Showing papers in "Selecta Mathematica-new Series in 1997"

Periodic Korteweg de Vries equation with measures as initial data

Abstract: The main result of the paper is that the periodic KdV equation $y_t + \partial^3_x y + yy_x = 0$ has a unique global solution for initial data y(0) given by a measure $\mu\in M({\Bbb T})$ of sufficiently small norm $\parallel\mu\parallel$ . There are two main ingredients in the proof. The first is the study of the local well-posedness problem in terms of the space-time Fourier-norms as exploited in [Bo] and also subsequent work such as [K-P-V2]. At the end the estimates eventually depend on a uniform estimate in terms of the Fourier coefficients¶¶ $ {{\rm sup}\atop{n\in{\Bbb Z},\,t\in{\Bbb R}}}|\hat{y}(n)(t)| < C .$ .¶¶Such a priori bound (in the space of pseudo-measures) on the solution may be derived from spectral theory and more precisely from the preservation of the periodic spectrum of a potential evolving according to KdV, which is the second ingredient. Thus the result at this stage depends heavily on integrability features of this particular equation. We also sketch an argument establishing almost periodicity properties of these solutions. This work is in spirit closely related to [Bo]. Natural questions suggested by these investigations is an extension of the result (at least for the IVP local in time) to a more general nonintegrable setting as well as to what extent the estimates on Fourier coefficients by spectral invariants and vice versa remains valid in distributional spaces.

Hyper-Kähler geometry and invariants of three-manifolds

Abstract: We study a 3-dimensional topological sigma-model, whose target space is a hyper-Kahler manifold X. A Feynman diagram calculation of its partition function demonstrates that it is a finite type invariant of 3-manifolds which is similar in structure to those appearing in the perturbative calculation of the Chern-Simons partition function. The sigma-model suggests a new system of weights for finite type invariants of 3-manifolds, described by trivalent graphs. The Riemann curvature of X plays the role of Lie algebra structure constants in Chern-Simons theory, and the Bianchi identity plays the role of the Jacobi identity in guaranteeing the so-called IHX relation among the weights. We argue that, for special choices of X, the partition function of the sigma-model yields the Casson-Walker invariant and its generalizations. We also derive Walker's surgery formula from the SL(2, Z) action on the finite-dimensional Hilbert space obtained by quantizing the sigma-model on a two-dimensional torus.

Kostka polynomials and energy functions in solvable lattice models

Abstract: The relation between the charge of Lascoux-Schutzenberger and the energy function in solvable lattice models is clarified. As an application, A.N. Kirillov's conjecture on the expression of the branching coefficient of \({\widehat{{\frak {sl}}_n}}/{\frak {sl}}_n\) of Kostka polynomials is proved.

Projectively flat Finsler 2-spheres of constant curvature

Abstract: After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure". I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1.

Intertwining operators of double affine Hecke algebras

Abstract: A systematic study of the representation theory of double affine Hecke algebras and related harmonic analysis is started in this paper. Continuing the previous papers we use the technique of intertwining operators to create Macdonald polynomials, estimate their denominators, generalize the classical representations of p-adic affine Hecke algebras in the spaces of functions on affine Weyl groups, and to find out when induced representations are irreducible and co-spherical. The connection with recent results by Sahi and Knop on the integrality of the Macdonald polynomials is established.

Characteristic Classes in Symplectic Topology

Abstract: From the cohomological point of view the symplectomorphism group $Sympl (M)$ of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary classes are first manifestations of this principles. On a deeper level live characteristic classes of symplectic actions in periodic cohomology and symplectic Hodge decompositions. The present paper is called to introduce theories and constructions listed above and to suggest numerous concrete applications. These includes: nonvanishing results for cohomology of symplectomorphism groups (as a topological space, as a topological group and as a discrete group), symplectic rigidity of Chern classes, lower bounds for volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and Oh, new characters for Torelli group and generalizations for automorphism groups of one-relator groups, arithmetic properties of special values of Witten zeta-function and solution of a conjecture of Brylinski. The Appendix, written by L. Katzarkov, deals with fixed point sets of finite group actions in moduli spaces.

Verlinde algebras and the intersection form on vanishing cycles

Abstract: We prove Zuber's conjecture [Z] establishing connections of the fusion rules of the su( N )k WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential.

The Cohen-Macaulay property of the category of$ ({\frak g}, K) $-modules

Abstract: Let (g,K) be a Harish-Chandra pair. In this paper we prove that if P and P' are two projective (g,K) -modules, then Hom(P, P') is a Cohen-Macaulay module over the algebra Z(g,K) of K-invariant elements in the center of U(g) . This fact implies that the category of (g,K) -modules is locally equivalent to the category of modules over a Cohen-Macaulay algebra, where by a Cohen-Macaulay algebra we mean an associative algebra that is a free finitely generated module over a polynomial subalgebra of its center.

