Volume 5 (2005), Number 2. Abstracts

S. Anisov. Exact Values of Complexity for an Infinite Number of 3-Manifolds [PDF]

We find the exact values of complexity for an infinite series of 3-manifolds. Namely, by calculating hyperbolic volumes, we show that c(N_n)=2n, where c is the complexity of a 3-manifold and N_n is the total space of the punctured torus bundle over S¹ with monodromy \begin{pmatrix}2&1\\1&1\end{pmatrix}ⁿ. We also apply a recent result of Matveev and Pervova to show that c(M_n)≥2Cn with C≈0.598, where a compact manifold M_n is the total space of the torus bundle over S¹ with the same monodromy as N_n, and discuss an approach to the conjecture c(M_n)=2n+5 based on the equality c(N_n)=2n.

Keywords. Complexity of 3-manifolds, figure eight knot complement, Gromov norm.

2000 Mathematics Subject Classification. 51M25, 57Q15, 57M50.

V. Buchstaber and A. Lazarev. The Gelfand Transform in Commutative Algebra [PDF]

We consider the transformation ev which associates to any element in a K-algebra A a function on the the set of its K-points. This is the analogue of the fundamental Gelfand transform. Both ev and its dual ev^* are the maps from a discrete K-module to a topological K-module and we investigate in which case the image of each map is dense. This question arises in the classical problem of the reconstruction of a function by its values at a given set of points. The answer is nontrivial for various choices of K and A already for A=K[x], the polynomial ring in one variable. Applications to the structure of algebras of cohomology operations are given.

Keywords. Linear topology, rings of divided powers, numerical polynomials, Landweber–Novikov algebra, Steenrod algebra.

2000 Mathematics Subject Classification. Primary: 13B25, 13A05; Secondary: 55N20.

B. Enriquez and V. Rubtsov. Quantizations of the Hitchin and Beauville–Mukai Integrable Systems [PDF]

Spectral transformation is known to set up a birational morphism between the Hitchin and Beauville–Mukai integrable systems. The corresponding phase spaces are: (a) the cotangent bundle of the moduli space of bundles over a curve C, and (b) a symmetric power of the cotangent surface T^*(C). We conjecture that this morphism can be quantized, and we check this conjecture in the case where C is a rational curve with marked points and rank 2 bundles. We discuss the relation of the resulting isomorphism of quantized algebras with Sklyanin's separation of variables.

Keywords. Hilbert scheme, quantization, lagrangian fibration, Lie–Reinhart algebra, Gaudin model.

2000 Mathematics Subject Classification. Primary: 14H70, 17B80, 17B63, 81R12; Secondary: 81R12.

P. Etingof and Sh. Gelaki. On Radically Graded Finite-Dimensional Quasi-Hopf Algebras [PDF]

In this paper we continue the structure theory of finite dimensional quasi-Hopf algebras started in our previous papers. First, we completely describe the class of radically graded finite dimensional quasi-Hopf algebras over C, whose radical has prime codimension. As a corollary we obtain that if p>2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have Frobenius–Perron dimension p^N, N=1, 2, 3, 4, 5 or 7. Second, we construct new examples of finite dimensional quasi-Hopf algebras which are not twist equivalent to a Hopf algebra. For instance, to every finite dimensional simple Lie algebra g and a positive integer n, we attach a quasi-Hopf algebra of dimension n^{dim g}.

Keywords. Quasi-Hopf algebras, finite tensor categories.

2000 Mathematics Subject Classification. 16W30, 17B37.

T. Ito and B. Scárdua. On Holomorphic Foliations Transverse to Spheres [PDF]

We study the problem of existence and classification of holomorphic foliations transverse to a real submanifold in the complex affine space. In particular we investigate the existence of a codimension one holomorphic foliation transverse to a sphere in Cⁿ for n≥3.

Keywords. Holomorphic foliation, transverse section, foliation with singularities.

