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Of all publications in the section: 323
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Working paper
Arsie A., Buryak A., Lorenzoni P. et al. math. arxive. Cornell University, 2020
In this paper, we generalize the Givental theory for Frobenius manifolds and cohomological field theories to flat F-manifolds and F-cohomological field theories. In particular, we define a notion of Givental cone for flat F-manifolds, and we provide a generalization of the Givental group as a matrix loop group acting on them. We show that this action is transitive on semisimple flat F-manifolds. We then extend this action to F-cohomological field theories in all genera. We show that, given a semisimple flat F-manifold and a Givental group element connecting it to the constant flat F-manifold at its origin, one can construct a family of F-CohFTs in all genera, parameterized by a vector in the associative algebra at the origin, whose genus~\$0\$ part is the given flat F-manifold. If the flat F-manifold is homogeneous, then the associated family of F-CohFTs contains a subfamily of homogeneous F-CohFTs. However, unlike in the case of Frobenius manifolds and CohFTs, these homogeneous F-CohFTs can have different conformal dimensions, which are determined by the properties of a certain metric associated to the flat F-manifold.
Working paper
Gorsky E., Hogancamp M., Mellit A. et al. math. arxive. Cornell University, 2019
We prove that the full twist is a Serre functor in the homotopy category of type A Soergel bimodules. As a consequence, we relate the top and bottom Hochschild degrees in Khovanov-Rozansky homology, categorifying a theorem of Kálmán.
Working paper
Kolesnikov A., Milman E. math. arxive. Cornell University, 2016
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.
Working paper
Finkelberg M. V., Tsymbaliuk A. math. arxive. Cornell University, 2018
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).
Working paper
Prokhorov Y. math. arxive. Cornell University, 2015. No. 1508.04371.
We study singular Fano threefolds of type V22.
Working paper
Blokh A., Oversteegen L., Timorin V. 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.
Working paper
Elagin Alexey, Lunts Valery, Schnürer O. math. arxive. Cornell University, 2018
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.
Working paper
Lvovsky S. math. arxive. Cornell University, 2016
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.
Working paper
Tyurin N. A. math. arxive. Cornell University, 2016
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..
Working paper
Amerik E., Campana F. math. arxive. Cornell University, 2018
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.
Working paper
Shchur L. math. arxive. Cornell University, 2021. No. 2107.02406.
In the late 80s and 90s, theoretical physicists of the Landau Institute for Theoretical Physics designed and developed several specialized computers for challenging computational problems in the physics of phase transitions. These computers did not have a central processing unit. They optimize algorithms to handle elementary operations on integers - read, write, compare, and count. The approach allowed them to achieve recording run times. Computers performed calculations three orders of magnitude faster than similar calculations on the world's best supercomputers. The approach made it possible to obtain fundamentally new results, some of which have not yet been surpassed in the accuracy of calculations. The report will present the main ideas for the development of specialized computers and the scientific results obtained with their help. The lessons of planning and execution of long-term complex scientific projects will also be discussed.
Working paper
Vedenin A., Galkin V., Karatetskaia E. et al. math. arxive. Cornell University, 2020
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.
Working paper
Kozhina A. math. arxive. Cornell University, 2016. No. 1602.04770.
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.
Working paper
Konakov V., Kozhina A., Menozzi S. math. arxive. Cornell University, 2016. No. 1506.08758v2.
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.
Working paper
Stankevich N. math. arxive. Cornell University, 2021
The dynamics of a multiplex heterogeneous network of oscillators is studied. Two types of similar models based on the Hodgkin-Huxley formalism are used as the basic elements of the networks. The first type model demonstrates bursting oscillations. The second model demonstrates bistability between bursting attractor and stable steady state. Basin of attraction of the stable equilibrium in the model is very small. Bistabilty is a result taking into account an additional ion channel, which has a non-monotonic characteristic and can be interpreted as a channel with a communication defect. Suggested multiplex networks assumed more active communication between models with a defect as a result in such networks it is enough to have one element with a communication defect in the subnetworks in order to stabilize the state of equilibrium in the entire network.
Working paper
Golota A. math. arxive. Cornell University, 2015
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.
Working paper
Verbitsky M., Mj M., Biswas I. math. arxive. Cornell University, 2018
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.
Working paper
Bogomolov F. A., Lukzen E. math. arxive. Cornell University, 2020
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.