improved bounds for the transition temperature of directed polymers in a finite dimensional random medium we consider the problem of directed polymers in a random medium of a finitedimensional lattice in the high temperature phase of this system it is known that the annealed and quenched free energies coincide upper bounds on the transition temperature to a low temperature phase had previously been obtained by calculating the first two moments z and z2 of the partition function we improve these bounds by estimating noninteger moments z for 1 2

quantum cohomology of flag manifolds and toda lattices we discuss relations of vafa s quantum cohomology with floer s homology theory introduce equivariant quantum cohomology formulate some conjectures about its general properties and on the basis of these conjectures compute quantum cohomology algebras of the flag manifolds the answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a toda lattice

the set of vertices with positive curvature in a planar graph with nonnegative curvature abstract in this paper we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature based on this we show that the automorphism group of a planar possibly infinite graph with nonnegative combinatorial curvature and positive total curvature is a finite group and give an upper bound estimate for the order of the group

on the lp spectrum of laplacians on graphs abstract we study the p independence of spectra of laplace operators on graphs arising from regular dirichlet forms on discrete spaces here a sufficient criterion is given solely by a uniform subexponential growth condition moreover under a mild assumption on the measure we show a one sided spectral inclusion without any further assumptions we study applications to normalized laplacians including symmetries of the spectrum and a characterization for positivity of the cheeger constant furthermore we consider laplacians on planar tessellations for which we relate the spectral p independence to assumptions on the curvature

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

on the toda and kac van moerbeke systems given a solution of the toda lattice we explicitly construct a solution of the kac van moerbeke system related to each other by a miura type transformation as an illustration of our method we derive the n soliton solutions of the kac van moerbeke lattice

a stationary planar random graph with singular stationary dual dyadic lattice graphs dyadic lattice graphs and their duals are commonly used as discrete approximations to the hyperbolic plane we use them to give examples of random rooted graphs that are stationary for simple random walk but whose duals have only a singular stationary measure this answers a question of curien and shows behaviour different from the unimodular case the consequence is that planar duality does not combine well with stationary random graphs we also study harmonic measure on dyadic lattice graphs and show its singularity

long time asymptotics for the toda lattice in the soliton region we apply the method of nonlinear steepest descent to compute the long time asymptotics of the toda lattice for decaying initial data in the soliton region in addition we point out how to reduce the problem in the remaining region to the known case without solitons

chromatic quasisymmetric functions abstract we introduce a quasisymmetric refinement of stanley s chromatic symmetric function we derive refinements of both gasharov s schur basis expansion of the chromatic symmetric function and chow s expansion in gessel s basis of fundamental quasisymmetric functions we present a conjectural refinement of stanley s power sum basis expansion which we prove in special cases we describe connections between the chromatic quasisymmetric function and both the q eulerian polynomials introduced in our earlier work and conjecturally representations of symmetric groups on cohomology of regular semisimple hessenberg varieties which have been studied by tymoczko and others we discuss an approach using the results and conjectures herein to the e positivity conjecture of stanley and stembridge for incomparability graphs of 3 1 free posets

statistical mechanics of a single particle in a multiscale random potential parisi landscapes in finite dimensional euclidean spaces we construct an n dimensional gaussian landscape with multiscale translation invariant logarithmic correlations and investigate the statistical mechanics of a single particle in this environment in the limit of high dimension n the free energy of the system and overlap function are calculated exactly using the replica trick and parisi s hierarchical ansatz in the thermodynamic limit we recover the most general version of the derrida s generalized random energy model grem the low temperature behaviour depends essentially on the spectrum of length scales involved in the construction of the landscape if the latter consists of k discrete values the system is characterized by a k step replica symmetry breaking solution we argue that our construction is in fact valid in any finite spatial dimensions n 1 we discuss the implications of our results for the singularity spectrum describing multifractality of the associated boltzmann gibbs measure finally we discuss several generalizations and open problems such as the dynamics in such a landscape and the construction of a generalized multifractal random walk

symplectic connections and the linearisation of hamiltonian systems this paper uses symplectic connections to give a hamiltonian structure to the first variation equation for a hamiltonian system along a given dynamic solution this structure generalises that at an equilibrium solution obtained by restricting the symplectic structure to that point and using the quadratic form associated with the second variation of the hamiltonian plus casimir as energy this structure is different from the well known and elementary tangent space construction our results are applied to systems with symmetry and to lie poisson systems in particular

