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Of all publications in the section: 482
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Working paper
Ivan Cheltsov, Martinez-Garcia J. arxiv.org. math. Cornell University, 2014
For every smooth del Pezzo surface $S$, smooth curve $C\in|-K_{S}|$ and $\beta\in(0,1]$, we compute the $\alpha$-invariant of Tian $\alpha(S,(1-\beta)C)$ and prove the existence of K\"ahler--Einstein metrics on $S$ with edge singularities along $C$ of angle $2\pi\beta$ for $\beta$ in certain interval. In particular we give lower bounds for the invariant $R(S,C)$, introduced by Donaldson as the supremum of all $\beta\in(0,1]$ for which such a metric exists.
Working paper
Kolesnikov A., Klartag B. arxiv.org. math. Cornell University, 2013. No. 1402.2636.
We investigate the Brenier map \nabla \Phi between the uniform measures on two convex domains in \mathbb{R}^n or more generally, between two log-concave probability measures on \mathbb{R}^n. We show that the eigenvalues of the Hessian matrix D^2 \Phi exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension n.
Working paper
Vladimir L. Popov. arxiv.org. math. Cornell University, 2021. No. 2106.02072.
For each integer n>0, we construct a series of irreducible algebraic varieties X, for which the automorphism group Aut(X) contains as a subgroup the automorphism group Aut(F_n) of a free group F_n of rank n. For n > 1, such groups Aut(X) are nonamenable, and for n > 2, they are nonlinear and contain the braid group B_n. Some of these varieties X are affine, and among affine, some are rational and some are not, some are smooth and some are singular. The byproduct is that for n > 2, each Cremona group of rank > 3n-1 contains Aut(F_n) and the braid group B_n.
Working paper
Buchstaber V., Limonchenko I. arxiv.org. math. Cornell University, 2018. No. 1808.08851.
We introduce the notions of algebraic and geometric direct families of polytopes and develop a theory of such families. The theory is then applied to the problem of existence of nontrivial higher Massey products in cohomology of moment-angle-complexes.
Working paper
Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.
Working paper
Bigeni A. arxiv.org. math. Cornell University, 2017. No. 1705.03804.
Fang and Fourier defined the symplectic Dellac configurations in order to parametrize the torus fixed points of the symplectic degenerated flag varieties, and conjectured that their numbers are the elements of a sequence of integers (1, 2, 10, 98, 1594, ...) which appears in the study by Randrianarivony and Zeng of the median Euler numbers. In this paper, we prove the conjecture by considering a combinatorial interpretation of the latter integers in terms of the surjective pistols (which form a well-known combinatorial model of the Genocchi numbers), and constructing an appropriate surjection from the symplectic Dellac configurations to the surjective pistols.
Working paper
Verbitsky M. arxiv.org. math. Cornell University, 2013
Let M be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on M up to the action of the group of isotopies. The group Γ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic complex structure is the one with a Γ-orbit dense in $\Teich$. Let M be a complex torus of complex dimension ≥2 or a hyperkahler manifold with b_2>3. We prove that M is ergodic, unless M has maximal Picard rank (there is a countable number of such M). This is used to show that all hyperkahler manifolds are Kobayashi non-hyperbolic.
Working paper
Romanov A. arxiv.org. math. Cornell University, 2013. No. 1309.6283.
For a continuous semicascade on a metrizable compact set Ω, we consider the weak* convergence of generalized operator ergodic means in    End C*(Ω). We discuss conditions on the dynamical system under which: (a) every ergodic net contains a convergent subsequence; (b) all ergodic nets converge; (c) all ergodic sequences converge. We study the relationships between the convergence of ergodic means and the properties of transitivity of the proximality relation on Ω, minimality of supports of ergodic measures, and uniqueness of minimal sets in the closure of trajectories of a semicascade. These problems are solved in terms of three algebraic-topological objects associated with the dynamical system: the Ellis enveloping semigroup, the Kohler operator semigroup Г, and the semigroup G that is the weak* closure of the convex hull of Г in End C*(Ω). The main results are stated for ordinary semicascades (whose Ellis semigroup is metrizable) and tame semicascades. For a dynamics, being ordinary is equivalent to being “nonchaotic” in an appropriate sense. We present a classification of compact dynamical systems in terms of topological properties of the above-mentioned semigroups.
