extended kepler coulomb quantum superintegrable systems in three dimensions the quantum kepler coulomb system in three dimensions is well known to be second order superintegrable with a symmetry algebra that closes polynomially under commutators this polynomial closure is also typical for second order superintegrable systems in 2d and for second order systems in 3d with nondegenerate four parameter potentials however the degenerate three parameter potential for the 3d kepler coulomb system also second order superintegrable is an exception as its symmetry algebra does not close polynomially the 3d four parameter potential for the extended kepler coulomb system is not even second order superintegrable but verrier and evans 2008 j math phys 49 022902 showed it was fourth order superintegrable and tanoudis and daskaloyannis 2011 arxiv 11020397v1 showed that if a second fourth order symmetry is added to the generators the symmetry algebra closes polynomially here based on the tremblay turbiner and winternitz construction we consider an infinite class of quantum extended kepler coulomb three and four parameter systems indexed by a pair of rational numbers k1 k2 and reducing to the usual systems when k1 k2 1 we show these systems to be superintegrable of arbitrarily high order and determine the structure of their symmetry algebras we demonstrate that the symmetry algebras close algebraically only for systems admitting extra discrete symmetries is polynomial closure achieved underlying the structure theory is the existence of raising and lowering operators not themselves symmetry operators or even defined independent of basis that can be employed to construct the symmetry operators and their structure relations

good filtrations and strong f regularity of the ring of u p invariants abstract let k be an algebraically closed field of positive characteristic g a reductive group over k and v a finite dimensional g module let p be a parabolic subgroup of g and u p its unipotent radical we prove that if s sym v has a good filtration then s u p is strongly f regular

theta caracteristiques des courbes tracees sur une surface lisse nous generalisons la notion de theta caracterisque sur une courbe lisse a une courbe de gorenstein puis construisons un espace de modules t au sens grossier pour les theta caracteristiques des courbes de degre d sur une surface algebrique projective lisse nous montrons que le schema t est projectif et determinons l ensemble de ses points fermes ensuite nous generalisons le theoreme d atiyah et mumford sur l invariance sous deformation de la dimension modulo 2 de l espace des sections globales d une theta caracterique a notre situation ce qui donne en corollaire que t est reunion de deux fermes disjoints puis nous donnons un critere de lissite et calculons l espace tangent en un point stable a t enfin nous considerons le cas ou la surface est le plan projectif nous montrons que t n a que des singularites rationnelles et calculons sa dimension ensuite nous decomposons t en comosantes irreductibles en utilisant la decomposition sur l ouvert des courbes lisses de beauville et catanese puis donnons une description generique de ces composantes irreductibles

superintegrability and higher order integrals for quantum systems we refine a method for finding a canonical form of symmetry operators of arbitrary order for the schrodinger eigenvalue equation h 2 v e on any 2d riemannian manifold real or complex that admits a separation of variables in some orthogonal coordinate system the flat space equations with potentials v x iy k 1 x iy k 1 in cartesian coordinates and v r2 r2cos 2k r2sin 2k the tremblay turbiner and winternitz system in polar coordinates have each been shown to be classically superintegrable for all rational numbers k we apply the canonical operator method to give a constructive proof that each of these systems is also quantum superintegrable for all rational k we develop the classical analog of the quantum canonical form for a symmetry it is clear that our methods will generalize to other hamiltonian systems

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

1 1 mode locking and generalized synchronization in mechanical oscillators we describe the relation between the complete phase and generalized synchronization of the mechanical oscillators response system driven by the chaotic signal generated by the driven system we identified the close dependence between the changes in the spectrum of lyapunov exponents and a transition to different types of synchronization the strict connection between the complete synchronization imperfect complete synchronization of response oscillators and their phase or generalized synchronization with the driving system the 1 1 mode locking is shown we argue that the observed phenomena are generic in the parameter space and preserved in the presence of a small parameter mismatch

a new superintegrable hamiltonian we identify a new superintegrable hamiltonian in three degrees of freedom obtained as a reduction of pure keplerian motion in six dimensions the new hamiltonian is a generalization of the keplerian one and has the familiar 1 r potential with three barrier terms preventing the particle crossing the principal planes in three degrees of freedom there are five functionally independent integrals of motion and all bound classical trajectories are closed and strictly periodic the generalization of the laplace runge lenz vector is identified and shown to provide functionally independent isolating integrals they are quartic in the momenta and do not arise from separability of the hamilton jacobi equation a formulation of the system in action angle variables is presented

