Volume 12 (2012), Number 2. Abstracts

F. Aicardi. Rational Tangles and the Modular Group [PDF]

There is a natural way to define an isomorphism between the group of transformations of isotopy classes of rational tangles and the modular group. This isomorphism allows to give a simple proof of the Conway theorem, stating the one-to-one correspondence between isotopy classes of rational tangles and rational numbers. Two other simple ways to define this isomorphism, one of which suggested by Arnold, are also shown.

Keywords. Tangles, rational tangles, modular group, continued fractions, braids group, spherical braids group.

2000 Mathematics Subject Classification. 57M27, 20F36.

R.-O. Buchweitz and D. van Straten. An Index Theorem for Modules on a Hypersurface Singularity [PDF]

A topological interpretation of Hochster's Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was conjectured by H. Dao.

Keywords. Matrix factorisation, hypersurface singularity, maximal Cohen–Macaulay module, intersection form, linking number, K-Theory.

2000 Mathematics Subject Classification. 32S25, 32S55, 14C17, 19D10, 19L10, 57R99.

D. Daigle and A. Melle-Hernández. Linear Systems of Rational Curves on Rational Surfaces [PDF]

Given a curve C on a projective nonsingular rational surface S, over an algebraically closed field of characteristic zero, we are interested in the set Ω_C of linear systems 𝕃 on S satisfying C∈𝕃, dim 𝕃 ≥ 1, and the general member of 𝕃 is a rational curve. The main result of the paper gives a complete description of Ω_C and, in particular, characterizes the curves C for which Ω_C is non empty.

Keywords. Rational curves, rational surfaces, linear systems, weighted cluster of singular points.

2000 Mathematics Subject Classification. Primary: 14C20, 14J26.

A. Davydov and A. Platov. Optimal Stationary Solution in Forest Management Model by Accounting Intra-Species Competition [PDF]

We consider a model of exploitation of a size-structured population when the birth, growth and mortality rates depend on the individual size and interspecies competition, while the exploitation intensity is a function of the size only. For a given exploitation intensity and under natural assumptions on the rates, we establish existence and uniqueness of a nontrivial stationary state of the population. In addition, we prove existence of an exploitation intensity which maximizes a selected profit functional of exploitation.

Keywords. Calculus of variations and optimal control, stationary solution.

2000 Mathematics Subject Classification. Primary: 49J15; Secondary: 34H05, 93C15, 93C95.

L. Gatto and I. Scherbak. “On One Property of One Solution of One Equation” or Linear ODE's, Wronskians and Schubert Calculus [PDF]

For a linear ODE with indeterminate coefficients, we explicitly exhibit a fundamental system of solutions in terms of the coefficients. We show that the generalized Wronskians of the fundamental system are given by an action of the Schur functions on the usual Wronskian, and thence enjoy Pieri's and Giambelli's formulae. As an outcome, we obtain a natural isomorphism between the free module generated by the generalized Wronskians and the singular homology module of the Grassmannian.

Keywords. Linear ODEs, fundamental solutions, generalized Wronskians, Schur functions, homology of the Grassmannian.

2000 Mathematics Subject Classification. 14M15, 34A30, 55R40.

M. Gekhtman, M. Shapiro, and A. Vainshtein. Cluster Structures on Simple Complex Lie Groups and Belavin–Drinfeld Classification [PDF]

We study natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. According to our main conjecture, each class in the Belavin–Drinfeld classification of Poisson–Lie structures on G corresponds to a cluster structure in O(G). We prove a reduction theorem explaining how different parts of the conjecture are related to each other. The conjecture is established for SL_n, n<5, and for any G in the case of the standard Poisson–Lie structure.

Keywords. Poisson–Lie group, cluster algebra, Belavin–Drinfeld triple.

2000 Mathematics Subject Classification. 53D17, 13F60.

V. Goryunov and J. Haddley. Invariant Symmetries of Unimodal Function Singularities [PDF]

We classify finite order symmetries g of the 14 exceptional unimodal function singularities f in 3 variables, which satisfy a so-called splitting condition. This means that the rank 2 positive subspace in the vanishing homology of f should not be contained in one eigenspace of g_★. We also obtain a description of the hyperbolic complex reflection groups appearing as equivariant monodromy groups acting on the hyperbolic eigensubspaces arising.

Keywords. Exceptional unimodal function singularities, symmetry, equivariant monodromy, complex hyperbolic reflection groups.

