A symplectic jeu de taquin bijection between the tableaux of King and of De Concini

The definitions, methods, and results are entirely combinatorial. The symplectic jeu de taquin algorithm developed here is an extension of Schützenberger's original jeu de taquin and acts on a skew form of De Concini's symplectic standard tableaux. This algorithm is used to construct a weight preserving bijection between the two most widely known sets of symplectic tableaux. Anticipated applications to Knuth relations and to decomposing symplectic tensor products are indicated.

     

Self-duality equation: Monodromy matrices and algebraic curves

It is shown that the general local solution of the self-duality equation with SU(1,1) and SU(2) gauge groups is associated with some algebraic curve with moving branch points if the related “monodromy matrix” is rational. The “multisoliton” solutions including monopoles and instantons, correspond to degenerate curves when the branch cuts collapse to double points. Bibliography:18 titles. Dedicated to L. D. Faddeev on the occasion of his 60th birthday Published inZapiski Nauchnykh Seminarov POMI, Vol. 215, 1994, pp. 197–216. Translated by D. A. Korotkin.     

Jeu de taquin and a monodromy problem for Wronskians of polynomials

The Wronskian associates to d linearly independent polynomials of degree at most n, a non-zero polynomial of degree at most d(n−d). This can be viewed as giving a flat, finite morphism from the Grassmannian Gr(d,n) to projective space of the same dimension. In this paper, we study the monodromy groupoid of this map. When the roots of the Wronskian are real, we show that the monodromy is combinatorially encoded by Schützenberger's jeu de taquin; hence we obtain new geometric interpretations and proofs of a number of results from jeu de taquin theory, including the Littlewood-Richardson rule.     

Algebraic K-theory of generalized free products and functors Nil

In this paper, we extend Waldhausen's results on algebraic K-theory of generalized free products in a more general setting and we give some properties of the Nil functors. As a consequence, we get new groups with trivial Whitehead groups.     

Witt vectors and K-theory of automorphisms via noncommutative motives

We prove that the functor ring-of-rational-Witt-vectors W ₀(?) becomes co-representable in the category of noncommutative motives. As an application, we obtain an immediate extension of W ₀(?) from commutative rings to schemes. Then, making use of the theory of noncommutative motives, we classify all natural transformations of the functor K-theory-of-automorphisms.     

Multiplicative K-theory and K-theory of Functors

In this article we give a construction of Max Karoubi’s multiplicative K-theory as the K-theory of an appropriate functor between two categories. We use this construction to explain why the two definitions of relative multiplicative K-theory for a compact pair of manifolds we give in the article agree. Part of this work has been done while I was holding an EEC postdoctoral position funded by the network ‘Algebraic K-theory, linear algebraic groups and related structures’ at University College Dublin. I would like to thank Professor David Lewis for inviting me to Dublin. During the later stages of the work, I was supported by EPSERC grant GR/S08046/01.     

Periods of generalized Fermat curves

Let $k,n\ge 2$ be integers. A generalized Fermat curve of type $(k,n)$ is a compact Riemann surface S that admits a subgroup of conformal automorphisms $H\le \text{Aut}(S)$ isomorphic to ${\mathbb{Z}}_k^n$, such that the quotient surface $S/H$ is biholomorphic to the Riemann sphere $\stackrel{?}{\mathbb{C}}$ and has $n+1$ branch points, each one of order k. There exists a good algebraic model for these objects, which makes them easier to study. Using tools from algebraic topology and integration theory on Riemann surfaces, we find a set of generators for the first homology group of a generalized Fermat curve. Finally, with this information, we find a set of generators for the period lattice of the associated Jacobian variety.     

Convex hulls of generalized moment curves

We consider the family of curves inR ⁴: {fx115-1}, wherep andq are positive integers, and determine the facial structure of the convex hull of these curves.     

Monodromy Groups and Bilinear Monodromy Invariant Combinations of Generalized Hypergeometric Type Integrals

The problem of determining bilinear combinations of holomorphic and antiholomorphic generalized hypergeometric type integrals left invariant under the action of the monodromy groups of the integrals is studied. In the special cases of simple Pochhammer type integrals and of twofold hypergeometric type integrals the existence and uniqueness of the bilinear invariants are proved, and the bilinear invariants are explicitly computed. Preparing the tools it is shown how to linearize and iterate representations of the braid group B_n as automorphism groups of certain free subgroups of the braid group B_n+1, and how the resulting iterated linear representations of the braid group in a natural way provide an algorithm to compute the monodromy group of generalized hypergeometric type integrals. Explicit formulae for different types of integration contours are given in the case of simple and twofold integrals.     

ANOTHER TYPE OF GENERALIZED BALL CURVES AND SURFACES

In 2000,Wu presented two new types of generalized Ball curves,one of which is called an NB1 curve located between the Wang Ball curve and the Said Ball curve.In this article,the authors aim to discuss properties of NB1 curves and surfaces,including the recursive algorithms,conversion algorithms between NB1 and Bézier curves and surfaces, etc.In addition the authors compare the computation efficiency of recursive algorithms for the NB1 and above mentioned two generalized Ball curves and surfaces.     

Geometry of curves in generalized flag varieties

The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.     

Singularities of the boundary of fundamental systems,flat points of projective curves,and Schubert cells

The paper studies the bifurcations of the flat points of projective curves and describes the singularities of the boundaries of the regions where typical three-parameter families of curves have no flat points. A flat-point-preserving transformation group of projective curves is introduced. Finite determinacy and versality theorems are proved for this group. A duality theorem is proved and some results concerning the connection of this problem with other problems of singularity theory are presented.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 215–232, 1988.     

Characterizations of generalized convexities via generalized directional derivatives

The paper gives characterizations of convexity, quasiconvexity, invexity and pseudoconvexity for a (radially) upper-semicontinuous function f in a topological vector space via appropriate properties of a bifunction which is majorized by the upper radial derivative of f and which stands for a generalized derivative of some sort.     

Limits of dual curves via foliations

We develop a method to compute limits of dual plane curves in Zeuthen families of any kind. More precisely, we compute the limit 0-cycle of the ramification scheme of a general linear system on the generic fiber, only assumed geometrically reduced, of a Zeuthen family of any kind.     

Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P³ defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph with each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.