Working paper
A.V. Romanov. arxiv.org. math. Cornell University, 2018. No. 1806.09132.
We study the problem on the weak-star decomposability of a topological N0-dynamical system (Ω, '), where ' is an endomorphism of a metric compact set Ω, into ergodic components in terms of the associated enveloping semigroups. In the tame case (where the Ellis semigroup E(Ω,') consists of B1-transformations Ω → Ω), we show that (i) the desired decomposition exists for an appropriate choice of the generalized sequential averaging method; (ii) every sequence of weighted ergodic means for the shift operator x → x ◦ ', x ∈ C(Ω), contains a pointwise convergent subsequence. We also discuss the relationship between the statistical properties of (Ω, ') and the mutual structure of minimal sets and ergodic measures.
Working paper
Fedor Bogomolov, Böhning C. arxiv.org. math. Cornell University, 2014
We compare the notions of essential dimension and stable cohomological dimension of a finite group G, prove that the latter is bounded by the length of any normal series with cyclic quotients for G, and show that, however, this bound is not sharp by showing that the stable cohomological dimension of the finite Heisenberg groups H_p, p any prime, is equal to two.
Working paper
Amerik E., Kurlberg P., Nguyen K. et al. arxiv.org. math. Cornell University, 2013. No. 1305.4398.
Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.
Working paper
Dunin-Barkowski P., Popolitov A., Popolitova S. arxiv.org. math. Cornell University, 2018. No. 1812.00858.
We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family of knots which "grows" from the figure-eight knot and contains both two-strand torus knots and twist knots. We prove that parameter space splits into four chambers, each with its own evolution, and two isolated points. Remarkably, the evolution in the Khovanov case features an extra eigenvalue, which drops out in the Jones (t -> -1) limit.
Working paper
Romanov I., Shamaev A. arxiv.org. math. Cornell University, 2018. No. arXiv:1603.01212v3.
The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in a finite time.
Working paper
Romanov I., Alexey Shamaev. arxiv.org. math. Cornell University, 2016. No. arXiv:1503.04461.
We will consider the exact controllability of the distributed system governed by wave equation with memory. It will be proved that this mechanical system can be driven to an equilibrium point in a finite time, the absolute value of the distributed control function being bounded. In this case the memory kernel is a linear combination of exponentials.
Working paper
Yuri Prokhorov, Zaidenberg M. arxiv.org. math. Cornell University, 2014
We construct 4 different families of smooth Fano fourfolds with Picard rank 1, which contain cylinders, i.e., Zariski open subsets of the form Z x A1, where Z is a quasiprojective variety. The affi ne cones over such a fourfold admit eff ective Ga-actions. Similar constructions of cylindrical Fano threefolds were done previously in our papers jointly with Takashi Kishimoto.
Working paper
Galkin S., Shinder E. arxiv.org. math. Cornell University, 2012. No. 1210.3339.
We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface S. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on S. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.
Working paper
Kuznetsov A., Polishchuk A. arxiv.org. math. Cornell University, 2011. No. 1110.5607 .
We introduce a new construction of exceptional objects in the derived category of coherent sheaves on a compact homogeneous space of a semisimple algebraic group and show that it produces exceptional collections of the length equal to the rank of the Grothendieck group on homogeneous spaces of all classical groups.
Working paper
Lee K., Shabalin T. arxiv.org. math. Cornell University, 2014
We construct exceptional collections of maximal length on four families of surfaces of general type with $p_g=q=0$ which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as their orthogonal complements.