We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given.
Added: Jan 3, 2014
Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we prove a theorem on invariant subfields of $1$-extensions. We also obtain a general construction of all rationally triangulable subgroups of the Cremona groups and, as an application, classify rationally triangulable connected one-dimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy.
Added: Apr 16, 2015
We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of Cartier divisors in weighted projective space has a very weak Landau–Ginzburg model coming from a toric degeneration.
Added: Jul 2, 2013
Kuyumzhiyan K., Arzhantsev I., Zaidenberg M. Sbornik Mathematics. 2012. Vol. 203. No. 7.
P. 923-949.
We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.
Added: Feb 4, 2013
Smirnov E. In bk.: Recent Developments in Representation Theory. Vol. 673. Providence: AMS, 2016.
P. 179-226.
The aim of these notes is to give an introduction into Schubert calculus on Grassmannians and flag varieties. We discuss various aspects of Schubert calculus, such as applications to enumerative geometry, structure of the cohomology rings of Grassmannians and flag varieties, Schur and Schubert polynomials. We conclude with a survey of results of V. Kiritchenko, V. Timorin and the author on a new approach to Schubert calculus on full flag varieties via combinatorics of Gelfand-Zetlin polytopes.
Added: Oct 13, 2015
Galkin S. arxiv.org. math. Cornell University, 2014. No. 1404.7388.
Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer to a question of Ostrover and Tyomkin: the quantum cohomology algebra of a toric Fano manifold contains a field as a direct summand. Moreover, it gives a good evidence that the same statement holds for any Fano manifold.
Added: May 4, 2014
Пальвелев Р., Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 277. С. 199-214.
Added: Feb 23, 2013
Kuyumzhiyan K. Siberian Mathematical Journal. 2012. Vol. 53. No. 6.
P. 1089-1104.
Let T be a maximal torus in a classical linear group G. In this paper we find all simple rational G-modules V such that for each vector v a V the closure of the T-orbit of v is a normal affine variety. For every G-module without this property we present a T-orbit with nonnormal closure. To solve this problem, we use a combinatorial criterion of normality which is formulated in terms of the set of weights of a simple G-module. The same problem for G = SL(n) was solved by the author earlier.
Added: Feb 5, 2013
Boiti M., Pempinelli F.,
Pogrebkov A. Journal of Mathematical Physics. 2011. Vol. 52. No. 083506.
P. 1-21.
Properties of Jost and dual Jost solutions of the heat equation, F (x,k)
and Y(x,k), in the case of a pure solitonic potential are studied in
detail.We describe their analytical properties on the spectral parameter k
and their asymptotic behavior on the x-plane and we show that the values
of e(−qx)F (x, k) and the residues of exp(qx )Y(x,k) at special discrete
values of k are bounded functions of x in a polygonal region of the
q-plane. Correspondingly, we deduce that the extended version L(q) of the
heat operator with a pure solitonic potential has left and right
annihilators for q belonging to these polygonal regions.
Added: Feb 16, 2013
Chania: SMTDA, 2012.
Full papers (articles) of 2nd Stochastic Modeling Techniques and Data Analysis (SMTDA-2012) International Conference are represented in the proceedings. This conference took place from 5 June by 8 June 2012 in Chania, Crete, Greece.
Added: Jan 29, 2013
Oxford: Oxford University Press, 2011.
Added: Feb 26, 2013
Timorin V., Blokh A., Oversteegen L. et al. arxiv.org. math. Cornell University, 2013. No. 1305.5788.
According to a recent paper \cite{bopt13}, polynomials from the closure $\ol{\phd}_3$ of the {\em Principal Hyperbolic Domain} ${\rm PHD}_3$ of the cubic connectedness locus have a few specific properties. The family $\cu$ of all polynomials with these properties is called the \emph{Main Cubioid}. In this paper we describe the set $\cu^c$ of laminations which can be associated to polynomials from $\cu$.
Added: Oct 6, 2013
Bufetov A. I., Mkrtchyan S., Scherbina M. et al. arxiv.org. math. Cornell University, 2013. No. 1301.0342.
Added: Feb 21, 2013
Т. Дискретная оптимизация. Иркутск: ИДСТУ СО РАН, 2011.
Added: Mar 4, 2013
Сергеев А. Г. Труды Математического института им. В.А. Стеклова РАН. 2012. Т. 279. С. 193-205.
Added: Feb 23, 2013
Goncharuk N. B., Kudryashov Y. arxiv.org. math. Cornell University, 2015. No. 1504.03313.
In this article we prove in a new way that a generic polynomial vector field in ℂ² possesses countably many homologically independent limit cycles. The new proof needs no estimates on integrals, provides thinner exceptional set for quadratic vector fields, and provides limit cycles that stay in a bounded domain.
Added: Apr 15, 2015
Конаков В. Д. STI. WP BRP. Издательство попечительского совета механико-математического факультета МГУ, 2012. № 2012.
Added: Dec 5, 2012
Colliot-Thélène J., Kunyavskiĭ B.,
Vladimir L. Popov et al. Compositio Mathematica. 2011. Vol. 147. No. 2.
P. 428-466.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group
over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k-
rational functions on G, respectively, g. The conjugation action of G on itself induces
the adjoint action of G on g. We investigate the question whether or not the field
extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the
answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the
case where G is simple. For simple groups we show that the answer is positive if G is
split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A
key ingredient in the proof of the negative result is a recent formula for the unramified
Brauer group of a homogeneous space with connected stabilizers. As a byproduct of
our investigation we give an affirmative answer to a question of Grothendieck about the
existence of a rational section of the categorical quotient morphism for the conjugating
action of G on itself.
Added: Mar 17, 2013
Iss. 20. NY: Springer, 2012.
This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.
Added: Dec 19, 2012
We obtain a partial solution of the problem on the growth of the norms of exponential functions with a continuous phase in the Wiener algebra. The problem was posed by J.-P. Kahane at the International Congress of Mathematicians in Stockholm in 1962. He conjectured that (for a nonlinear phase) one can not achieve the growth slower than the logarithm of the frequency. Though the conjecture is still not confirmed, the author obtained first nontrivial results.
Added: Apr 12, 2012
We give an explicit formula for a quasi-isomorphism between the operads Hycomm (the homology of the moduli space of stable genus 0 curves) and BV/Δ (the homotopy quotient of Batalin-Vilkovisky operad by the BV-operator). In other words we derive an equivalence of Hycomm-algebras and BV-algebras enhanced with a homotopy that trivializes the BV-operator. These formulas are given in terms of the Givental graphs, and are proved in two different ways. One proof uses the Givental group action, and the other proof goes through a chain of explicit formulas on resolutions of Hycomm and BV. The second approach gives, in particular, a homological explanation of the Givental group action on Hycomm-algebras.
Added: Aug 29, 2012
V. L. Popov. Transformation Groups. 2011. Vol. 16. No. 3.
P. 827-856.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the universal covering isogeny
Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In
particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a
cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions
on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating
set of k[G]G and that of the representation ring of G and answer two Grothendieck's
questions on constructing generating sets of k[G]G. We prove the existence of a rational
(i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational
cross-section in G (for char k = 0, this has been proved earlier); this answers the other
question cited in the epigraph. We also prove that the existence of a rational section
is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a
maximal torus of G and W the Weyl group.
Added: Mar 16, 2013
Edited by: А. Михайлов Вып. 14. М.: Социологический факультет МГУ, 2012.
Added: Mar 14, 2013