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Of all publications in the section: 482
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Working paper
Feigin E., Makedonskyi I. arxiv.org. math. Cornell University, 2017. No. 1709.05674.
The goal of this paper is twofold. First, we write down the semi-infinite Pl\"ucker relations, describing the Drinfeld-Pl\"ucker embedding of the (formal version of) semi-infinite flag varieties in type A. Second, we study the homogeneous coordinate ring, i.e. the quotient by the ideal generated by the semi-infinite Pl\"ucker relations. We establish the isomorphism with the algebra of dual global Weyl modules and derive a new character formula. Co
Working paper
Alexander Kuznetsov. arxiv.org. math. Cornell University, 2014
In this review we discuss what is known about semiorthogonal decompositions of derived categories of algebraic varieties. We review existing constructions, especially the homological projective duality approach, and discuss some related issues such as categorical resolutions of singularities.
Working paper
Feigin E., Makhlin I. arxiv.org. math. Cornell University, 2020. No. 2008.13243.
We construct a family of flat semitoric degenerations for the Hibi variety of every finite distributive lattice. The irreducible components of each degeneration are the toric varieties associated with polytopes forming a regular subdivision of the order polytope of the underlying poset. These components are themselves Hibi varieties. For each degeneration in our family we also define the corresponding weight polytope and embed the degeneration into the associated toric variety as the union of orbit closures given by a set of faces. Every such weight polytope projects onto the order polytope with the chosen faces projecting into the parts of the corresponding regular subdivision. We apply these constructions to obtain a family of flat semitoric degenerations for every type A Grassmannian and complete flag variety.
Working paper
Panov V., Sirotkin I. arxiv.org. math. Cornell University, 2015. No. 1503.02214.
In this paper, we analyze a L{\'e}vy model based on two popular concepts - subordination and L{\'e}vy copulas. More precisely, we consider a two-dimensional L{\'e}vy process such that each component is a time-changed (subordinated) Brownian motion and the dependence between processes, which are used for stochastic time change, is described via some L{\'e}vy copula. We prove a series representation for our model, which can be efficiently used for simulation proposes, and provide some practical examples based on real data.
Working paper
V. Lebedev. arxiv.org. math. Cornell University, 2018. No. arXiv:1805.02254v2.
We show that if ϕ: R → R is a continuous mapping and the set of nonlinearity of ϕ has nonzero Lebesgue measure, then ϕ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.
Working paper
Netay I. V., Савватеев А. В. arxiv.org. math. Cornell University, 2016
The paper is devoted to the description of family of scalene triangles for which the triangle formed by the intersection points of bisectors with opposite sides is isosceles. We call them Sharygin triangles. It turns out that they are parametrized by an open subset of an elliptic curve. Also we prove that there are infinitely many non-similar integer Sharygin triangles.
Working paper
Veretennikov A., Zverkina G. arxiv.org. math. Cornell University, 2013. No. arXiv:1306.2359v2.
A simple proof of Dynkin's formula is derived for a general Erlang type single server queueing model with discontinuous characteristics; polynomial convergence rate to stationarity is established.
Working paper
Gavrilovich M. arxiv.org. math. Cornell University, 2019
We consider simplicial sets equipped with a notion of smallness, and observe that this slight "topological" extension of the "algebraic" simplicial language allows a concise reformulation of a number of classical notions in topology, e.g. continuity, limit of a map or a sequence along a filter, various notions of equicontinuity and uniform convergence of a sequence of functions; completeness and compactness; in algebraic topology, locally trivial bundles as a direct product after base-change and geometric realisation as a space of discontinuous paths. In model theory, we observe that indiscernible sequences in a model form a simplicial set with a notion of smallness which can be seen as an analogue of the Stone space of types. These reformulations are presented as a series of exercises, to emphasise their elementary nature and that they indeed can be used as exercises to make a student familiar with computations in basic simplicial and topological language. (Formally, we consider the category of simplicial objects in the category of filters in the sense of Bourbaki.) This work is unfinished and is likely to remain such for a while, hence we release it as is, in the small hope that our reformulations may provide interesting examples of computations in basic simplicial and topological language on material familiar to a student in a first course of topology or category theory.
