Volume 9 (2009), Number 2. Abstracts

A. Barg and P. Purkayastha. Bounds on Ordered Codes and Orthogonal Arrays [PDF]

We derive new estimates of the size of codes and orthogonal arrays in the ordered Hamming space (the Niederreiter–Rosenbloom–Tsfasman space). We also show that the eigenvalues of the ordered Hamming scheme, the association scheme that describes the combinatorics of the space, are given by the multivariate Krawtchouk polynomials, and establish some of their properties.

Keywords. Ordered Hamming space, association schemes, multivariate Krawtchouk polynomials, Delsarte method, asymptotic bounds.

2000 Mathematics Subject Classification. Primary: 05E30; Secondary: 94B65.

A. Bufetov. Logarithmic Asymptotics for the Number of Periodic Orbits of the Teichmüller Flow on Veech's Space of Zippered Rectangles [PDF]

A logarithmic asymptotics is obtained for the number of periodic orbits of the Teichmüller flow on Veech's space of zippered rectangles, such that the norm of the corresponding renormalization matrix does not exceed a given value. The exponential growth rate of the number of such orbits is equal to the entropy of the flow.

Keywords. Periodic orbits, Teichmüller flow, suspension flows, moduli spaces, countable shifts.

2000 Mathematics Subject Classification. 37D25, 37A50, 37B40, 37C40.

B. Deroin, V. Kleptsyn, and A. Navas. On the Question of Ergodicity for Minimal Group Actions on the Circle [PDF]

This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated.

Keywords. Ergodic theory, group actions, circle diffeomorphisms, Lyapunov exponents, random dynamical systems, stationary measures.

2000 Mathematics Subject Classification. Primary: 37C85; Secondary: 37A50, 37D25, 37E10, 37F15.

E. Gorsky. Adams Operations and Power Structures [PDF]

We study the relations between Adams operation on a lambda-ring and the power structure on it, introduced by S. Gusein-Zade, I. Luengo and A. Melle-Hernández. We give the explicit equations expressing them by each other. An interpretation of the formula of E. Getzler for the equivariant Euler characteristics of configuration spaces is also given.

Keywords. λ-rings, Adams operations, plethysms, power structures, moduli of curves.

2000 Mathematics Subject Classification. 55S15, 19L20, 05E05, 14H10.

Sh. Izumiya. Legendrian Dualities and Spacelike Hypersurfaces in the Lightcone [PDF]

We show four Legendrian dualities between pseudo-spheres in Minkowski space as a basic theorem. We can apply such dualities for constructing extrinsic differential geometry of spacelike hypersurfaces in pseudo-spheres. In this paper we stick to spacelike hypersurfaces in the lightcone and establish an extrinsic differential geometry which we call the lightcone differential geometry.

Keywords. Legendrian duality, spacelike hypersurfaces, the lightcone, conformally flat Riemannian manifolds.

2000 Mathematics Subject Classification. Primary: 53A35, 58C27, 58C28.

I. Losev. Lifting Central Invariants of Quantized Hamiltonian Actions [PDF]

Let G be a connected reductive group over an algebraically closed field K of characteristic 0, X an affine symplectic variety equipped with a Hamiltonian action of G. Further, let * be a G-invariant Fedosov star-product on X such that the Hamiltonian action is quantized. We establish an isomorphism between the center of the quantum algebra K[X][[ℏ]]^G and the algebra of formal power series with coefficients in the Poisson center of K[X]^G.

Keywords. Reductive groups, Hamiltonian actions, central invariants, quantization.

2000 Mathematics Subject Classification. 53D20, 53D55, 14R20.

P. Mnëv. Notes on Simplicial BF Theory [PDF]

In this work we discuss a construction of “simplicial BF theory” (a field theory with finite-dimensional space of fields, associated to a triangulated manifold), that is in a sense equivalent to topological BF theory on the manifold (with infinite-dimensional space of fields). This is done in framework of the simplicial program, i.e., the program of constructing discrete topological field theories. We also discuss the relation of these constructions to homotopy algebra.

Keywords. Topological quantum field theory, exact discretization, Batalin–Vilkovisky formalism, BV effective action, quantum homological perturbation theory.

2000 Mathematics Subject Classification. 57R56, 81Q30, 81T25, 55P62, 18G55, 57Q10.

S. Shadrin. BCOV Theory via Givental Group Action on Cohomological Fields Theories [PDF]

In a previous paper, Losev, the author, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa.

In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all tautological equations becomes a trivial observation.

Keywords. Cohomological field theory, mirror symmetry, Batalin–Vilkovisky algebras, tautological relations, Givental's quantization of Frobenius manifolds.

2000 Mathematics Subject Classification. Primary: 14J32; Secondary: 14N35, 53D45.

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