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Working paper

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure γ enjoys 1n-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex K and L,
γ(λK+(1−λ)L)12n≥λγ(K)12n+(1−λ)γ(L)12n.
Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to p-concavity, with p>0, with respect to the addition of symmetric convex se

Added: Jul 31, 2018

Working paper

We define the Kobayashi quotient of a complex variety by identifying points with vanishing Kobayashi pseudodistance between them and show that if a compact complex manifold has an automorphism whose order is infinite, then the fibers of this quotient map are nontrivial. We prove that the Kobayashi quotients associated to ergodic complex structures on a compact manifold are isomorphic. We also give a proof of Kobayashi's conjecture on the vanishing of the pseudodistance for hyperk\"ahler manifolds having Lagrangian fibrations without multiple fibers in codimension one. For a hyperbolic automorphism of a hyperk\"ahler manifold, we prove that its cohomology eigenvalues are determined by its Hodge numbers, compute its dynamical degree and show that its cohomological trace grows exponentially, giving estimates on the number of its periodic points.

Added: Sep 6, 2016

Working paper

We present an upeer bound of the number of solutions (x,y) of a polynomial equation P(x,y)=0 over a field F_p in the case where x,y from G, G is a subgroup of F_p^*.

Added: Jun 22, 2015

Working paper

Bendikov A., Grigor'yan A.,

math. arxive. Cornell University, 2018. No. 1811.05210.
Added: Nov 9, 2018

Working paper

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration Ω∗_(r)(X) are unions of finitely presented 𝕃-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules 𝕃/I(p,n)x where deg x≥p^n−1/(p−1). As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free 𝕃-resolutions of Ω∗(X), and show that algebraic cobordism of a smooth surface can be described in terms of K_0 together with a topological filtration.

Added: Dec 6, 2018

Working paper

We prove that the set of n-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most 2n−1. Moreover, we show that the Hausdorff dimension of n-points configurations on which some locally minimal trees have the same length is also at most 2n−1. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them.

Added: Oct 21, 2019

Working paper

We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.

Added: Dec 3, 2018

Working paper

We define stationary descendent integrals on the moduli space of stable maps from disks to $(\CP^1,\rr\pp^1)$. We prove a localization formula for the stationary theory involving contributions from the fixed points and from all the corner-strata. We use the localization formula to prove a recursion relation and a closed formula for all genus $0$ disk cover invariants in the stationary case. For all higher genus invariants, we propose a conjectural formula.

Added: Oct 5, 2020

Working paper

We lay the foundation for a version of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define the notion of $r$-spin disks, their moduli space, and the Witten bundle, we show that the moduli space is a compact smooth orientable orbifold with corners, and we prove that the Witten bundle is canonically relatively oriented relative to the moduli space. In the sequel to this paper, we use these constructions to define open $r$-spin intersection theory and relate it to the Gelfand--Dickey hierarchy, thus providing an analogue of Witten's $r$-spin conjecture in the open setting.

Added: Oct 5, 2020

Working paper

We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand--Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan--Jarvis--Ruan--Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed theory.

Added: Oct 5, 2020

Working paper

The article demonstrates rather general approach to problems of discrete geometry: treat them as global optimization problems to be solved by one of general purpose solver implementing branch-and-bound algorithm (B&B). This approach may be used for various types of problems, i.e. Tammes problems, Thomson problems, search of minimal potential energy of micro-clusters, etc. Here we consider a problem of densest packing of equal circles in special geometrical object, so called square flat torus ℝ2/ℤ2 with the induced metric. It is formulated as Mixed-Integer Nonlinear Problem with linear and non-convex quadratic constraints.
The open-source B&B-solver SCIP, this http URL, and its parallel implementation ParaSCIP, this http URL, had been used in computing experiments to find "very good" approximations of optimal arrangements. The main result is a confirmation of the conjecture on optimal packing for N=9 that was published in 2012 by O. Musin and A. Nikitenko. To do that, ParaSCIP took about 2000 CPU*hours (16 hours x 128 CPUs) of cluster HPC4/HPC5, National Research Centre "Kurchatov Institute", this http URL

Added: Dec 20, 2018

Working paper

In recent years PBW degenerations of Demazure modules and Schubert varieties were defined and studied in several papers. Various interesting properties (such as these PBW degenerations embedding naturally into the corresponding degenerate representations and flag varieties) were obtained in type A but only with restrictions on the Weyl group element or the highest weight. We show that these properties cannot hold in full generality due to the following issue with the definition. The degenerate variety depends on the highest weight used to define it and not only on its Weyl group stabilizer (as is the case for PBW degenerate flag varieties as well as classical Schubert varieties). Perhaps surprisingly, the minimal counterexamples appear only for sl6. The counterexamples are constructed with the help of a study of the Cartan components appearing in this context.

