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Working paper

We interpret the combinatorial Mandelbrot set in terms of \it{quadratic laminations} (equivalence relations ∼ on the unit circle invariant under σ2). To each lamination we associate a particular {\em geolamination} (the collection L∼ of points of the circle and edges of convex hulls of ∼-equivalence classes) so that the closure of the set of all of them is a compact metric space with the Hausdorff metric. Two such geolaminations are said to be {\em minor equivalent} if their {\em minors} (images of their longest chords) intersect. We show that the corresponding quotient space of this topological space is homeomorphic to the boundary of the combinatorial Mandelbrot set. To each equivalence class of these geolaminations we associate a unique lamination and its topological polynomial so that this interpretation can be viewed as a way to endow the space of all quadratic topological polynomials with a suitable topology.

Added: Jul 7, 2015

Working paper

It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field $k$ of characteristic $p$, a lifting of $A$ over the ring of second Witt vectors $W_2(k)$ specifies the structure of a Fontaine-Laffaille module on the periodic cyclic homology of $A$. The purpose of this paper is to develop a relative version of Kaledin's theory for DG algebras over a base $k$-algebra $R$ incorporating in the picture the Gauss-Manin connection on the relative periodic cyclic homology constructed by Getzler. Our main result asserts that, under some assumptions on $A$, the Gauss-Manin connection on its periodic cyclic homology can be recovered from the Hochschild homology of $A$ equipped with the action of the Kodaira-Spencer operator as the inverse Cartier transform (in the sense of Ogus-Vologodsky). As an application, we prove, using the reduction modulo $p$ technique, that, for a smooth and proper DG algebra over a complex punctured disk, the monodromy of the Gauss-Manin connection on its periodic cyclic homology is quasi-unipotent.

Added: Nov 8, 2017

Working paper

The paper extends the classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide the uniform asymptotic expansion in terms of the Hermite polynomials. We explicitly obtain expressions for a few initial terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation.

Added: Sep 12, 2015

Working paper

What is the optimal way to cut a convex bounded domain $K$ in Euclidean space $(\Real^n,\abs{\cdot})$ into two halves of equal volume, so that the interface between the two halves has least surface area? A conjecture of Kannan, Lov\'asz and Simonovits asserts that, if one does not mind gaining a universal numerical factor (independent of $n$) in the surface area, one might as well dissect $K$ using a hyperplane. This conjectured essential equivalence between the former non-linear isoperimetric inequality and its latter linear relaxation, has been shown over the last two decades to be of fundamental importance to the understanding of volumetric and spectral properties of convex domains.
In this work, we address the conjecture for the subclass of generalized Orlicz balls
$$
K = \{x \in \Real^n \; ; \; \sum_{i=1}^n V_i(x_i) \leq E \} ,
$$
confirming its validity for certain levels $E \in \Real$ under a mild technical assumption on the growth of the convex functions $V_i$ at infinity. In sharp contrast to previous approaches for tackling the KLS conjecture, we emphasize that no symmetry assumptions are assumed on $K$. This significantly enlarges the subclass of convex bodies for which the conjecture is confirmed.

Added: Dec 27, 2016

Working paper

There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing the limiting individual behavior of particles inside a large ensemble of identical interacting particles. The resulting equations are generally referred to as nonlinear Scr¨odinger equations or Hartree equations, or Gross-Pitaevski equations. In this paper we extend some of these convergence results to a stochastic framework. Concretely we work with the Belavkin stochastic filtering of many particle quantum systems. The resulting limiting equation is an equation of a new type, which can be seen as a complex-valued infinite dimensional nonlinear diffusion of McKean-Vlasov type. This result is the key ingredient for the theory of quantum mean-field games developed by the author in a previous paper.

Added: Oct 30, 2020

Working paper

The multistsochastic Monge--Kantorovich problem on the product $X = \prod_{i=1}^n X_i$ of $n$ spaces is a generalization of the multimarginal Monge--Kantorovich problem. For a given integer number $1 \le k<n$ we consider the minimization problem $\int c d \pi \to \inf$ of the space of measures with fixed projections onto every $X_{i_1} \times \dots \times X_{i_k}$ for arbitrary set of $k$ indices $\{i_1, \dots, i_k\} \subset \{1, \dots, n\}$. In this paper we study basic properties of the multistochastic problem, including well-posedness, existence of a dual solution, boundedness and continuity of a dual solution.

Added: Aug 21, 2020

Working paper

Let (X,C) be a germ of a threefold X with terminal singularities along an irreducible reduced complete curve C with a contraction f:(X,C)→(Z,o) such that C=f−1(o)red and −KX is ample. Assume that (X,C) contains a point of type (IIA). This paper continues our study of such germs containing a point of type (IIA) started in our previous paper.