lyapunov exponents of large sparse random matrices and the problem of directed polymers with complex random weights we present results on two different problems the lyapunov exponent of large sparse random matrices and the problem of polymers on a cayley tree with random complex weights we give an analytic expression for the largest lyapunov exponent of products of random sparse matrices with random elements located at random positions in the matrix this expression is obtained through an analogy with the problem of random directed polymers on a cayley tree i e in the mean field limit which itself can be solved using its relationship with random energy models rem and grem for the random polymer problem with complex weights we find that in addition to the high and the low temperature phases which were already known in the case of positive weights the mean field theory predicts a new phase phase iii which is dominated by interference effects

unit interval orders and the dot action on the cohomology of regular semisimple hessenberg varieties abstract motivated by a 1993 conjecture of stanley and stembridge shareshian and wachs conjectured that the characteristic map takes the character of the dot action of the symmetric group on the cohomology of a regular semisimple hessenberg variety to x g t where x g t is the chromatic quasisymmetric function of the incomparability graph g of the corresponding natural unit interval order and is the usual involution on symmetric functions we prove the shareshian wachs conjecture our proof uses the local invariant cycle theorem of beilinson bernstein deligne to obtain a surjection which we call the local invariant cycle map from the cohomology of a regular hessenberg variety of jordan type to a space of local invariant cycles as ranges over all partitions the local invariant cycles collectively contain all the information about the dot action on a regular semisimple hessenberg variety we then prove a result showing that under suitable hypotheses the local invariant cycle map is an isomorphism if and only if the special fiber has palindromic cohomology this is a general theorem which is independent of the hessenberg variety context applying this result to the universal family of hessenberg varieties we show that in our case the surjections are actually isomorphisms thus reducing the shareshian wachs conjecture to computing the cohomology of a regular hessenberg variety but this cohomology has already been described combinatorially by tymoczko and using a new reciprocity theorem for certain quasisymmetric functions we show that tymoczko s description coincides with the combinatorics of the chromatic quasisymmetric function

quantum cohomology of flag manifolds g b and quantum toda lattices let g be a connected semi simple complex lie group b its borel subgroup t a maximal complex torus contained in b and lie t its lie algebra this setup gives rise to two constructions the generalized nonperiodic toda lattice 28 29 and the flag manifold g b the toda lattice for g b t is the dynamical system on the cotangent bundle t lie t endowed with the canonical holomorphic symplectic form and the holomorphic hamiltonian function we consider in this paper

polymers on disordered hierarchical lattices a nonlinear combination of random variables the problem of directed polymers on disordered hierarchical and hypercubic lattices is considered for the hierarchical lattices the problem can be reduced to the study of the stable laws for combining random variables in a nonlinear way we present the results of numerical simulations of two hierarchical lattices finding evidence of a phase transition in one case for a limiting case we extend the perturbation theory developed by derrida and griffiths to nonzero temperature and to higher order and use this approach to calculate thermal and geometrical properties overlaps of the model in this limit we obtain an interpolation formula allowing one to obtain the noninteger moments of the partition function from the integer moments we obtain bounds for the transition temperature for hierarchical and hypercubic lattices and some similarities between the problem on the two different types of lattice are discussed

heat kernel eigenfunctions and conditioned brownian motion in planar domains abstract some recent results concerning uniform convergence of the shape of the heat kernel to that of the first eigenfunction and the lifetime of h processes in bounded lipschitz domains in r n are extended to all planar domains of finite area

potential theory for the schrodinger equation soit a a ij x x i x j un operateur uniformement elliptique dont les coefficients sont des fonctions bornees mesurables sur un domaine de lipschitz borne d r d on definit la classe de kato k d on demontre des resultats sur l operateur l a v en utilisant des resultats connus pour a et en etudiant la jauge conditionnelle

schubert polynomials and classes of hessenberg varieties abstract regular semisimple hessenberg varieties are a family of subvarieties of the flag variety that arise in number theory numerical analysis representation theory algebraic geometry and combinatorics we give a giambelli formula expressing the classes of regular semisimple hessenberg varieties in terms of chern classes in fact we show that the cohomology class of each regular semisimple hessenberg variety is the specialization of a certain double schubert polynomial giving a natural geometric interpretation to such specializations we also decompose such classes in terms of the schubert basis for the cohomology ring of the flag variety the coefficients obtained are nonnegative and we give closed combinatorial formulas for the coefficients in many cases we introduce a closely related family of schemes called regular nilpotent hessenberg schemes and use our results to determine when such schemes are reduced

a conformal inequality related to the conditional gauge theorem we prove the inequality h x 1 g x y h y cg x y c where g is the green function of a plane domain d h is positive and harmonic on d and c is a constant whose value depends on the topological nature of the domain in particular for the class of proper simply connected domains c may be taken to be an absolute constant as an application we prove the conditional gauge theorem for plane domains of finite area for which the constant c in the above inequality is finite

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

sectional curvature of polygonal complexes with planar substructures in this paper we introduce a class of polygonal complexes for which we consider a notion of sectional combinatorial curvature these complexes can be viewed as generalizations of 2 dimensional euclidean and hyperbolic buildings we focus on the case of non positive and negative combinatorial curvature as geometric results we obtain a hadamard cartan type theorem thinness of bigons gromov hyperbolicity and estimates for the cheeger constant we employ the latter to get spectral estimates show discreteness of the spectrum in the sense of a donnelly li type theorem and present corresponding eigenvalue asymptotics moreover we prove a unique continuation theorem for eigenfunctions and the solvability of the dirichlet problem at infinity

exact solutions for the statistics of extrema of some random 1d landscapes application to the equilibrium and the dynamics of the toy model abstract the real space renormalization group rsrg method introduced previously for the brownian landscape is generalized to obtain the joint probability distribution of the subset of the important extrema at large scales of other one dimensional landscapes for a large class of models we give exact solutions obtained either by the use of constrained path integrals in the continuum limit or by solving the rsrg equations via an ansatz which leads to the liouville equation we apply in particular our results to the toy model energy landscape which consists in a quadratic potential plus a brownian potential which describes among others the energy of a single domain wall in a 1d random field ising model rfim in the presence of a field gradient the measure of the renormalized landscape is obtained explicitly in terms of airy functions and allows to study in detail the boltzmann equilibrium of a particle at low temperature as well as its non equilibrium dynamics for the equilibrium we give results for the statistics of the absolute minimum which dominates at zero temperature and for the configurations with nearly degenerate minima which govern the thermal fluctuations at very low temperature for the dynamics we compute the distribution over samples of the equilibration time or equivalently the distribution of the largest barrier in the system we also study the properties of the rare configurations presenting an anomalously large equilibration time which govern the long time dynamics we compute the disorder averaged diffusion front which interpolates between the kesten distribution of the sinai model at short rescaled time and the reaching of equilibrium at long rescaled time finally the method allows to describe the full coarsening i e many domain walls of the 1d rfim in a field gradient as well as its equilibrium

diffusion of directed polymers in a random environment we consider a system of random walks or directed polymers interacting weakly with an environment which is random in space and time in spatial dimensionsd 2 we establish that the behavior is diffusive with probability one the diffusion constant is not renormalized by the interaction

extremal paths on a random cayley tree we investigate the statistics of extremal path s both the shortest and the longest from the root to the bottom of a cayley tree the lengths of the edges are assumed to be independent identically distributed random variables drawn from a distribution rho l besides the number of branches from any node is also random exact results are derived for arbitrary distribution rho l in particular for the binary 0 1 distribution rho l pdelta l 1 1 p delta l 0 we show that as p increases the minimal length undergoes an unbinding transition from a localized phase to a moving phase at the critical value p p c 1 b 1 where b is the average branch number of the tree as the height n of the tree increases the minimal length saturates to a finite constant in the localized phase p p c where the velocity v min p is determined via a front selection mechanism at p p c the minimal length grows with n in an extremely slow double logarithmic fashion the length of the maximal path on the other hand increases linearly as v max p n for all p the maximal and minimal velocities satisfy a general duality relation v min p v max 1 p 1 which is also valid for directed paths on finite dimensional lattices

conditioned brownian motion in planar domains on etudie un mouvement brownien complexe standard qui part d un point x dans un domaine greenien connexe du plan complexe c et soit conditionne pour sortir en un point donne y de sa frontiere de martin soit conditionne pour frapper un point y dans avant de laisser

energy landscape for directed polymers we consider the directed polymer problem in two dimensions on a torus at zero temperature for each site x we can compute the free energy e x of the conformation of the polymer which goes by this site we analyse the geometrical properties of this free energy landscape e x which has the original property of being translationally invariant in terms of the one point probability density of free energy in particular we study the statistical distribution of energies e as a function of the system size and the spatial correlations parallel and perpendicular to the mean orientation of the polymer the free energy field is shown to display an anisotropic self affine structure the scaling exponents obtained numerically are in good agreement with predicted values

cheeger constants growth and spectrum of locally tessellating planar graphs in this article we study relations between the local geometry of planar graphs combinatorial curvature and global geometric invariants namely the cheeger constants and the exponential growth we also discuss spectral applications

the lifetime of conditioned brownian motion in planar domains of infinite area in this paper it is shown that all expected lifetimes ofh processes ind are finite if and only if the area ofd is finite ifd x y o x y o x x where o x o are two lipschitz functions we show that if is a bounded convex region in the plane there is anh process in with expected lifetime at leastc area we also give an example of a planar domaind of infinite area such that the expected lifetime of eachh process ind is finite

two time distribution function in one dimensional random directed polymers the explicit expression for the two time free energy distribution function in one dimensional random directed polymers is derived in terms of the bethe ansatz replica technique it is shown that such type of the distribution function can be represented in the form of a generalized two dimensional fredholm determinant

the h path distribution of the lifetime of conditioned brownian motion for non smooth domains consider the h paths of doob and let d be the lifetime of brownian motion killed at d where d is a bounded lipschitz or nta domain in n then log p x h tau d t sim lambda d t as t to infty where d is the first positive eigenvalue of 1 2 on d

a variational approach to directed polymers the authors develop a variational approach for directed polymers in random media in the framework of the replica formalism they derive the functional expression for the free energy in terms of the probability distribution of the probability of finding a polymer at a given point they compare these results with those coming from the broken replica symmetry approach and present some new computations in this framework

the full kostant toda hierarchy on the positive flag variety we study some geometric and combinatorial aspects of the solution to the full kostant toda f kt hierarchy when the initial data is given by an arbitrary point on the totally non negative tnn flag variety of sl n mathbb r the f kt flows on the tnn flag variety are complete and we show that their asymptotics are completely determined by the cell decomposition of the tnn flag variety given by rietsch total positivity and real flag varieties ph d thesis massachusetts institute of technology cambridge 1998 our results represent the first results on the asymptotics of the f kt hierarchy and even the f kt lattice moreover our results are not confined to the generic flow but cover non generic flows as well we define the f kt flow on the weight space via the moment map and show that the closure of each f kt flow forms an interesting convex polytope which we call a bruhat interval polytope in particular the bruhat interval polytope for the generic flow is the permutohedron of the symmetric group mathfrak s n we also prove analogous results for the full symmetric toda hierarchy by mapping our f kt solutions to those of the full symmetric toda hierarchy in the appendix we show that bruhat interval polytopes are generalized permutohedra in the sense of postnikov int math res not imrn 6 1026 1106 2009

totally positive toeplitz matrices and quantum cohomology of partial flag varieties we show that the set of totally positive unipotent lower triangular toeplitz matrices in gln form a real semi algebraic cell of dimension n 1 furthermore we prove a natural cell decomposition for its closure the proof uses properties of the quantum cohomology rings of the partial flag varieties of gln c relying in particular on the positivity of the structure constants which are enumerative gromov witten invariants we also give a characterization of total positivity for toeplitz matrices in terms of the quantum schubert classes this work builds on some results of dale peterson s which we explain with proofs in the type a case

random walks in random media with random signs abstract monte carlo calculations indicate that the transverse displacement and free energy fluctuations of 1 1 dimensional directed random walks in media with sign randomness are characterized by the same exponents as directed walks in ordinary random media with positive statistical weights

on the bethe ansatz for random directed polymers we show that the inclusion of the gapless center of mass motion together with a functional integral representation of the bethe wave function allows one to predict exactly the critical exponents for random directed polymers in 1 1 dimensions the corresponding amplitudes are computed they compare satisfactorily with existing numerical data within a replica symmetric theory we find that the green function of the polymer has the form recently proposed by parisi

