These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

http://cds.cern.ch/record/265648/files/9407018.pdf

Geometry of 2D topological field theories

0. Introduction. An n-dimensional convex polytope is simple if the number of codimension-one faces meeting at each vertex is n. In this paper we investigate certain group actions on manifolds, which have a simple convex polytope as orbit space. Let P\" denote such a simple polytope. We have two situations in mind. (1) The group is Z, M\" is n-dimensional and (2) The group is T\", M is 2n-dimensional and M2\"/T\" P\". Up to an automorphism of the group, the action is required to be locally isomorphic to the standard representation of Z. on in the second case. In the first case, we call M a \"small cover\" of P\"; in the second, it is a \"toric manifold\" over P. First examples are provided by the natural actions of Z. and T\" on RP\" and CP\", respectively. In both cases the orbit space is an n-simplex. Associated to a small cover of P\", there is a homomorphism 2\" Z’ Z, where m is the number of codimension-one faces of P\". The homomorphism 2 specifies an isotropy subgroup for each codimension-one face. We call it a \"characteristic function\" of the small cover. Similarly, the characteristic function ofa toric manifold over pn is a map Z --) 7/\". A basic result is that small covers and toric manifolds over P\" are classified by their characteristic functions (see Propositions 1.7 and 1.8). The algebraic topology of these manifolds is very beautiful. The calculation of their homology and cohomology groups is closely related to some well-known constructions in commutative algebra and the combinatorial theory of convex polytopes. We discuss some of these constructions below. Let f denote the number of/-faces of P\" and let h denote the coefficient of \"in f(t 1). Then (fo, f,) is called the f-vector and (ho, h,) the h-vector of P\". The f-vector and the h-vector obviously determine one another. The Upper Bound Theorem, due to McMullen, asserts that the inequality h < (,-,{-x), holds for all n-dimensional convex polytopes with m faces of codimension one. In 1971 McMullen conjectured simple combinatorial conditions on a sequence (h0, h,) of integers necessary and sufficient for it to be the h-vector of a simple convex polytope. The sufficiency of these conditions was proved by Billera and Lee and necessity by Stanley (see [Bronsted] for more details and references). Research on

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Convex polytopes, Coxeter orbifolds and torus actions

Contents: Matrix Eigenvalue Methods.- Double Bracket Isospectral Flows.- Singular Value Decomposition.- Linear Programming.- Approximation and Control.- Balanced Matrix Factorizations.- Invariant Theory and System Balancing.- Balancing via Gradient Flows.- Sensitivity Optimization.- Linear Algebra.- Dynamical Systems.- Global Analysis.

Optimization and Dynamical Systems

Noble metal nanoparticles for water purification: A critical review

The Korteweg–de Vries equation \[ u_t + uu_x + u_{xxx} = 0\] is a nonlinear partial differential equation arising in the study of a number of different physical systems, e.g., water waves, plasma physics, anharmonic lattices, and elastic rods. It describes the long time evolution of small-but-finite amplitude dispersive waves. From detailed studies of properties of the equation and its solutions, the concept of solitons was introduced and the method for exact solution of the initial-value problem using inverse scattering theory was developed. A survey of these and other results for the Korteweg–deVries equation are given, including conservation laws, an alternate method for exact solution, soliton solutions, asymptotic behavior of solutions, Backlund transformation, and a nonlinear WKB method. The recent literature contains many extensions of these ideas to a number of other nonlinear evolution equations of physical interest and to other classes of equations. Some of these equations and results are indica...

The Korteweg–deVries Equation: A Survey of Results

Inverse spectral theory is used to prescribe and study equations for the slow modulations of N-phase wave trains for the Korteweg-de Vries (KdV) equation. An invariant representation of the modulational equations is deduced. This representation depends upon certain differentials on a Riemann surface. When evaluated near ∞ on the surface, the invariant representation reduces to averaged conservations laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Riemann invariants for the modulational equations. Integrals of the invariant representation over certain cycles on the Riemann surface yield “conservation of waves.” Explicit formulas for the characteristic speeds of the modulational equations are derived. These results generalize known results for a single-phase traveling wave, and indicate that complete integrability can induce enough structure into the modulational equations to diagonalize (in the sense of Riemann invariants) their first-order terms.

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

A new set of canonically conjugate variables 1s introduced for the penoclic Korteweg-de Vries equation and the periodic Toda lattice. These vanables are used for reducing both equations to a nonlinear system which can be integrated in terms of theta functions. It becomes clear that the discrete and the continuous problems are, in a sense, Jsomorph1c. Action variables are defined by loop integrals, and the basic oscillation frequencies are computed. In the infinite-period limit, these action variables tend to the ones used in the canonical description of the inverse-scattering solution method.

Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)

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H. Flaschka

Papers

Multiphase averaging and the inverse spectral solution of the Korteweg—de Vries equation

Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)

Kac-moody lie algebras and soliton equations: II. Lax equations associated with A1(1)

A convexity theorem for isospectral manifolds of Jacobi matrices in a compact Lie algebra

Shock waves in the Toda lattice: Analysis