homogeneous Poisson point process and we are interested in the global behavior of the process (X(t)), namely, we are looking for conditions under which the processes (Y(T,t), T is non negative), Y(T,t) is X(tT) normalized by B(T), t in (0, 1), weakly converges in C(0, 1) to some process Y when T tends to infinity. In the second part of the paper the process X(t) is considered as a Markov chain. We construct diffusion approximations for this process and investigate their accuracy. The main tool in this part is the paramertix method.

Added: Sep 23, 2016