2000 Mathematics Subject Classification. Primary: 32S30; Secondary: 20H10.

M. Granger, D. Mond, and M. Schulze. Partial Normalizations of Coxeter Arrangements and Discriminants [PDF]

We study natural partial normalization spaces of Coxeter arrangements and discriminants and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin's Frobenius manifold structure, which is lifted (without unit), to the space of the arrangement. We also describe an independent approach to these structures via duality of maximal Cohen–Macaulay fractional ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter group. Finally, we show that our partial normalizations give rise to new free divisors.

Keywords. Coxeter group, logarithmic vector field, free divisor.

2000 Mathematics Subject Classification. 20F55, 17B66, 13B22.

K. Kaveh and A. Khovanskii. Convex Bodies Associated to Actions of Reductive Groups [PDF]

We associate convex bodies to a wide class of graded G-algebras where G is a connected reductive group. These convex bodies give information about the Hilbert function as well as multiplicities of irreducible representations appearing in the graded algebra. We extend the notion of Duistermaat–Heckman measure to graded G-algebras and prove a Fujita type approximation theorem and a Brunn–Minkowski inequality for this measure. This in particular applies to arbitrary G-line bundles giving an equivariant version of the theory of volumes of line bundles. We generalize the Brion–Kazarnowskii formula for the degree of a spherical variety to arbitrary G-varieties. Our approach follows some of the previous works of A. Okounkov. We use the asymptotic theory of semigroups of integral points and Newton–Okounkov bodies developed in the authors' paper `Okounkov bodies, semigroups of integral points, graded algebras and intersection theory'.

Keywords. Reductive group action, multiplicity of a representation, Duistermaat–Heckman measure, moment map, graded G-algebra, G-line bundle, volume of a line bundle, semigroup of integral points, convex body, mixed volume, Brunn–Minkowski inequality.

2000 Mathematics Subject Classification. Primary: 14L30, 53D20; Secondary: 52A39.

M. Kazarian and S. Lando. Topological Relations on Witten–Kontsevich and Hodge Potentials [PDF]

Let \overline{M}_g;n denote the moduli space of genus g stable algebraic curves with n marked points. It carries the Mumford cohomology classes κ_i. A homology class in H_∗(\overline{M}_g;n) is said to be κ-zero if the integral of any monomial in the κ-classes vanishes on it. We show that any κ-zero class implies a partial differential equation for generating series for certain intersection indices on the moduli spaces. The genus homogeneous components of the Witten–Kontsevich potential, as well as of the more general Hodge potential, which include, in addition to ψ-classes, intersection indices for λ-classes, are special cases of these generating series, and the well-known partial differential equations for them are instances of our general construction.

Keywords. Moduli spaces, Deligne–Mumford compactification, Witten–Kontsevich potential, Hodge integrals.

2000 Mathematics Subject Classification. 14H10, 14H70.

B. Khesin. Symplectic Structures and Dynamics on Vortex Membranes [PDF]

We present a Hamiltonian framework for higher-dimensional vortex filaments (or membranes) and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively, i.e., singular elements of the dual to the Lie algebra of divergence-free vector fields. It turns out that the localized induction approximation (LIA) of the hydrodynamical Euler equation describes the skew-mean-curvature flow on vortex membranes of codimension 2 in any ℝⁿ, which generalizes to any dimension the classical binormal, or vortex filament, equation in ℝ³.

This framework also allows one to define the symplectic structures on the spaces of vortex sheets, which interpolate between the corresponding structures on vortex filaments and smooth vorticities.

Keywords. Vortex filament equation, Euler equation, vortex sheet, mean curvature flow, localized induction approximation, symplectic structure, vortex membrane.

2000 Mathematics Subject Classification. Primary: 35Q35; Secondary: 53C44, 58E40.

M. Sevryuk. KAM Theory for Lower Dimensional Tori within the Reversible Context 2 [PDF]

The reversible context 2 in KAM theory refers to the situation where dim Fix G < ½ codim T, where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus one deals with. Up to now, the persistence of invariant tori in the reversible context 2 has been only explored in the extreme particular case where dim Fix G=0 [M. B. Sevryuk, Regul. Chaotic Dyn. 16 (2011), no. 1–2, 24–38]. We obtain a KAM-type result for the reversible context 2 in the general situation, where the dimension of Fix G is arbitrary. As in the case where dim Fix G=0, the main technical tool is J. Moser's modifying terms theorem of 1967.

Keywords. KAM theory, Moser's modifying terms theorem, reversible systems, reversible context 2, fixed point manifold, lower dimensional invariant torus.

2000 Mathematics Subject Classification. 70K43, 70H33.

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