directed paths on percolation clusters we consider a variant of the problem of directed polymers on a disordered lattice in which the disorder is geometrical in nature in particular we allow a finite probability for each bond to be absent from the lattice we show through the use of numerical and scaling arguments on both euclidean and hierarchical lattices that the model has two distinct scaling behaviors depending upon whether the concentration of bonds on the lattice is at or above the directed percolation threshold we are particularly interested in the exponents and defined by f t and x t describing the free energy and transverse fluctuations respectively above the percolation threshold the scaling behavior is governed by the standard random energy exponents 1 3 and 2 3 in 1 1 dimensions at the percolation threshold we predict and verify numerically in 1 1 dimensions the exponents 1 2 and v v where v and v are the directed percolation exponents in addition we predict the absence of a free phase in any dimension at the percolation threshold

interference of directed paths we study the problem of interference between directed paths in two dimensions on a euclidean square lattice and on a hierarchical structure the connection between this problem and directed polymers in a random medium at zero temperature is presented through a direct construction this mapping leads to a quantitative description of the exponential vanishing of the amplitude as a function of distance travelled within the medium a phase transition occurring for a small concentration of impurities which has been reported in the past is analysed using a direct numerical study although we cannot reach a definitive conclusion the critical point seems to vanish in the thermodynamic limit a renormalization argument is proposed to account for this effect finally the same problem treated on a hierarchical lattice is shown to give rise to pathological behaviour for large disorder for small disorder the connection with directed polymers is recovered

sharp martingale inequalities and applications to riesz transforms on manifolds lie groups and gauss space abstract we prove new sharp l p logarithmic and weak type inequalities for martingales under the assumption of differential subordination the l p estimates are feynman kac type versions of burkholder s celebrated martingale transform inequalities from the martingale l p inequalities we obtain that riesz transforms on manifolds of nonnegative bakry emery ricci curvature have exactly the same l p bounds as those known for riesz transforms in the flat case of r d from the martingale logarithmic and weak type inequalities we obtain similar inequalities for riesz transforms on compact lie groups and spheres combining the estimates for spheres with poincare s limiting argument we deduce the corresponding results for riesz transforms associated with the ornstein uhlenbeck semigroup thus providing some extensions of p a meyer s l p inequalities

ground state energy of a non integer number of particles with δ attractive interactions we show how to dene and calculate the ground state energy of a system of quantum particles with attractive interactions when the number of particles n is non integer the question is relevant to obtain the probability distribution of the free energy of a directed polymer in a random medium when one expands the ground state energy in powers of the interaction all the coecients of the perturbation series are polynomials in n allowing to dene the perturbation theory for non integer n we develop a procedure to calculate all the cumulants of the free energy of the directed polymer and we give explicit although complicated expressions of the rst three cumulants c 2000 elsevier science b v all rights reserved

on the long time behavior of the doubly infinite toda lattice under initial data decaying at infinity we provide rigorous analysis of the long time behavior of the doubly infinite toda lattice under initial data that decay at infinity in the absence of solitions we solve approximately and for large times the riemann hilbert matrix factorization problem equivalent to the related inverse scattering problem with the help of the beals coifman formula by reducing it to a simpler one through a series of contour deformations in the spirit of the deift zhou method

geometric analysis aspects of infinite semiplanar graphs with nonnegative curvature we apply alexandrov geometry methods to study geometric analysis aspects of infinite semiplanar graphs with nonnegative combinatorial curvature we obtain the metric classification of these graphs and construct the graphs embedded in the projective plane minus one point moreover we show the volume doubling property and the poincare inequality on such graphs the quadratic volume growth of these graphs implies the parabolicity finally we prove the polynomial growth harmonic function theorem analogous to the case of riemannian manifolds

an analogue of the descartes euler formula for infinite graphs and higuchi s conjecture let r be a connected 2 manifold without boundary obtained from a possibly infinite collection of polygons by identifying them along edges of equal length let v be the set of vertices and for every v v let k v denote the gaussian curvature of v 2 minus the sum of incident polygon angles descartes showed that v k 4 whenever r may be realized as the surface of a convex polytope in r 3 more generally if r is made of finitely many polygons euler s formula is equivalent to the equation v 1 r where r is the euler characteristic of r our main theorem shows that whenever v 0 for every vertex v we apply our main theorem to deduce that r is made of finitely many polygons and is homeomorphic to either the 2 sphere or to the projective plane further we show that unless is a prism antiprism or the projective planar analogue of one of these that v 3444 this resolves a recent conjecture of higuchi

on an estimate of cranston and mcconnell for elliptic diffusions in uniform domains we show that ifd n n 3 n 3 is a bounded uniform domain then the lifetime of the doobh paths ind for elliptic diffusions in divergence form is finite this result holds for any bounded domaind in the plane

the toda lattice dynkin diagrams singularities and abelian varieties the periodic l 1 particle toda lattices coresponding to extended dynkin diagrams have l 1 polynomial constants of the motion namely as many as there are dots in the dynkin diagram for most values of the constants their intersection defines as l dimensional affine invariant manifold which completes into a complex algebraic torus abelian variety by glueing on a divisor entirely specified by the extended dynkin diagram therefore the global geometry of the complex invariant tori such as polarization dimension of certain linear systems divisor equivalences etc is entirely given by the extended dynkin diagram in particular an explicit geometric description for the case of three particle toda lattices is given which describes the divisors their singularities and how the various divisors intersect

minimal path on the hierarchical diamond lattice we consider the minimal paths on a hierarchical diamond lattice where bonds are assigned a random weight depending on the initial distribution of weights we find all possible asymptotic scaling properties the different cases found are the small disorder case the analog of levy s distributions with a power law decay at and finally a limit of large disorder which can be identified as a percolation problem the asymptotic shape of the stable distributions of weights of the minimal path are obtained as well as their scaling properties as a side result we obtain the asymptotic form of the distribution of effective percolation thresholds for finite size hierarchical lattices

lifetime of conditioned brownian motion in lipschitz domains if d d d 3 is bounded and has lipschitz boundary then the expected lifetime of any brownian h path process in d is finite

directed paths in a complex random potential effects of long range correlations of the disorder the authors apply the replica method supplemented by a variational approach to directed paths which acquire random amplitudes and random phases this problem is pertinent to the hopping of electrons in disordered materials in the variable range hopping regime here they consider the long range case in which both the amplitude and phase correlations of the disorder are long ranged and a mixed case in which amplitude correlations are long ranged and phase correlations are short ranged the variational approach is expected to become exact in the limit of infinite dimensions and it yields the flory exponents for the transverse fluctuations stationary solutions which extremize the free energy and are characterized by non trivial patterns of replica symmetry breaking are obtained both analytically and numerically the mixed case has a non trivial phase diagram marked by abrupt changes in the pattern of replica symmetry breaking in this case the imaginary part of the random potential plays an important role

hessenberg varieties slodowy slices and integrable systems this work is intended to contextualize and enhance certain well studied relationships between hessenberg varieties and the toda lattice thereby building on the results of kostant peterson and others one such relationship is the fact that every lagrangian leaf in the toda lattice is compactified by a suitable choice of hessenberg variety it is then natural to imagine the toda lattice as extending to an appropriate union of hessenberg varieties we fix a simply connected complex semisimple linear algebraic group g and restrict our attention to a particular family of hessenberg varieties a family that includes the peterson variety and all toda leaf compactifications the total space of this family x h 0 is shown to be a poisson variety with a completely integrable system defined in terms of mishchenko fomenko polynomials this leads to a natural embedding of completely integrable systems from the toda lattice to x h 0 we also show x h 0 to have an open dense symplectic leaf isomorphic to g z times s text reg where z is the centre of g and s text reg is a regular slodowy slice in the lie algebra of g this allows us to invoke results about integrable systems on g times s text reg as developed by rayan and the second author lastly we witness some implications of our work for the geometry of regular hessenberg varieties

quantum cohomology rings of lagrangian and orthogonal grassmannians and total positivity we verify in an elementary way a result of peterson for the maximal orthogonal and lagrangian grassmannians and then find vafa intriligator type formulas which compute their 3 point genus zero gromov witten invariants finally we study the total positivity of the related peterson s varieties and show that rietsch s conjecture about the total positivity holds for these cases

nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices we prove the moments of the directed polymer partition function gz using an exact position space renormalization group scheme on a hierarchical lattice after sufficient iteration the characteristic functionf n ln gzn of the probability z converges to a stable limitf n for smalln the limiting behavior is independent of the initial distribution while for largen f n is completely determined by it and is thus nonuniversal there is a smooth crossover between the two regimes for small effective dimensions and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution for large effective dimensions there is a sharp transition between the two regimes and analytical continuation from integer moments ton 0 is not possible replica arguments can account for most features of the observed results