Volume 10 (2010), Number 2. Abstracts

V.I. Arnold. Mathematics of Chaos [PDF]

In September 2008, V.I. Arnold was awarded the prestigious Shaw Prize. This prize was established in 2002 by the Shaw Prize Foundation named after philanthropist Sir Run Run Shaw. Here we publish the lecture given by V.I. Arnold at the awards ceremony in Hong Kong and a short Autobiography (CV) prepared for this occasion.

Keywords. Chaos, prime vectors, continued fractions, random numbers

2000 Mathematics Subject Classification. 00A99, 65P20

V. Batyrev and D. Juny. Classification of Gorenstein Toric Del Pezzo Varieties in Arbitrary Dimension [PDF]

An n-dimensional Gorenstein toric Fano variety X is called Del Pezzo variety if the anticanonical class −K_X is an (n−1)-multiple of a Cartier divisor. Our purpose is to give a complete biregular classfication of Gorenstein toric Del Pezzo varieties in arbitrary dimension n &ge 2. We show that up to isomorphism there exist exactly 37 Gorenstein toric Del Pezzo varieties of dimension n which are not cones over (n−1)-dimensional Gorenstein toric Del Pezzo varieties. Our results are closely related to the classification of all Minkowski sum decompositions of reflexive polygons due to Emiris and Tsigaridas and to the classification up to deformation of n-dimensional almost Del Pezzo manifolds obtained by Jahnke and Peternell.

Keywords. Toric varieties, Fano varieties, lattice polytopes

2000 Mathematics Subject Classification. 14M25, 14J45, 52B20

Yu. Ilyashenko and J. Llibre. A Restricted Version of Hilbert's 16th Problem for Quadratic Vector Fields [PDF]

The restricted version of Hilbert's 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields “σ-distant from centers and κ-distant from singular quadratic vector fields” provided that the limit cycles are “δ-distant from singular points and infinity”.

Keywords. Limit cycles, quadratic systems

2000 Mathematics Subject Classification. Primary: 34C40, 51F14; Secondary: 14D05, 14D25

A. Kanel-Belov, A. Dyskin, Y. Estrin, E. Pasternak, and I. Ivanov-Pogodaev. Interlocking of Convex Polyhedra: towards a Geometric Theory of Fragmented Solids [PDF]

The article presents arrangements of identical regular polyhedra with very special and curious properties. Namely, the solids are situated in a sort of a layer and are interlocked in the sense that no one of them can be moved out without disturbing others. This situation cannot happen in the plane. First examples of this sort (composed of irregular convex polyhedra) were complicated and were constructed in a non regular way by G. Galperin. The examples presented here were constructed in framework of applied studies by the authors, C. Khor and M. Glickman and were not described in mathematical publications. The full version of this paper is presented here: http://arxiv.org/abs/0812.5089.

Keywords. Interlocking structures, combinatorial geometry, convex polyhedron, tilling

2000 Mathematics Subject Classification. 52B10, 74R

K. Kaveh and A. Khovanskii. Mixed Volume and an Extension of Intersection Theory of Divisors [PDF]

Let K_rat(X) be the collection of all non-zero finite dimensional subspaces of rational functions on an n-dimensional irreducible variety X. For any n-tuple L₁, …, L_n ∈ K_rat(X), we define an intersection index [L₁, …, L_n] as the number of solutions in X of a system of equations f₁ = … = f_n = 0 where each f_i is a generic function from the space L_i. In counting the solutions, we neglect the solutions x at which all the functions in some space L_i vanish as well as the solutions at which at least one function from some subspace L_i has a pole. The collection K_rat(X) is a commutative semigroup with respect to a natural multiplication. The intersection index [L₁, …, L_n] can be extended to the Grothendieck group of K_rat(X). This gives an extension of the intersection theory of divisors. The extended theory is applicable even to non-complete varieties. We show that this intersection index enjoys all the main properties of the mixed volume of convex bodies. Our paper is inspired by the Bernstein–Kushnirenko theorem from the Newton polytope theory.

Keywords. System of algebraic equations, mixed volume of convex bodies, Bernstein–Kushirenko theorem, linear system on a variety, Cartier divisor, intersection index

2000 Mathematics Subject Classification. 14C20, 52A39

M. Losik. On the Continuous Cohomology of Diffeomorphism Groups [PDF]

Suppose that M is a connected orientable n-dimensional manifold and m>2n. If Hⁱ(M,R) = 0 for i>0, it is proved that for each m there is a monomorphism H^m(W_n,O(n)) → H^m_cont(Diff M,R). If M is closed and oriented, it is proved that for each m there is a monomorphism H^m(W_n,O(n)) → H^m−n_cont(Diff₊ M,R), where Diff₊ M is the group of orientation preserving diffeomorphisms of M.

Keywords. Diffeomorphism group, group cohomology, diagonal cohomology

2000 Mathematics Subject Classification. 22E41, 58D05, 57R32, 22E65, 17B66

Ch. Maire. Cohomology of Number Fields and Analytic Pro-p-Groups [PDF]

In this work, we are interested in the tame version of the Fontaine–Mazur conjecture. By viewing the pro-p-proup G_S as a quotient of a Galois extension ramified at p and S, we obtain a connection between the conjecture studied here and a question of Galois structure. Moreover, following a recent work of A. Schmidt, we give some evidence of links between this conjecture, the étale cohomology and the computation of the cohomological dimension of the pro-p-groups G_S that appear.

Keywords. Extensions with restricted ramification, cohomology of number fields and p-adic analytic structures

2000 Mathematics Subject Classification. 11R37, 11R23

I. Nakai and K. Yanai. Relations of Formal Diffeomorphisms and the Center Problem [PDF]

A word of germs of holomorphic diffeomorphisms of (C,0) is a composite of some time-1 maps of formal vector fields fixing 0, in other words, a noncommutative integral of a piecewise constant time depending formal vector field. We calculate its formal-vector-field-valued logarithm applying the Campbell–Hausdorff type formula of the Lie integral due to Chacon and Fomenko to the time depending formal vector field. For words of two time 1-maps we define Cayley diagrams in the plane spanned by the generating two vector fields in the Lie algebra of formal vector fields, and we show that some principal parts in the Taylor coefficients of the logarithm are given in terms of the higher moments of the Cayley diagrams. Solving the so-called center problem, the vanishing of the Lie integral, we show the various results on the existence and non-existence of relations of non-commuting formal diffeomorphisms in terms of the characteristic curves associated to the Cayley diagram.

Keywords. Holomorphic diffeomorphism, relation, free group, Campbell–Hausdorff

2000 Mathematics Subject Classification. 37F75, 53C12, 81Q70

A. Varchenko. A Selberg Integral Type Formula for an sl₂ One-Dimensional Space of Conformal Blocks [PDF]

For distinct complex numbers z₁,…,z_2N, we give a polynomial P(y₁,…,y_2N) in the variables y₁,…,y_2N which is homogeneous of degree N, linear with respect to each variable, sl₂-invariant with respect to a natural sl₂-action, and is of order N−1 at (y₁,…,y_2N) = (z₁,…,z_2N).

We give also a Selberg integral type formula for the associated one-dimensional space of conformal blocks.

Keywords. Conformal blocks, invariant polynomials

2000 Mathematics Subject Classification. Primary: 81T40, 33C70; Secondary: 32S40, 52B30

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