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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

Added: Nov 12, 2018

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

Working paper

Given a Lie algebra with a scalar product, one may consider the latter as a symplectic structure on a dg-scheme, which is the spectrum of the Chevalley--Eilenberg algebra. In the first section we explicitly calculate the first order deformation of the differential on the Hochschild complex of the Chevalley--Eilenberg algebra. The answer contains the Duflo character. This calculation is used in the last section. There we sketch the definition of the Wilson loop invariant of knots, which is hopefully equal to the Kontsevich integral, and show that for unknot they coincide. As a byproduct we get a new proof of the Duflo isomorphism for a Lie algebra with a scalar product.

Added: Sep 23, 2015

Working paper

We apply Weyl n-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

Added: Oct 16, 2020

Working paper

For every commutative ring A, one has a functorial commutative ring W(A) of p-typical Witt vectors of A, an iterated extension of A by itself. If A is not commutative, it has been known since the pioneering work of L. Hesselholt that W(A) is only an abelian group, not a ring, and it is an iterated extension of the Hochschild homology group HH0(A) by itself. It is natural to expect that this construction generalizes to higher degrees and arbitrary coefficients, so that one can define "Hochschild-Witt homology" WHH∗(A,M) for any bimodule M over an associative algebra A over a field k. Moreover, if one want the resulting theory to be a trace theory in the sense of arXiv:1308.3743, then it suffices to define it for A=k. This is what we do in this paper, for a perfect field k of positive characteristic p. Namely, we construct a sequence of polynomial functors Wm, m≥1 from k-vector spaces to abelian groups, related by restriction maps, we prove their basic properties such as the existence of Frobenius and Verschiebung maps, and we show that Wm are trace functors in the sense of arXiv:1308.3743. The construction is very simple, and it only depends on elementary properties of finite cyclic groups.

Added: Oct 29, 2016