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Of all publications in the section: 319
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Working paper
Barannikov S. math. arxive. Cornell University, 2017
The equivariantly closed matrix integrals introduced in [B06], see also [B10], are studied in the case of the graded associative algebras with odd or even scalar product.
Working paper
Anikin A., Dvurechensky P., Gasnikov A. et al. math. arxive. Cornell University, 2015
Working paper
Anikin A., Dvurechensky P., Gasnikov A. et al. math. arxive. Cornell University, 2015
Working paper
Ianovski E. math. arxive. Cornell University, 2019
We consider the problem of electing a committee of k candidates, subject to some constraints as to what this committee is supposed to look like. In our framework, the candidates are given labels as an abstraction of gender, religion, ethnicity, and other attributes, and the election outcome is constrained by interval constraints – of the form “Between 3 and 5 candidates with label X” – and dominance constraints – “At least as many candidates with label X as with label Y”. While in general this problem would require us to rethink how we determine which election outcomes are good, in the case of a committee scoring rule this becomes a constrained optimisation problem – simply find a valid committee with the highest score. In the case of weakly separable rules we show the existence of a polynomial time solution in the case of tree-like constraints, and a fixed-parameter tractable algorithm for the general case, which is otherwise NP-hard
Working paper
Bogomolov F. A., Fu H. math. arxive. Cornell University, 2017
We explicitly construct pairs of elliptic curves defined over the algebraic numbers with large intersection of projective torsion points.
Working paper
Kuznetsov A. math. arxive. Cornell University, 2018
We show that the derived category of a general Enriques surface can be realized as a semiorthogonal component in the derived category of a smooth Fano variety with a diagonal Hodge diamond.
Working paper
Ornea L., Verbitsky M. math. arxive. Cornell University, 2017
An locally conformally Kahler (LCK) manifold with potential is a complex manifold with a cover which admits an automorphic Kahler potential. An LCK manifold with potential can be embedded to a Hopf manifold, if its dimension is at least 3. We give a functional-analytic proof of this result based on Riesz-Schauder theorem and Montel theorem. We give an alternative argument for complex surfaces, deducing embedding theorem from the Spherical Shell Conjecture.
Working paper
Amerik E., Kuznetsova A. math. arxive. Cornell University, 2016
Let B be a simply-connected projective variety such that the first cohomology groups of all line bundles on B are zero. Let E be a vector bundle over B and X=ℙ(E). It is easily seen that a power of any endomorphism of X takes fibers to fibers. We prove that if X admits an endomorphism which is of degree greater than one on the fibers then E splits into a direct sum of line bundles.
Working paper
Kondyrev G., Prikhodko A. math. arxive. Cornell University, 2019
Working paper
Verbitsky M. math. arxive. Cornell University, 2017
Let M be a hyperkahler manifold, Γ its mapping class group, and Teich the Teichmuller space of complex structures of hyperkahler type. After we glue together birationally equivalent points, we obtain the so-called birational Teichmuller space Teichb. Every connected component of Teichb is identified with its period space P by global Torelli theorem. The mapping class group of M acts on P as a finite index subgroup of the group of isometries of the integer cohomology lattice, that is, satisfies assumptions of Ratner theorem. We prove that there are three classes of orbits, closed, dense and the intermediate class which corresponds to varieties with Re(H2,0(M)) containing a given rational vector. The closure of the later orbits is a fixed point set of an anticomplex involution of P. This fixes an error in the paper 1306.1498, where this third class of orbits was overlooked. We explain how this affects the works based on 1306.1498.
Working paper
Gorsky E., Wedrich P. math. arxive. Cornell University, 2019
We describe the universal target of annular Khovanov-Rozansky link homology functors as the homotopy category of a free symmetric monoidal category generated by one object and one endomorphism. This categorifies the ring of symmetric functions and admits categorical analogues of plethystic transformations, which we use to characterize the annular invariants of Coxeter braids. Further, we prove the existence of symmetric group actions on the Khovanov-Rozansky invariants of cabled tangles and we introduce spectral sequences that aid in computing the homologies of generalized Hopf links. Finally, we conjecture a characterization of the horizontal traces of Rouquier complexes of Coxeter braids in other types.
Working paper
Kochetkov Y. math. arxive. Cornell University, 2016. No. 1611.01010.
Let T(n,m) be the set of plane labelled bipartite trees with n white vertices and m -- black. If the number n+m of vertices is even, then the set T(n,m) is a union of two disjoined subsets -- subset of "even" trees and subset of "odd" trees. This partition has a clear geometric meaning.
Working paper
Busjatskaja I., Kochetkov Y. math. arxive. Cornell University, 2018. No. 1811.10357.
In this paper we at first consider plane trees with the root vertex and a marked directed edge, outgoing from the root vertex. For such trees we introduce a new characteristic-- the parity, using the bracket code. It turns out that the parity depends only on the root vertex (not on the marked edge). And in the case of an even number of vertices the parity does not depend on the root vertex also.   Then we consider rotation groups of bipartite trees, study their properties and prove that in the case of even number of vertices rotation groups of even and odd trees are different.
Working paper
Kolesnikov A., Zaev D. math. arxive. Cornell University, 2015
We study the Monge and Kantorovich transportation problems on R∞ within the class of exchangeable measures. With the help of the de Finetti decomposition theorem the problem is reduced to an unconstrained optimal transportation problem on the Hilbert space. We find sufficient conditions for convergence of finite-dimensional approximations to the Monge solution. The result holds, in particular, under certain analytical assumptions involving log-concavity of the target measure. As a by-product we obtain the following result: any uniformly log-concave exchangeable sequence of random variables is i.i.d.
Working paper
Buryak A., Rossi P. math. arxive. Cornell University, 2018
In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.
Working paper
Сафроненко Е. В. math. arxive. Cornell University, 2019
Working paper
Kolesnikov A., Klartag B. math. arxive. Cornell University, 2017
Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{\"a}hler-Einstein equation $e^{-\Phi} = \det D^2 \Phi$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 \Phi$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2 \Phi$ for arbitrary $K$ is uniformly bounded by $\frac{n-1}{4(n+1)}$ and verify this conjecture in the two-dimensional case. The general case remains open.
Working paper
Konakov V., Panov V., Piterbarg V. math. arxive. Cornell University, 2020. No. 2005.11249.
In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. Unlike previously known facts in this field, our main theorem yields the asymptotic representation of the cor- responding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maxi- mal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.