Working paper
Bolsinov A., Izosimov A. arxiv.org. math. Cornell University, 2013
We study the relationship between singularities of bi-Hamiltonian systems and algebraic properties of compatible Poisson brackets. As the main tool, we introduce the notion of linearization of a Poisson pencil. From the algebraic viewpoint, a linearized Poisson pencil can be understood as a Lie algebra with a fixed 2-cocycle. In terms of such linearizations, we give a criterion for non-degeneracy of singular points of bi-Hamiltonian systems and describe their types.
Working paper
Smirnov E. arxiv.org. math. Cornell University, 2016. No. 1609.07771.
We give an alternative proof of a recent result by Pasquier stating that for a generalized flag variety X=G/P and an effective Q-divisor D stable with respect to a Borel subgroup the pair (X,D) is Kawamata log terminal if and only if [D]=0.
Working paper
Staal A. P., Karu K. arxiv.org. math. Cornell University, 2012
We study logarithmic jet schemes of a log scheme and generalize a theorem of M. Mustata from the case of ordinary jet schemes to the logarithmic case. If X is a normal local complete intersection log variety, then X has canonical singularities if and only if the log jet schemes of X are irreducible.
Working paper
Smirnov E., Тутубалина А. А. arxiv.org. math. Cornell University, 2020. No. 2006.16995.
Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gröbner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres.
Working paper
Galkin S. arxiv.org. math. Cornell University, 2018. No. 1809.02705.
We classify smooth Fano threefolds that admit degenerations to toric Fano threefolds with ordinary double points.
Working paper
Blokh A., Oversteegen L., Ptacek R. et al. arxiv.org. math. Cornell University, 2014
A crucial fact established by Thurston in his 1985 preprint is that distinct \emph{minors} of quadratic laminations do not cross inside the unit disk; this led to his construction of a combinatorial model of the Mandelbrot set. Thurston's argument is based upon the fact that \emph{majors} of a quadratic lamination never enter the region between them, a result that fails in the cubic case. In this paper, devoted to laminations of any degree, we use an alternative approach in which the fate of sets of intersecting leaves of two distinct laminations is studied. It turns out that, under some natural assumptions, these sets of intersecting leaves behave like gaps of a lamination. Relying upon this, we rule out certain types of mutual location of critical sets of distinct laminations (this can be viewed as a partial generalization of the theorem that quadratic minors do not cross inside the unit disk). The main application is to the cubic case.
Working paper
Bogachev V., Kolesnikov A. arxiv.org. math. Cornell University, 2011. No. 1110.1822..
Given the standard Gaussian measure $\gamma$ on the countable product of lines $\mathbb{R}^{\infty}$ and a probability measure $g \cdot \gamma$ absolutely continuous with respect to $\gamma$, we consider the optimal transportation $T(x) = x + \nabla \varphi(x)$ of $g \cdot \gamma$ to $\gamma$. Assume that the function $|\nabla g|^2/g$ is $\gamma$-integrable. We prove that the function $\varphi$ is regular in a certain Sobolev-type sense and satisfies the classical change of variables formula $g = {\det}_2(I + D^2 \varphi) \exp \bigl(\mathcal{L} \varphi - 1/2 |\nabla \varphi|^2 \bigr)$. We also establish sufficient conditions for the existence of third order derivatives of $\varphi$.
Working paper
Gontsov R. R., Vyugin I. V. arxiv.org. math. Cornell University, 2013
The paper conserns the solvability by quadratures of linear diferential systems, which is one of the qestions of the differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular ones and propose some criteria of summubility for systems whose (formal) exponents are sufficiently small.
Working paper
Golubeva V. A., Lavrov A. arxiv.org. math. Cornell University, 2017
The present paper provides a method for finding partial differential equations satisfied by the Feynman integrals for diagrams of various types, using the Griffiths theorem on the reduction of poles of rational differential forms. As an application, an algorithm for computing partial differential equations satisfied by Feynman integrals for diagrams of a ladder type is described.
Working paper
Amerik E. arxiv.org. math. Cornell University, 2014
This is not a research paper, but a survey submitted to a proceedings volume.
Closed classes of three-valued logic generated by periodic symmetric funtions that equal $1$ in tuples from $\{1,2\}^n$ and equal $0$ on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained.