harmonic maps into the orthogonal group and null curves we find algebraic parametrizations of extended solutions of harmonic maps of finite uniton number from a surface to the orthogonal group mathrm o n in terms of free holomorphic data which lead to formulae for all such harmonic maps our work reveals an interesting correspondence between certain harmonic maps and the free weierstrass representation of null curves and minimal surfaces in 3 and 4 space

totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

the lie poisson structure of integrable classical non linear sigma models the canonical structure of classical non linear sigma models on riemannian symmetric spaces which constitute the most general class of classical non linear sigma models known to be integrable is shown to be governed by a fundamental poisson bracket relation that fits into ther s matrix formalism for non ultralocal integrable models first discussed by maillet the matricesr ands are computed explicitly and being field dependent satisfy fundamental poisson bracket relations of their own which can be expressed in terms of a new numerical matrixc it is proposed that all these poisson brackets taken together are representation conditions for a new kind of algebra which for this class of models replaces the classical yang baxter algebra governing the canonical structure of ultralocal models the poisson brackets for the transition matrices are also computed and the notorious regularization problem associated with the definition of the poisson brackets for the monodromy matrices is discussed

quiver varieties and lusztig s algebra abstract we study preprojective algebras of graphs and their relationship to module categories over representations of quantum sl 2 as an application ade quiver varieties of nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by lusztig

approximate solution methods for linear stochastic difference equations i moments the cumulant expansion for linear stochastic differential equations is extended to the case of linear stochastic difference equations we consider a vector difference equation which contains a deterministic matrix a0 and a random perturbation matrix a1 t the expansion proceeds in powers of c where c is the correlation time of the fluctuations in a1 t and a measure for their strength compared to the differential case additional cumulants occur in the expansion moreover one has to distinguish between a nonsingular and a singular a0 we also discuss a limiting situation in which the stochastic difference equation can be replaced by a stochastic differential equation the derivation is not restricted to the case where in the limit the stochastic parameters in the difference equation are replaced by white noise

discrete pluriharmonic functions as solutions of linear pluri lagrangian systems pluri lagrangian systems are variational systems with the multi dimensional consistency property this notion has its roots in the theory of pluriharmonic functions in the z invariant models of statistical mechanics in the theory of variational symmetries going back to noether and in the theory of discrete integrable systems a d dimensional pluri lagrangian problem can be described as follows given a d form l on an m dimensional space m d whose coefficients depend on a function u of m independent variables called field find those fields u which deliver critical points to the action functionals s sigma int sigma l for any d dimensional manifold in the m dimensional space we investigate discrete 2 dimensional linear pluri lagrangian systems i e those with quadratic lagrangians l the action is a discrete analogue of the dirichlet energy and solutions are called discrete pluriharmonic functions we classify linear pluri lagrangian systems with lagrangians depending on diagonals they are described by generalizations of the star triangle map examples of more general quadratic lagrangians are also considered

harmonic maps of finite uniton number into g2 we establish explicit formulae for canonical factorizations of extended solutions corresponding to harmonic maps of finite uniton number into the exceptional lie group g2 in terms of the grassmannian model for the group of based algebraic loops in g2 a description of the frenet frame data for such harmonic maps is given in particular we show that harmonic spheres into g2 correspond to solutions of certain algebraic systems of quadratic and cubic equations

lie groups and twistor spaces on considere un espace symetrique de riemann simplement connexe m la connexion de levi civita et le fibre torseur metrique 7b f m j m dont les fibres 7b f x m pour x m est constitue de structures complexes orthogonales

flows of constant mean curvature tori in the 3 sphere the equivariant case we present a deformation for constant mean curvature tori in the 3 sphere we show that the moduli space of equivariant constant mean curvature tori in the 3 sphere is connected and we classify the minimal the embedded and the alexandrov embedded tori therein

on the homotopy of simplicial algebras over an operad according to a result of h cartan cf 5 the homotopy of a simplicial commutative algebra is equipped with divided power operations in this paper we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad to be precise we work over a fixed field f and we consider operads in the category of f modules an operad is an algebraic device that specifies a type of algebras there are operads com as lie and pois whose algebras are respectively commutative algebras associative algebras lie algebras and poisson algebras in general if p denotes an operad then we call p algebras the associated algebras first we generalize the notion of a divided power in the context of algebras over an operad this is done as follows recall that the free commutative algebra is given by the formula t com v n v sn for v modf

periodic and homoclinic solutions of the modified 2 1 chiral model we use algebraic backlund transformations bts to construct explicit solutions of the modified 2 1 chiral model from t2 r to su n where t2 is a 2 torus algebraic bts are parametrized by z c poles and holomorphic maps from t2 to gr k cn we apply backlund transformations with carefully chosen poles and s to construct infinitely many solutions of the 2 1 chiral model that are i doubly periodic in space variables and periodic in time i e triply periodic ii homoclinic in the sense that the solution u has the same stationary limit u0 as t and is tangent to a stable linear mode of u0 as t and is tangent to an unstable mode of u0 as t

representation theory in complex rank i p deligne defined interpolations of the tensor category of representations of the symmetric group s n to complex values of n namely he defined tensor categories rep s t for any complex t this construction was generalized by f knop to the case of wreath products of s n with a finite group generalizing these results we propose a method of interpolating representation categories of various algebras containing s n such as degenerate affine hecke algebras symplectic reflection algebras rational cherednik algebras etc to complex values of n we also define the group algebra of s n for complex n study its properties and propose a schur weyl duality for rep s t

inhomogeneous linear random differential equations with mutual correlations between multiplicative additive and initial value terms abstract the cumulant expansion for linear stochastic differential equations is extended to the general case in which the coefficient matrix the inhomogeneous part and the initial condition are all random and moreover statistically interdependent the expansion now involves not only the autocorrelation functions of the coefficient matrix as in the homogeneous case but also crosscorrelation functions of the coefficient matrix with the inhomogeneous part and with the initial value term as a first illustration we consider an exactly solvable stochastic differential equation with initial correlations and compare the exact solution with that of the cumulant expansion secondly we show in general how the method can be used for the calculation of second moments and treat the harmonic oscillator with random frequency and driving term as an example

twistors 4 symmetric spaces and integrable systems an order four automorphism of a lie algebra gives rise to an integrable system introduced by terng we show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a riemannian symmetric space as applications we find that surfaces with holomorphic mean curvature in 4 dimensional real or complex space forms constitute an integrable system as do hamiltonian stationary lagrangian surfaces in 4 dimensional hermitian symmetric spaces this last providing a conceptual explanation of a result of helein romon

examples of tensor categories let g be a tensor category over a field k of characteristic 0 in his paper i w 7 delinge had established rather general conditions for g to be the category of representations of an algebraic group he has shown that if any object nec satisfies the condition a n 0 for a sufficiently large n then 0 is equivalent to the category rep g of representations of a proalgebraic group scheme g over k for tensor categories over fields of positive characteristic this result is no longer true in the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of deligne s theorem does not hold to describe these categories denote by gz an algebraic chevalley group over z and by g the corresponding group over fp for a sufficiently large p our category c will be a certain subquotient of the category r g of finite dimensional representations of g more precisely let s be the full subcategory of r g whose objects called small representations are direct sums of the irreducible representations of g with highest weights inside the fundamental alcove let also sl 2 c g be the principal slz subgroup in g and slz im c sl2 be the infinitesimal neighbourhood of the cartan subgroup h in sl2 denote by o the full subcategory of r g consisting of representations m of g called quasiprojective representations such that the restriction of m to sl2 m is projective the category 0 is a quotient of the subcategory s pcr g in the last section of the paper we show that the same approach can be applied to representations of quantum groups let u be the quantum group at a root of prime order p of 1 which is a hopf algebra over the ring of integers in a cyclotomic field see 2 3 starting from the category of representations of u we construct a semisimple braided category g p with the finite number of simple objects on the other hand in wess zumino witten conformal field theory one encounters the category o g m of integrable representations of the correspond

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

a loop group method for affine harmonic maps into lie groups we generalize the uhlenbeck segal theory for harmonic maps into compact semi simple lie groups to general lie groups equipped with torsion free bi invariant connection

isotropic geometry and twistors in higher dimensions i the generalized klein correspondence and spinor flags in even dimensions an equivariant diffeomorphism is given between flag varieties of totally isotropic subspaces of a complex inner product space and certain orbits under the complex spin group within flag varieties of the associated spinor modules this allows a generalization of the klein correspondence underlying twistor theory applicable to arbitrary even dimensional conformally compactified minkowski space and its isotropic subspaces

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

the dimension and structure of the space of harmonic 2 spheres in the m sphere we prove the conjecture posed in 1993 by bolton and woodward that the dimension of the space of harmonic maps from the 2 sphere to the 2n sphere is 2d n we also give an explicit algebraic method to construct all harmonic maps from the 2 sphere to the m sphere

intersection numbers on moduli spaces and symmetries of a verlinde formula we investigate the geometry and topology of a standard moduli space of stable bundles on a riemann surface and use a generalization of the verlinde formula to derive results on intersection pairings

multi time lagrangian 1 forms for families of backlund transformations relativistic toda type systems we establish the pluri lagrangian structure for families of backlund transformations of relativistic toda type systems the key idea is a novel embedding of these discrete time one dimensional systems into certain two dimensional 2d pluri lagrangian lattice systems this embedding allows us to identify the corner equations which are the main building blocks of the multi time euler lagrange equations with local superposition formulae for backlund transformations these superposition formulae in turn are key ingredients necessary to understand and to prove commutativity of the multi valued backlund transformations furthermore we discover a 2d generalization of the spectrality property known for families of backlund transformations this result produces a family of local conservations laws for 2d pluri lagrangian lattice systems with densities being derivatives of the discrete 2 form with respect to the backlund spectral parameter thus a relation of the pluri lagrangian structure with more traditional integrability notions is established

a cumulant expansion for the time correlation functions of solutions to linear stochastic differential equations it is shown that the cumulant expansion for linear stochastic differential equations hitherto used to compute one time averages of the solution process is also capable of yielding the two time correlation and probability density functions the general case with a coefficient matrix an inhomogeneous part and an initial condition which are all random and mutually correlated is discussed two examples are given the latter of which treats the harmonic oscillator with stochastic frequency and driving term studied before finally we investigate the relation of our method with the so called smoothing method

aks hierarchy and bi hamiltonian geometry of gelfand zakharevich type a bi hamiltonian system is a system of differential equations which can be written in hamiltonian form in two distinct ways the applications of gelfand zakharevich bi hamiltonian structure which is an extension of a poisson nijenhuis structure on phase space has been extensively explored by falqui magri and pedroni in the context of separation of variables it is well known that the integrable hamiltonian systems defined by the adler kostant symes aks scheme contains bi hamiltonian structure in this paper we unveil the connection between adler kostant symes formalism applied to loop algebra and the gelfand zakharevich bi hamiltonian structure by superposition the results of fordy and kulish in the aks scheme we also study the commuting flows of the aks hierachy and its connection to the zakharov shabat hierarchy

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

the space of harmonic maps of s 2 into s 4 toute surface superminimale ramifiee d aire 4 d dans s 4 provient d une paire de fonctions mesomorphes f 1 f 2 de bidegre d d telles que f 1 et f 2 ont le meme diviseur de ramification on deduit des conditions sous lesquelles des surfaces superminimales ramifiees peuvent etre engendrees a partir de telles paires de fonctions

on the fundamental group of the space of harmonic 2 spheres in the n sphere harmonic maps s s are of interest from the point of view of differential geometry as branched minimal immersions and from the point of view of gauge theory as classical solutions of the s non linear sigma model from the work of chern calabi barbosa and others there is now a complete description of such harmonic maps in terms of holomorphic data however very little is known about the global properties of the space harm s s of all harmonic maps since the analogous object in yang mills theory the space of solutions to the yang mills equations has proved to be of great significance the space of harmonic maps is also of some interest in this paper we shall give an approach to the study of the topology of harm s s

deformation quasi hopf algebras of non semisimple type from cochain twists given a symmetric decomposition mathfrak g mathfrak h oplus mathfrak p of a semisimple lie algebra mathfrak g we define the notion of a mathfrak p contractible quantized universal enveloping algebra quea for these queas the contraction mathfrak g rightarrow mathfrak g 0 making mathfrak p abelian is nonsingular and yields a quea of mathfrak g 0 for a certain class of symmetric decompositions we prove by refining cohomological arguments due to drinfel d that every quea of mathfrak g 0 so obtained is isomorphic to a cochain twist of the undeformed envelope mathcal u mathfrak g 0 to do so we introduce the mathfrak p contractible chevalley eilenberg complex and prove for this class of symmetric decompositions a version of whitehead s lemma for this complex by virtue of the existence of the cochain twist there exist triangular quasi hopf algebras based on these contracted queas and in the approach due to beggs and majid the dual quantized coordinate algebras admit quasi associative differential calculi of classical dimensions as examples we consider poincare in 3 and 4 spacetime dimensions

filtrations factorizations and explicit formulae for harmonic maps we use filtrations of the grassmannian model to produce explicit algebraic formulae for all harmonic maps of finite uniton number from a riemann surface and so all harmonic maps from the 2 sphere to the unitary group for a general class of factorizations by unitons we show how these specialize to give explicit formulae for such harmonic maps to each of the classical compact lie groups and their inner symmetric spaces the nonlinear model of particle physics our methods also give an explicit iwasawa decomposition of the algebraic loop group

generators for rational loop groups uhlenbeck proved that a set of simple elements generates the group of rational loops in gl n that satisfy the u n reality condition for an arbitrary complex reductive group a choice of representation defines a notion of rationality and enables us to write a natural set of simple elements using these simple elements we prove generator theorems for the fundamental representations of the remaining neo classical groups and most of their symmetric spaces we also obtain explicit dressing and permutability formulae

koszul duality and the pbw theorem in symmetric tensor categories in positive characteristic abstract we generalize the theory of koszul complexes and koszul algebras to symmetric tensor categories in characteristic zero the generalization is routine while in characteristic p there is a subtlety the symmetric algebra of an object is not always koszul i e its koszul complex is not always exact namely this happens in the verlinde category ver p in any characteristic p 5 we call an object koszul if its symmetric algebra is koszul and show that the only koszul objects of ver p are usual supervector spaces i e a non invertible simple object l m 2 m p 2 is not koszul we show however that the symmetric algebra s l m is almost koszul in the sense of brenner butler and king namely p m m koszul and compute the corresponding internal yoneda algebra i e the internal ext algebra from the trivial module to itself we then proceed to discuss the pbw theorem for operadic lie algebras i e algebras over the operad lie this theorem is well known to fail for vector spaces in characteristic 2 as one needs to require that x x 0 and for supervector spaces in characteristic 3 as one needs to require that x x x 0 for odd x but it holds in these categories in any characteristic p 5 there is a well known proof based on koszul duality however we show that in the category ver p because of failure of koszul duality the pbw theorem can fail in any characteristic p 5 namely one needs to impose the p jacobi identity a certain generalization to characteristic p of the identities x x 0 and x x x 0 on the other hand our main result is that once the p jacobi identity is imposed the pbw theorem holds this shows that the correct definition of a lie algebra in ver p is an algebra over lie which satisfies the p jacobi identity this also applies to any symmetric tensor category that admits a symmetric tensor functor to ver p e g a symmetric fusion category see 19 theorem 1 5 finally we prove the pbw theorem for lie algebras in any quasi semisimple symmetric tensor category

the geometric complexity of special lagrangian t2 cones we prove a number of results on the geometric complexity of special lagrangian slg t2 cones in 3 every slg t2 cone has a fundamental integer invariant its spectral curve genus we prove that the spectral curve genus of an slg t2 cone gives a lower bound for its geometric complexity i e the area the stability index and the legendrian index of any slg t2 cone are all bounded below by explicit linearly growing functions of the spectral curve genus we prove that the cone on the clifford torus which has spectral curve genus zero in s5 is the unique slg t2 cone with the smallest possible legendrian index and hence that it is the unique stable slg t2 cone this leads to a classification of all rigid index 1 slg cone types in dimension three for cones with spectral curve genus two we give refined lower bounds for the area the legendrian index and the stability index one consequence of these bounds is that there exist s1 invariant slg torus cones of arbitrarily large area legendrian and stability indices we explain some consequences of our results for the programme due to joyce to understand the most common three dimensional isolated singularities of generic families of slg submanifolds in almost calabi yau manifolds

localized induction equation and pseudospherical surfaces we describe a close connection between the localized induction equation hierarchy of integrable evolution equations on space curves and surfaces of constant negative gauss curvature

explicit construction of harmonic two spheres into the complex grassmannian we present an explicit description of all harmonic maps of finite uniton number from a riemann surface into a complex grassmannian namely starting from a constant map q and a collection of meromorphic functions and their derivatives we show how to algebraically construct all harmonic maps from the two sphere into a given grassmannian g p mathbb c n in this setting the uniton number depends on q and p and we obtain a sharp estimate for it

approximate solution methods for linear stochastic difference equations ii inhomogeneous equations multi time averages biological application abstract the cumulant expansion for linear stochastic difference equations introduced in part i is applied to the general case where the equation contains multiplicative additive and initial value terms which are all random and statistically interdependent also the two time correlation functions of the solution are discussed finally the expansion for the probability density functions is studied in the case that the coefficient matrix constitutes a markov process an exact equation for the joint probability density function of the solution of the difference equation and the random coefficient matrix is derived from this equation the moments of the solution can be obtained in a simple way as an application we consider the growth of a biological population with two age classes in a random environment which itself is modelled by a two state markov chain the exact results are compared with those of the cumulant expansion and with previous findings of tuljapurkar

all harmonic 2 spheres in the unitary group completely explicitly we give a completely explicit formula for all harmonic maps of finite uniton number from a riemann surface to the unitary group u n in any dimension and so all harmonic maps from the 2 sphere in terms of freely chosen meromorphic functions on the surface and their derivatives using only combinations of projections and avoiding the usual bar partial problems or loop group factorizations we interpret our constructions using segal s grassmannian model giving an explicit factorization of the algebraic loop group and showing how to obtain harmonic maps into a grassmannian