Added: Aug 3, 2017

Working paper

The general goal of this paper is to gather and review several methods from homotopy and combinatorial topology and formal concepts analysis (FCA) and analyze their connections. FCA appears naturally in the problem of combinatorial simplification of simplicial complexes and allows to see a certain duality on a class of simplicial complexes. This duality generalizes Poincare duality on cell subdivisions of manifolds. On the other hand, with the notion of a topological formal context, we review the classical proofs of two basic theorems of homotopy topology: Alexandrov Nerve theorem and Quillen--McCord theorem, which are both important in the applications. A brief overview of the applications of the Nerve theorem in brain studies is given. The focus is made on the task of the external stimuli space reconstruction from the activity of place cells. We propose to use the combination of FCA and topology in the analysis of neural codes. The lattice of formal concepts of a neural code is homotopy equivalent to the nerve complex, but, moreover, it allows to analyse certain implication relations between collections of neural cells.

Added: Nov 15, 2019

Working paper

We study toric G-solid Fano threefolds that have at most terminal singularities, where G is an algebraic subgroup of the normalizer of a maximal torus in their automorphism groups.

Added: Aug 19, 2020

Working paper

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.

Added: Jul 31, 2017

Working paper

We propose a remarkably simple and explicit conjectural formula for a bihamiltonian structure of the double ramification hierarchy corresponding to an arbitrary homogeneous cohomological field theory. Various checks are presented to support the conjecture.

Added: Oct 1, 2020

Working paper

In previous papers we introduced the notion of special Bohr - Sommerfeld lagrangian cycles on a compact simply connected symplectic manifold with integer symplectic form, and presented the main interesting case: compact simply connected algebraic variety with an ample line bundle such that the space of Bohr - Sommerfeld lagrangian cycles with respect to a compatible Kahler form of the Hodge type and holomorphic sections of the bundle is finite. The main problem appeared in this way is singular components of the corresponding lagrangian shadows (or sceletons of the corresponding Weinstein domains) which are hard to distinguish or resolve. In the present text we avoid this difficulty presenting the points of the moduli space of special Bohr - Sommerfeld lagrangian cycles by exact compact lagrangian submanifolds on the complements $X \backslash D_{\alpha}$ modulo Hamiltonian isotopies, where $D_{\alpha}$ is the zero divisor of holomorphic section $\alpha$. In a sense it corresponds to the usage of gauge classes of hermitian connections instead of pure holomorphic structures in the theory of the moduli space of (semi) stable vector bundles.

Added: Aug 6, 2017

Working paper

Local perturbations of an infinitely long rod go away to infinity. On the contrary, in the case of a finite length of the rod the perturbations reach its boundary and are reflected from them. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local in time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high "transparency" of these boundary conditions.

Added: Jan 13, 2020

Working paper

It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Alaniya's theorem, showing that the Dolbeault cohomology of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. This implies that the twisted version of Console-Fino theorem is false (Console-Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold M is a Kähler structure on its cover M̃ such that the deck transform map acts on M̃ by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.

Added: Jun 9, 2019

Working paper

Let E be a smooth elliptic curve over ℂ. For E embedded into ℙ2 as Hesse cubic and V an Ulrich bundle on E we derive an explicit presentation of V using Moore matrices and theta functions.

Added: Oct 9, 2018

Working paper

Buchstaber and Terzic introduced a notion of universal space of parameters F for a manifold M^2n, which has an effective action of compact torus T^k , k≤n with some additional properties. with special properties. This space is needed to construction of factor M^2n/T^k. Buchstaber and Terzic constructed the universal space of parameters for G_5,2. In this work we construct universal space of parameters for complex Grassmann manifold G_q+1,2. Our construction is based on the construction of moduli space of stable curves of genus zero with q+1 marked points due to Salamon, McDuff and Hofer.

Added: Nov 20, 2019

Working paper

Entov M.,

math. arxive. Cornell University, 2017
Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as well as for any closed Kahler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kahler resolution of a Kahler orbifold with isolated singularities and relies on the results of Demailly-Paun and Miyaoka on Kahler cohomology classes.

Added: Apr 10, 2017

Working paper

We provide new examples of K-unstable polarized smooth del Pezzo surfaces using a flopped version first used by Cheltsov and Rubinstein of the test configurations introduced by Ross and Thomas. As an application, we provide new obstructions for the existence of constant scalar curvature Kahler metrics on polarized smooth del Pezzo surfaces.

Added: Dec 3, 2018

Working paper

In the present paper we prove that any family of hyperk¨ahler manifolds over a compact simply connected base can be pulled back from a family over a curve. A consimilar result about families of complex tori is also obtained.

Added: Dec 7, 2015

Working paper

Frikha N.,

, math. arxive. Cornell University, 2019. No. 1910.05945.
We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKeanVlasov driven by non-degenerate symmetric α-stable L´evy processes with values in R^d under some mild Holder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index α ∈ (0, 1). New strong well-posedness results are also derived from the previous analysis.

Added: Oct 15, 2019

Working paper

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to osp(1,2) and the nonsymmetric Macdonald polynomials of types $A_2^2$ and ${A_2}^{2\dagger}$ . We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the t=0 and t=\infty specializations of the nonsymmetric Macdonald polynomials. We show that the specializations can be described in terms of the Lie superalgebras action on the Weyl modules

Added: Jul 8, 2015