Added: Oct 22, 2019

Working paper

Blokh A., Oversteegen L.,

math. arxive. Cornell University, 2017
The combinatorial Mandelbrot set is a continuum in the plane, whose boundary can be defined, up to a homeomorphism, as the quotient space of the unit circle by an explicit equivalence relation. This equivalence relation was described by Douady and, in different terms, by Thurston. Thurston used quadratic invariant laminations as a major tool. As has been previously shown by the authors, the combinatorial Mandelbrot set can be interpreted as a quotient of the space of all limit quadratic invariant laminations. The topology in the space of laminations is defined by the Hausdorff distance. In this paper, we describe two similar quotients. In the first case, the identifications are the same but the space is smaller than that taken for the Mandelbrot set. The result (the quotient space) is obtained from the Mandelbrot set by "unpinching" the transitions between adjacent hyperbolic components. In the second case, we do not identify non-renormalizable laminations while identifying renormalizable laminations according to which hyperbolic lamination they can be "unrenormalised" to.

Added: Aug 2, 2017

Working paper

Let M be a complex manifold and L an oriented real line bundle on M equipped with a flat connection. An LCK ("locally conformally Kahler") form is a closed, positive (1,1)-form taking values in L, and an LCK manifold is one which admits an LCK form. Locally, any LCK form is expressed as an L-valued pluri-Laplacian of a function called LCK potential. We consider a manifold M with an LCK form admitting a global LCK potential, and prove that M admits a global, positive LCK potential. Then M admits a holomorphic embedding to a Hopf manifold.

Added: Jun 1, 2017

Working paper

We give an explicit construction of prime Fano threefolds of genus 12 with a G_m-action, describe their isomorphism classes and automorphism groups.

Added: Nov 29, 2017

Working paper

We apply the technique of the paper "The abelian/nonabelian correspondence and Frobenius manifolds" by I. Ciocan-Fontanine, B. Kim, C. Sabbah to construct Saito primitive forms for Gepner singularities.

Added: Nov 16, 2016

Working paper

Added: Sep 26, 2018

Working paper

Kamenova L.,

math. arxive. Cornell University, 2016
Let p: M -> B be a Lagrangian fibration on a hyperkahler manifold of maximal holonomy (also known as IHS), and H the generator of the Picard group of B. We prove that the pullback p∗(H) is a primitive class on M.

Added: Apr 10, 2017

Working paper

In this paper we define and investigate a class of groups characterized by a representation-theoretic property we call purely noncommuting or PNC. This property guarantees that the group has an action on a smooth projective variety with mild quotient singularities. It has intrinsic group-theoretic interest as well. The main results are as follows. (i) All supersolvable groups are PNC. (ii) No nonabelian finite simple groups are PNC. (iii) A metabelian group is guaranteed to be PNC if its commutator subgroup's cyclic prime-power-order factors are all distinct, but not in general. We also give a criterion guaranteeing a group is PNC if its nonabelian subgroups are all large, in a suitable sense, and investigate the PNC property for permutations.

Added: Dec 5, 2018

Working paper

Quantum games represent the really 21st century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In this paper we aim at initiating the truly dynamic theory with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox) the necessary new ingredient for quantum dynamic games must be the theory of non-direct observations and the corresponding quantum filtering. Apart from the technical problems in organising feedback quantum control in real time, the difficulty in applying this theory for obtaining mathematically amenable control systems is due partially to the fact that it leads usually to rather nontrivial jump-type Markov processes and/or degenerate diffusions on manifolds, for which the corresponding control is very difficult to handle. The starting point for the present research is the remarkable discovery (quite unexpected, at least to the author) that there exists a very natural class of homodyne detections such that the diffusion processes on projective spaces resulting by filtering under such arrangements coincide exactly with the standard Brownian motions (BM) on these spaces. In some cases one can even reduce the process to the plain BM on Euclidean spaces or tori. The theory of such motions is well studied making it possible to develop a tractable theory of related control and games, which can be at the same time practically implemented on quantum optical devices.

Added: Oct 30, 2020

Working paper

We classify quasi-simple finite groups of essential dimension 3.

Added: Aug 28, 2017