Geometry of conservation laws for a class of parabolic partial differential equations

Abstract: I consider the problem of computing the space of conservation laws for a second-order parabolic partial differential equation for one function of three independent variables. The PDE is formulated as an exterior differential system \({\cal I}\) on a 12-manifold M, and its conservation laws are identified with the vector space of closed 3-forms in the infinite prolongation of \({\cal I}\) modulo the so-called "trivial" conservation laws. I use the tools of exterior differential systems and Cartan's method of equivalence to study the structure of the space of conservation laws. My main result is: Theorem.Any conservation law for a second-order parabolic PDE for one function of three independent variables can be represented by a closed 3-form in the differential ideal ${\cal I}$ on the original 12-manifold M. I show that if a nontrivial conservation law exists, then \({\cal I}\) has a deprolongation to an equivalent system \({\cal J}\) on a 7-manifold N, and any conservation law for \({\cal I}\) can be expressed as a closed 3-form on N that lies in \({\cal J}\). Furthermore, any such system in the real analytic category is locally equivalent to a system generated by a (parabolic) equation of the formA (uxxuyy-u2xy)+Buxx+2Cuxy+Duyy+E = 0 where A, B, C, D, E are functions of x, y, t, u, ux, uy, ut. I compute the space of conservation laws for several examples, and I begin the process of analyzing the general case using Cartan's method of equivalence. I show that the non-linearizable equation \( u_t = {1 \over 2} e^{-u} (u_{xx}+u_{yy}) \) has an infinite-dimensional space of conservation laws. This stands in contrast to the two-variable case, for which Bryant and Griffiths showed that any equation whose space of conservation laws has dimension 4 or more is locally equivalent to a linear equation, i.e., is linearizable.

Inverse scattering up to smooth functions for the Dirac-ZS-AKNS system

Abstract: We consider the Dirac-ZS-AKNS system \(\psi_{1x} = -ik\psi_1 + q_1(x)\psi_2,\)(1)\(\psi_{2x} = -ik\psi_2 + q_2(x)\psi_1,\) where \( q_1(x), q_2(x)\in W^{n,1}({\Bbb R}) \) (the space of functions with n derivatives in L1), \( n\in{\Bbb N}\cup 0. \)(2) We consider for (1) the transition matrix \( T(k)=\biggl({a(k)\atop b(k)} {c(k)\atop d(k)}\biggr), k\in {\Bbb R},\) and, in addition, for the case of the Dirac system (i.e. for the selfadjoint case \( q_2(x)=\overline{q_1(x))} \) the scattering matrix \( S(k)=\biggl({s_{11}(k) \atop s_{21}(k)} {s_{12}(k)\atop s_{22}(k)}\biggr)= {1\over a(k)} \biggl({1\atop -{c(k)}} {b(k)\atop 1}\biggr),\, k\in {\Bbb R}. \) We can divide main results of the present work into three parts. I. We show that the inverse scattering transform and the inverse Fourier transform give the same solution, up to smooth functions, of the inverse scattering problem for (1). More preciseley, we show that, under condition (2) with \( n \in {\Bbb N} \), the following formulas are valid: \( q_1(x)=2\check{c}(2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}), \)(3)\( q_2(x)=2\check{b}(-2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}), \) and, in addition, for the case of the Dirac system \( q_1(x)=-2\check{s}_{21}(2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}), \)(4)\( q_2(x)=2\check{s}_{12}(-2x) \quad {\rm in} \quad W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}), \) where \( \check{\varphi}(x)=(2\pi)^{-1}\int_{\Bbb R} e^{-ikx}\varphi(k)dk, \, W^{n,1}({\Bbb R})/W^{n+1,1}({\Bbb R}) \) denotes the factor space. II. Using (3), (4), we give the characterization of the transition matrix and the scattering matrix for the case of the Dirac system under condition (2) with \( n\in {\Bbb N} \)III. As applications of the results mentioned above, we show that 1) for any real-valued initial data \( \theta(0,x)\in W^{n,1}({\Bbb R}), n\in{\Bbb N} \), the Cauchy problem for the sh-Gordon equation \( \theta_{xt}=sh\,\theta \) has a unique solution \( \theta(t,x) \) such that \(\theta_x(t,x)\in C([0,\infty[, W^{n-1,1}({\Bbb R}))\) and \(\theta(t,x)\rightarrow 0 \,{\rm as}\,|x|\rightarrow\infty\) for any t > 0, 2) in addition, for \(n\in{\Bbb N}\), for such a solution the following formula is valid: \( \theta(t,x)=\theta(0,x)\quad {\rm in} \quad W^{n,1}_{\rm loc} ({\Bbb R})/W^{n+1,1}_{\rm loc}({\Bbb R}), \) where \(W^{n,1}_{\rm loc}({\Bbb R})\) denotes the space of functions locally integrable with n derivatives. We give also a review of preceding results.

Geometry of conservation laws for a class of parabolic PDE's, II: Normal forms for equations with conservation laws

Abstract: We consider conservation laws for second-order parabolic partial differential equations for one function of three independent variables. An explicit normal form is given for such equations having a nontrivial conservation law. It is shown that any such equation whose space of conservation laws has dimension at least four is locally contact equivalent to a quasi-linear equation. Examples are given of nonlinear equations that have an infinite-dimensional space of conservation laws parameterized (in the sense of Cartan-Kahler) by two arbitrary functions of one variable. Furthermore, it is shown that any equation whose space of conservation laws is larger than this is locally contact equivalent to a linear equation.