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Working paper

A sharp Poincar\'e-type inequality is derived for the restriction of the Gaussian measure on the boundary of a convex set. In particular, it implies a Gaussian mean-curvature inequality and a Gaussian iso second-variation inequality. The new inequality is nothing but an infinitesimal form of Ehrhard's inequality for the Gaussian measure.

Added: Feb 23, 2016

Working paper

We define an integral form of shifted quantum affine algebras of type A and construct Poincaré-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized K-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type A. Finally, we obtain the rational (homological) analogues of the above results (proved earlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques).

Added: Dec 3, 2018

Working paper

We study singular Fano threefolds of type V22.

Added: Oct 9, 2015

Working paper

Blokh A., Oversteegen L.,

math. arxive. Cornell University, 2016
In this paper, we study slices of the parameter space of cubic polynomials, up to affine conjugacy, given by a fixed value of the multiplier at a non-repelling fixed point. In particular, we study the location of the maincubioid in this parameter space. The maincubioid is the set of affine conjugacy classes of complex cubic polynomials that have certain dynamical properties generalizing those of polynomials z2+c for c in the filled main cardioid.

Added: Sep 15, 2016

Working paper

We prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field, hereby answering a question of Iyama. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.

Added: Dec 1, 2018

Working paper

Let us say that a curve C⊂ℙ3 is osculating self-dual if it is projectively equivalent to the curve in the dual space (ℙ3)∗ whose points are osculating planes to~C. Similarly, we say that a k-dimensional subvariety X⊂ℙ2k+1 is osculating self-dual if its second osculating space at the general point is a hyperplane and X is projectively equivalent to the variety in (ℙ2k+1)∗ whose points are second osculating spaces to X. In this note we show that for each k≥1 there exist many osculating self-dual k-dimensional subvarieties in ℙ2k+1.

Added: Feb 28, 2016

Working paper

Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case..

Added: May 19, 2016

Working paper

We show that an everywhere regular foliation F with compact canonically polarized leaves on a quasi-projective manifold X has isotrivial family of leaves when the orbifold base of this family is special. By a recent work of Berndtsson, Paun and Wang, the same proof works in the case where the leaves have trivial canonical bundle. The specialness condition means that the p-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal Kodaira dimension p, for any p>0. This condition is satisfied, for example, in the very particular case when the Kodaira dimension of the determinant of the Logarithmic extension of the conormal bundle vanishes. Motivating examples are given by the `algebraically coisotropic' submanifolds of irreducible hyperkähler projective manifolds.

Added: Dec 4, 2018

Working paper

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.

Added: Oct 12, 2019

Working paper

We study the sensitivity of the densities of some Kolmogorov like degenerate diffusion processes with respect to a perturbation of the coefficients of the non-degenerate component. Under suitable (quite sharp) assumptions we quantify how the pertubation of the SDE affects the density. Natural applications of these results appear in various fields from mathematical finance to kinetic models.

Added: May 11, 2016

Working paper

We are interested in studying the sensitivity of diffusion processes or their approximations by Markov Chains with respect to a perturbation of the coefficients. As an important application, we give a first order expansion for the difference of the densities of a diffusion with H¨older coefficients and its approximation by the Euler scheme.

Added: Jul 1, 2015

Working paper

A locally conformally K¨ahler (LCK) manifold is a complex manifold whose universal cover is K¨ahler with monodromy group acting on the universal cover by holomorphic homotheties. A Vaisman manifold M is a compact non-K¨ahler LCK manifold admitting an action of a holomorphic conformal flow lifting to an action on a K¨ahler cover by nontrivial homotheties. When the orbits of the action on M are compact, it is known that every stable holomorphic vector bundle over M, dim(M) ≥ 3, is equivariant and filtrable. In the present paper we generalize this result to irregular Vaisman manifolds.

Added: Dec 7, 2015

Working paper

Let M be a compact complex manifold of dimension at least three and Π:M→X a positive principal elliptic fibration, where X is a compact Kähler orbifold. Fix a preferred Hermitian metric on M. In \cite{V}, the third author proved that every stable vector bundle on M is of the form L⊗Π^*B_0, where B0 is a stable vector bundle on X, and L is a holomorphic line bundle on M. Here we prove that every stable Higgs bundle on M is of the form (L⊗Π^*B_0,Π^*Φ_X), where (B_0,Φ_X) is a stable Higgs bundle on X and L is a holomorphic line bundle on M.

Added: Dec 5, 2018

Working paper

We offer a new approach to proving the Chen-Donaldson-Sun theorem which we demonstrate with a series of examples. We discuss the existence of a construction of a special metric on stable vector bundles over the surfaces formed by a families of curves and its relation to the one-dimensional cycles in the moduli space of stable bundles on curves.

Added: Oct 27, 2020

Working paper

Added: Oct 10, 2017

Working paper

Chaudru de Raynal P., Honoré I.,

math. arxive. Cornell University, 2018
We establish strong uniqueness for a class of degenerate SDEs of weak Hörmander type under suitable Hölder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here Hölder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold.

Added: Oct 31, 2020

Working paper

We consider a problem of manifold estimation from noisy observations. We suggest a novel adaptive procedure, which simultaneously reconstructs a smooth manifold from the observations and estimates projectors onto the tangent spaces. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We adjust the weights so they capture the manifold structure better. We propose a computationally efficient procedure, which iteratively refines the weights on each step, such that, after several iterations, we obtain the "oracle" weights, so the quality of the final estimates does not suffer even in the presence of relatively large noise. We also provide a theoretical study of the procedure and prove its optimality deriving both new upper and lower bounds for manifold estimation under the Hausdorff loss.

Added: Jun 12, 2019

Working paper

Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kahler supersymmetry algebra is associated to a Kahler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

Added: Nov 19, 2019

Working paper

We give a formula for the Heegaard Floer d-invariants of integral surgeries on two-component L--space links of linking number zero in terms of the h-function, generalizing a formula of Ni and Wu. As a consequence, we characterize L-space surgery slopes for such links in terms of the τ-invariant when the components are unknotted. For general links of linking number zero, we explicitly describe the relationship between the h-function, the Sato-Levine invariant and the Casson invariant. We give a proof of a folk result that the d-invariant of any nonzero rational surgery on a link of any number of components is a concordance invariant of links in the three-sphere with pairwise linking numbers zero. We also describe bounds on the smooth four-genus of links in terms of the h-function, expanding on previous work of the second author, and use these bounds to calculate the four-genus in several examples of links.

Added: Sep 3, 2019

Working paper

We study the asymmetric one-dimensional telegraph process in the bounded domain. Lower boundary is absorbing and upper boundary is reflecting with delay. Point stays in the upper boundary until switch of regime occurs. We obtain the distribution of this process in terms of Laplace trasforms

Added: Oct 22, 2015

Working paper

This paper is concerned with Random walk approximations of the Brownian motion on the Affine group Aff(R). We are in particular interested in the case where the innovations are discrete. In this framework, the return probability of the walk have fractional exponential decay in large time, as opposed to the polynomial one of the continuous object. We prove that integrating those return probabilities on a suitable neighborhood of the origin, the expected polynomial decay is restored. This is what we call a Quasi-local theorem.

Added: Sep 20, 2017