2000 Mathematics Subject Classification. Primary 32S65; Secondary 57R30.

T. Kappeler, V. Schroeder, and S. Kuksin. Poincaré Inequalities for Maps with Target Manifold of Negative Curvature [PDF]

We prove that for any given homotopic C¹-maps u, v: G → M in a nontrivial homotopy class from a metric graph into a closed manifold of negative sectional curvature, the distance between u and v can be bounded by 3(length(u) + length(v)) + C(κ, ρ/20), where ρ>0 is a lower bound of the injectivity radius and −κ<0 an upper bound for the sectional curvature of M. The constant C(κ, ε) is given by

C(κ, ε) = 8 sh⁻¹_κ(1) + 8 sh⁻¹_κ (1/sh_κ(ε))

with sh_κ(t) = sinh(√{κ} t). Various applications are given.

Keywords. Negative sectional curvature, short homotopies, Poincaré inequality.

2000 Mathematics Subject Classification. 53C21, 55P99, 26D10.

S. Oblezin. Isomonodromic Deformations of sl(2) Fuchsian Systems on the Riemann Sphere [PDF]

This paper is devoted to the two geometric constructions provided by the isomonodromic method for Fuchsian systems. We develop the subject in the sense of geometric representation theory following Drinfeld's ideas. Thus we identify the initial data space of the sl(2) Schlesinger system with the moduli space of the Frobenius–Hecke (FH-)sheaves originally introduced by Drinfeld. First, we perform the procedure of separation of variables in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Flashka–McLaughlin, Gaudin and Sklyanin formulas. In the second part of the paper, we construct the Drinfeld compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlevé-VI system.

Keywords. Isomonodromic deformation, separation of variables, the Drinfeld compactification, the Frobenius–Hecke sheaves, the Painlevé-VI equation.

2000 Mathematics Subject Classification. 15A54, 32G02, 34B02.

V. Pasol and A. Polishchuk. Universal Triple Massey Products on Elliptic Curves and Hecke's Indefinite Theta Series [PDF]

Generalizing the result of Polishchuk we express universal triple Massey products between line bundles on elliptic curves in terms of Hecke's indefinite theta series. We show that all Hecke's indefinite theta series arise in this way.

Keywords. Massey products, Fukaya products, elliptic curves, indefinite theta series, homological mirror symmetry.

2000 Mathematics Subject Classification. Primary 14H52; Secondary 55S30.

V. Rotger, A. Skorobogatov, and A. Yafaev. Failure of the Hasse Principle for Atkin–Lehner Quotients of Shimura Curves over Q [PDF]

We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin–Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin–Lehner involutions. A particular example is the quotient of the Shimura curve X_23⋅107 attached to the indefinite rational quaternion algebra of discriminant 23⋅107 by the Atkin–Lehner involution ω₁₀₇. The quadratic twist of X_23⋅107 by Q(√{−23}) with respect to this involution is also a counter-example to the Hasse principle over Q.

Keywords. Shimura curves, rational points, Hasse principle, descent.

2000 Mathematics Subject Classification. 11G18, 14G35.

S. Slavnov. On Completeness of Dynamic Topological Logic [PDF]

A classical result on topological semantics of modal logic due to McKinsey and Tarski (often called Tarski theorem) states that the logic S4 is complete with respect to interpretations in Rⁿ for each n. Recently several authors have considered dynamic topological logics, which are interpreted in dynamic spaces (abstract dynamic systems). A dynamic space is a topological space together with a continuous function on it. Artemov, Davoren, and Nerode introduced a bimodal logic S4C and proved it to be complete with respect to the class of all dynamic spaces. A number of polymodal logics for dynamic topological systems were considered by Kremer, Mints, and Rubakov. Earlier the author showed that the analogue of Tarski theorem does not hold for S4C; this result has also been established independently from the author by P. Kremer and later by J. van Benthem (private communication). In this paper we show that a certain generalization of Tarski theorem applies in the dynamic case. We prove that for any formula φ underivable in S4C there exists a countermodel in Rⁿ for n sufficiently large. We give also an upper bound on the dimension of a refuting model. It remains an open question whether our upper bound is exact.

Keywords. Topological semantics, modal logic, dynamic logic.

2000 Mathematics Subject Classification. 03B45, 03B44, 03B80.

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