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

infinite families of superintegrable systems separable in subgroup coordinates a method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly solvable it is shown that in two dimensional euclidean or pseudo euclidean spaces the method also preserves superintegrability two infinite families of classical and quantum superintegrable systems are obtained in two dimensional pseudo euclidean space whose classical trajectories and quantum eigenfunctions are investigated in particular the wave functions are expressed in terms of laguerre and generalized bessel polynomials

on the littlewood richardson polynomials we prove the equivalence of several descriptions of generators of rings of semiinvariants of quivers due to domokos and zubkov schofield and van den bergh and our earlier work we also show that the dimensions of semi invariants of weights n depend polynomially on n 2002 elsevier science usa all rights reserved

liaison and castelnuovo mumford regularity in this article we establish bounds for the castelnuovo mumford regularity of projective schemes in terms of the degrees of their defining equations the main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities when applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals

noise induced synchronization of uncoupled nonlinear systems we derive a recursion formulae of transition probability of the noise induced synchronization arising in a pair of identical uncoupled logistic maps linked by common noisy excitation only the formulae has a delta type stationary solution which represents the perfect synchronization with probability 1 the stationary solution maintains under chaotic bifurcation while the escape times to reach the perfect synchronization increase in the chaotic region the escape times analysis implies existence of lower dimensional dynamics around the perfect synchronization we also provide a physical implementation of the synchronization

cox rings and combinatorics given a variety x with a finitely generated total coordinate ring we describe basic geometric properties of x in terms of certain combinatorial structures living in the divisor class group of x for example we describe the singularities we calculate the ample cone and we give simple fano criteria as we show by means of several examples the results allow explicit computations as immediate applications we obtain an effective version of the kleiman chevalley quasiprojectivity criterion and the following observation on surfaces a normal complete surface with finitely generated total coordinate ring is projective if and only if any two of its non factorial singularities admit a common affine neighbourhood

the core of zero dimensional monomial ideals the core of an ideal is the intersection of all its reductions we describe the core of a zero dimensional monomial ideal i as the largest monomial ideal contained in a general reduction of i this provides a new interpretation of the core in the monomial case as well as an efficient algorithm for computing it we relate the core to adjoints and first coefficient ideals and in dimension two and three we give explicit formulas

on a non vanishing conjecture of kawamata and the core of an ideal we show under suitable hypothesis which are sharp in a certain sense that the core of an m primary ideal in a regular local ring of dimension d is equal to the adjoint or multiplier ideal of its d th power this generalizes the fundamental formula for the core of an integrally closed ideal in a two dimensional regular local ring due to huneke and swanson we also find a generalization of this result to singular non regular settings which we show to be intimately related to the problem of finding non zero sections of ample line bundles on projective varieties in particular we show that a graded analog of our formula for core would imply a remarkable conjecture of kawamata predicting that every adjoint ample line bundle on a smooth variety admits a non zero section

algebras defined by powers of determinantal ideals let x be an m x n matrix of indeterminates over a field k a k x and i z x the ideal generated by the t minors of x the main objects of this article are the rees algebra 2 y these characteristics will be called non exceptional we show how this result generalizes to an arbitrary integral domain of coefficients the intersection of primary ideals which gives i in non exceptional characteristics always is the integral closure of i it follows immediately from the primary decomposition that the powers i are integrally closed in non exceptional characteristics therefore is a normal domain and the primary decomposition of i turns out easy giving some insight into the structure of y an interesting observation the primary decomposition of the ideals i can be computed very quickly if one knows in advance that all these ideals are integrally closed the best results are obtained in characteristic 0 since one has a multiplicity free action of the linearly reductive group gl m k x gl n k on k x under which z is stable cf dep or bv sect 111 applying the theory of u invariants kraft kr one shows that g has rational singularities in particular and hence 9 are cohen macaulay rings we have no doubt that and 3 are cohen macaulay in arbitrary non exceptional characteristic it seems however that in exceptional characteristic they are as far as possible from this property the case t 2

necessary conditions for classical super integrability of a certain family of potentials in constant curvature spaces we formulate the necessary conditions for the maximal super integrability of a certain family of classical potentials defined in the constant curvature two dimensional spaces we give examples of homogeneous potentials of degree 2 on as well as their equivalents on and for which these necessary conditions are also sufficient we show explicit forms of the additional first integrals which can always be chosen as a polynomial with respect to the momenta and which can be of an arbitrary high degree with respect to the momenta

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

cohomology of the hilbert scheme of points on a surface with values in representations of tautological bundles let x a smooth quasi projective algebraic surface l a line bundle on x let x n the hilbert scheme of n points on x and l n the tautological bundle on x n naturally associated to the line bundle l on x we explicitely compute the image l n of the tautological bundle l n for the bridgeland king reid equivalence d b x n d bn x n in terms of a complex cof sn equivariant sheaves in d bn x n we give moreover a characterization of the image l n l n in terms of of the hyperderived spectral sequence e p q 1 associated to the derived k fold tensor power of the complex c l the study of the sn invariants of this spectral sequence allows to get the derived direct images of the double tensor power and of the general k fold exterior power of the tautological bundle for the hilbert chow morphism providing danila brion type formulas in these two cases this yields easily the computation of the cohomology of x n with values in l n l n andkl n

local cohomology with support in ideals of maximal minors abstract suppose that k is a field of characteristic zero x is an r s matrix of indeterminates where r s and r k x is the polynomial ring over k in the entries of x we study the local cohomology modules h i i r where i is the ideal of r generated by the maximal minors of x we identify the indices i for which these modules vanish compute h i i r at the highest nonvanishing index i r s r 1 and characterize all nonzero ones as submodules of certain indecomposable injective modules these results are consequences of more general theorems regarding linearly reductive groups acting on local cohomology modules of polynomial rings

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

polynomial constants of motion for calogero type systems in three dimensions we give an explicit and concise formula for higher degree polynomial first integrals of a family of calogero type hamiltonian systems in dimension three these first integrals together with the already known ones prove the maximal superintegrability of the systems

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

n 2 supersymmetric extension of the tremblay turbiner winternitz hamiltonians on a plane the family of tremblay turbiner winternitz hamiltonians hk on a plane corresponding to any positive real value of k is shown to admit an supersymmetric extension of the same kind as that introduced by freedman and mende for the calogero problem and based on an superalgebra the irreducible representations of the latter are characterized by the quantum number specifying the eigenvalues of the first integral of motion xk of hk bases for them are explicitly constructed the ground state of each supersymmetrized hamiltonian is shown to belong to an atypical lowest weight state irreducible representation

third order superintegrable systems separable in parabolic coordinates in this paper we investigate superintegrable systems which separate in parabolic coordinates and admit a third order integral of motion we give the corresponding determining equations and show that all such systems are multi separable and so admit two second order integrals the third order integral is their lie or poisson commutator we discuss how this situation is different from the cartesian and polar cases where new potentials were discovered which are not multi separable and which are expressed in terms of painleve transcendents or elliptic functions

families of classical subgroup separable superintegrable systems we describe a method for determining a complete set of integrals for a classical hamiltonian that separates in orthogonal subgroup coordinates as examples we use it to determine complete sets of integrals polynomial in the momenta for some families of generalized oscillator and kepler coulomb systems hence demonstrating their superintegrability the latter generalizes recent results of verrier and evans and rodriguez tempesta and winternitz another example is given of a superintegrable system on a non conformally flat space

complete sets of invariants for dynamical systems that admit a separation of variables consider a classical hamiltonian h in n dimensions consisting of a kinetic energy term plus a potential if the associated hamilton jacobi equation admits an orthogonal separation of variables then it is possible to generate algorithmically a canonical basis q p where p1 h p2 pn are the other second order constants of the motion associated with the separable coordinates and qi qj pi pj 0 qi pj ij the 2n 1 functions q2 qn p1 pn form a basis for the invariants we show how to determine for exactly which spaces and potentials the invariant qj is a polynomial in the original momenta we shed light on the general question of exactly when the hamiltonian admits a constant of the motion that is polynomial in the momenta for n 2 we go further and consider all cases where the hamilton jacobi equation admits a second order constant of the motion not necessarily associated with orthogonal separable coordinates or even separable coordinates at all in each of these cases we construct an addition

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

transformation of the stackel matrices preserving superintegrability if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions if we take a superintegrable stackel system and make variables faster or slower that is equivalent to a trivial transformation of the stackel matrix and potentials then we obtain an infinite family of superintegrable systems with explicitly defined additional integrals of motion we present some examples of such transformations associated with angle variables expressed via logarithmic functions

an infinite family of solvable and integrable quantum systems on a plane an infinite family of exactly solvable and integrable potentials on a plane is introduced it is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family the underlying algebraic structure of the new potentials is revealed as well as its hidden algebra we conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases a quasi exactly solvable and integrable generalization of the family is found

families of superintegrable hamiltonians constructed from exceptional polynomials we introduce a family of exactly solvable two dimensional hamiltonians whose wave functions are given in terms of laguerre and exceptional jacobi polynomials the hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems additional higher order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable

third order superintegrable systems separating in polar coordinates a complete classification of quantum and classical superintegrable systems in e2 is presented that allow the separation of variables in polar coordinates and admit an additional integral of motion of order 3 in the momentum new quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth painleve transcendent or in terms of the weierstrass elliptic function

absolute integral closures are big cohen macaulay algebras in characteristic p throughout this paper ring means commutative ring with identity and modules are unital our main interest is in local rings i e noetherian rings r m with a unique maximal ideal m in such a ring xx xn e m is a system of parameters if m has a power in the ideal x xn r and n is the krull dimension of r when r is complete and contains a field this means that r is module finite over a formal power series subring k xx xn a i is called cohen macaulay if some equivalently every system of parameters is a regular sequence in i which means that every xi x is a nonzerodivisor on r xx xt r for 0 i n 1 in the case where r is module finite over the formal power series subring a this means that r is a free 4 module for many theorems of commutative algebra and of algebraic geometry the cohen macaulay condition possibly on the local rings of a variety is just what is needed to make the theory work our main result which is given below in an algebraic form in theorem 1 and in a geometric form in theorem 2 asserts that under mild conditions on a local ring r of positive prime characteristic p one can correct the failure of the cohen macaulay condition in r itself by passing to a very large integral extension of r it is worth emphasizing that this is quite false in characteristic 0 and comes as a surprise we believe in characteristic p one can get an idea of how far these theorems are from the truth over a field of characteristic 0 from the following observation in the situation of theorem 2 when x is projectively normal the maps of cohomology are always injective by an easy trace argument despite the fact that both theorem 1 and theorem 2

character varieties of free groups are gorenstein but not always factorial abstract fix a rank g free group f g and a connected reductive complex algebraic group g let x f g g be the g character variety of f g when the derived subgroup d g g is simply connected we show that x f g g is factorial which implies it is gorenstein and provide examples to show that when dg is not simply connected x f g g need not even be locally factorial despite the general failure of factoriality of these moduli spaces using different methods we show that x f g g is always gorenstein

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

hankel determinantal rings have rational singularities abstract hankel determinantal rings i e determinantal rings defined by minors of hankel matrices of indeterminates arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves they may also be viewed as linear specializations of generic determinantal rings we prove that over fields of characteristic zero hankel determinantal rings have rational singularities in the case of positive prime characteristic we prove that they are f pure independent of the characteristic we give a complete description of the divisor class groups of these rings and show that each divisor class group element is the class of a maximal cohen macaulay module

quasi bi hamiltonian structures complex functions and superintegrability the tremblay turbiner winternitz ttw and the post winternitz pw systems the existence of quasi bi hamiltonian structures for the tremblay turbiner winternitz ttw and the post winternitz pw systems is studied we first recall that the superintegrability of these two systems is related with the existence of certain complex functions endowed with interesting poisson bracket properties and then we prove the existence of several quasi bi hamiltonian structures making use of these complex functions this is done in two steps first with complex 2 forms wedge product of the differentials of the complex functions and then with several real 2 forms the properties of these geometric structures and the associated recursion operators are also analyzed the paper can be considered as divided in two parts in the first part a we study the ttw system related to the harmonic oscillator and in the second part we study the pw system related to the kepler problem

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

generalized theta functions strange duality and odd orthogonal bundles on curves this paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial stiefel whitney class and the associated space of twisted spin bundles in particular we prove a verlinde type formula and a dimension equality that was conjectured by oxbury wilson modifying hitchin s argument we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection we furthermore address the issue of strange duality for odd orthogonal bundles and we demonstrate that the naive conjecture fails in general a consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the hitchin connection for these moduli spaces finally we answer a question of nakanishi tsuchiya about rank level duality for conformal blocks on the pointed projective line with spin weights

periodic orbits for an infinite family of classical superintegrable systems we show that all bounded trajectories in the two dimensional classical system with the potential are closed for all integer and rational values of k the period is and does not depend on k this agrees with our earlier conjecture suggesting that the quantum version of this system is superintegrable

theta characteristics on algebraic curves the theory of theta characteristics is developed algebraically so that it may be applied to possibly singular and or reducible algebraic curves the configuration of theta characteristics on a curve is described in terms of its singularities with applications to the geometry of plane quartic curves and the problem of appolonius some results on gorenstein local rings are appended 0 introduction the theory of theta characteristics has been studied from different points of view and often inadvertently cf for example the 19th century literature on bitangents to smooth plane quartic curves e g 4 and 10 until recently however virtually all treatments of theta characteristics per se involved the transcendental theory of the theta function associated to a compact riemann surface then in 1971 mumford in 8 gave a completely algebraic treatment of the subject formulating the main results and proving them without reference to the transcendental theory this in turn suggested that theta characteristics could be studied in a more general context for example on possibly singular and or reducible algebraic curves in this paper we undertake such an analysis in 1 following mumford we establish some of the basic facts about theta characteristics algebraically so as to apply to more general curves in fact 1 10 i and 1 13 in their present generality are due to mumford while he assumes throughout 8 that his curve x is smooth his proof seems to apply as given the argument for 1 13 here is somewhat different in order to make the proof more elementary and selfcontained one bonus is the result 1 11 on the subvariety of the moduli space 9thg consisting of curves with large semicanonical linear series this fact seemingly can be proved only via the algebraic set up in 2 we investigate the number and configuration of odd and even theta characteristics on a singular curve the results are expressed in the two main theorems 2 12 and 2 22 giving a more or less complete answer to these questions in terms of the singularities of the curve interestingly a recurrent motif in these arguments is one introduced by mumford in 8 the behavior of the maximal isotropic subspaces for a nondegenerate quadratic form on an even dimensional vector space here the mainspring of theorems 1 13 2 12 and 2 22 is assertion 1 4 below that the sum of the dimensions of the pairwise intersections of any three such subspaces is congruent to n mod 2 received by the editors july 11 1980 and in revised form october 22 1981 1980 mathematics subject classification primary 14h99 14h30 secondary 14h20 14h40 14k25 research partially supported by nsf grant mcs78 04008 aol dl 982 american mathematical society 0002 9947 8 1 0000 0371 10 25

representation growth and rational singularities of the moduli space of local systems let g be a semisimple algebraic group defined over mathbb q p and let gamma be a compact open subgroup of g mathbb q p we relate the asymptotic representation theory of gamma and the singularities of the moduli space of g local systems on a smooth projective curve proving new theorems about both 1 we prove that there is a constant c independent of g such that the number of n dimensional representations of gamma grows slower than n c confirming a conjecture of larsen and lubotzky in fact we can take c 3 cdot text dim e 8 1 745 we also prove the same bounds for groups over local fields of large enough characteristic 2 we prove that the coarse moduli space of g local systems on a smooth projective curve of genus at least lceil c 2 rceil 1 374 has rational singularities for the proof we study the analytic properties of push forwards of smooth measures under algebraic maps more precisely we show that such push forwards have continuous density if the algebraic map is flat and all of its fibers have rational singularities

exact solvability of superintegrable systems it is shown that all four superintegrable quantum systems on the euclidean plane possess the same underlying hidden algebra sl 3 the gauge rotated hamiltonians as well as their integrals of motion once rewritten in appropriate coordinates preserve a flag of polynomials this flag corresponds to highest weight finite dimensional representations of the sl 3 algebra realized by first order differential operators