Timorin, Vladlen A.
Artamkin, Igor V.
Esterov Alexander I.
Feigin Evgeny B.
119048Moscow, Usacheva str., 6
phone/fax: +7 (495) 624-26-16
Global bifurcations in the generic one-parameter families that unfold a vector field with a separatrix loop on the two-sphere are described. The sequence of bifurcation that occurs is in a sense in ono-to-one correspondence with finite sets on a circle having some additional structure on them. Families under study appear to be structurally stable. The main tool is the Leontovich-Mayer-Fedorov (LMF) graph, analog of the separatrix sceleton - an invariant of the orbital topological classification of the vector fields on the two-sphere. Its properties and applications are described.
A new characterization of Nikolskii–Besov classes via integration by parts is obtained
Let Γ be an arithmetic group of affine automorphisms of the n-dimensional future tube T. It is proved that the quotient space T/Γ is smooth at infinity if and only if the group Γ is generated by reflections and the fundamental polyhedral cone (“Weyl chamber”) of the group dΓ in the future cone is a simplicial cone (which is possible only for n ≤ 10). As a consequence of this result, a smoothness criterion for the Satake–Baily–Borel compactification of an arithmetic quotient of a symmetric domain of type IV is obtained.
A projective manifold is algebraically hyperbolic if the degree of any curve is bounded from above by its genus times a constant, which is independent from the curve. This is a property which follows from Kobayashi hyperbolicity. We prove that hyperkähler manifolds are not algebraically hyperbolic when the Picard rank is at least 3, or if the Picard rank is 2 and the SYZ conjecture on existence of Lagrangian fibrations is true. We also prove that if the automorphism group of a hyperkähler manifold is infinite then it is algebraically nonhyperbolic.
This is the second part of a 2-year course of abstract algebra for students beginning a professional study of higher mathematics.1 This textbook is based on courses given at the Independent University of Moscow and at the Faculty of Mathematics at the National Research University Higher School of Economics. In particular, it contains a large number of exercises that were discussed in class, some of which are provided with commentary and hints, as well as problems for independent solution that were assigned as homework.Working out the exercises is of crucial importance in understanding the subject matter of this book.
We discuss the correspondence between models solved by the Bethe ansatz and classical integrable systems of the Calogero type. We illustrate the correspondence by the simplest example of the inhomogeneous asymmetric six-vertex model parameterized by trigonometric(hyperbolic) functions.
We study the explicit formula (suggested by Gamayun, Iorgov and Lisovyy) for the Painlevé III(D 8) τ function in terms of Virasoro conformal blocks with a central charge of 1. The Painlevé equation has two types of bilinear forms, which we call Toda-like and Okamoto-like. We obtain these equations from the representation theory using an embedding of the direct sum of two Virasoro algebras in a certain superalgebra. These two types of bilinear forms correspond to the Neveu–Schwarz sector and the Ramond sector of this algebra. We also obtain the τ functions of the algebraic solutions of the Painlevé III(D 8) from the special representations of the Virasoro algebra of the highest weight (n + 1/4)2.
We consider d-fold branched coverings of the projective plane RP^2 andshow that the hypergeometric tau function of the BKP hierarchy of Kac and van deLeur is the generating function for weighted sums of the related Hurwitz numbers.In particular, we get the RP^2 analogues of the CP^1 generating functions proposedby Okounkov and by Goulden and Jackson. Other examples are Hurwitz numbersweighted by the Hall–Littlewood and by the Macdonald polynomials. We also considerintegrals of tau functions which generate Hurwitz numbers related to base surfaceswith arbitrary Euler characteristics e, in particular projective Hurwitz numbers e = 1
We consider $d$-fold branch covering of the real projective plane $RP^2$ and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In paticular we get the $RP^2$ analogue of the both $CP^1$ generating functions proposed by A.Okounkov and by Goulden-Jackson. Other examples are Hurwitz numbers weighted by Hall-Littlewood and by Macdonald polynomials. We also consider integrals of tau functions which generate projective Hurwitz numbers.
We prove that the characteristic foliation F on a nonsingular divisor D in an irreducible projective hyperk¨ahler manifold X cannot be algebraic, unless the leaves of F are rational curves or X is a surface. More generally, we show that if X is an arbitrary projective manifold carrying a holomorphic symplectic 2-form, and D and F are as above, then F can be algebraic with non-rational leaves only when, up to a finite ´etale cover, X is the product of a symplectic projective manifold Y with a symplectic surface and D is the pull-back of a curve on this surface.
When D is of general type, the fact that F cannot be algebraic unless X is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical class of the (orbifold) base of the family of leaves is zero. This implies, in particular, the isotriviality of the family of leaves of F. We show this, more generally, for regular algebraic foliations by curves defined by the vanishing of a holomorphic (d − 1)-form on a complex projective manifold of dimension d.
W. Thurston constructed a combinatorial model of the Mandelbrot set M2M2such that there is a continuous and monotone projection of M2M2to this model. We propose the following related model for the space MD3MD3of critically marked cubic polynomials with connected Julia set and all cycles repelling. If (P,c1,c2)∈MD3(P,c1,c2)∈MD3, then every point z in the Julia set of the polynomial P defines a unique maximal finite set AzAzof angles on the circle corresponding to the rays, whose impressions form a continuum containing z . Let G(z)G(z)denote the convex hull of AzAz. The convex sets G(z)G(z)partition the closed unit disk. For (P,c1,c2)∈MD3(P,c1,c2)∈MD3let <img height="16" border="0" style="vertical-align:bottom" width="14" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si6.gif">c1⁎be the co-critical point of c1c1. We tag the marked dendritic polynomial (P,c1,c2)(P,c1,c2)with the set <img height="18" border="0" style="vertical-align:bottom" width="159" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si14.gif">G(c1⁎)×G(P(c2))⊂D‾×D‾. Tags are pairwise disjoint; denote by <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combtheir collection, equipped with the quotient topology. We show that tagging defines a continuous map from MD3MD3to <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combso that <img height="18" border="0" style="vertical-align:bottom" width="57" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S1631073X1730119X-si10.gif">MD3combserves as a model for MD3MD3.
We prove, in a purely combinatorial way, the spectral curve topological recursion for the problem of enumeration of bi-colored maps, which are dual objects to dessins d'enfant. Furthermore, we give a proof of the quantum spectral curve equation for this problem. Then we consider the generalized case of 4-colored maps and outline the idea of the proof of the corresponding spectral curve topological recursion.
Let M be an irreducible holomorphic symplectic (hyperkähler) manifold. If b 2 (M ) > 5,
we construct a deformation M 0 of M which admits a symplectic automorphism of
infinite order. This automorphism is hyperbolic, that is, its action on the space of real
(1, 1)-classes is hyperbolic. If b 2 (M ) > 14, similarly, we construct a deformation which
admits a parabolic automorphism (and many other automorphisms as well).
In this paper we provide some affirmative results and some counterexamples for a solution of the splitting problem for n multivalued mappings, n>2.
In this paper, we consider the q → 0 limit of the deformed Virasoro algebra and that of the level 1, 2 representation of the Ding-Iohara-Miki algebra. Moreover, 5D AGT correspondence in this limit is discussed. This specialization corresponds to the limit from Macdonalds functions to Hall-Littlewood functions. Using the theory of Hall-Littlewood functions, some problems are solved. For example, the simplest case of 5D AGT conjectures is proven in this limit, and we obtain a formula for the 4-point correlation function of a certain operator.