a continuum of relative equilibria in the five body problem abstract it is generally believed that the set of relative equilibria equivalence classes in the newtonian n body problem for a given set of positive masses is finite however the result has only been proven for n 3 and remains a difficult open question for n 4 a wintner princeton math ser 5 215 princeton university press princeton nj 1941 s smale mathematical intelligencer 20 1998 7 15 we demonstrate that the condition for the masses being positive is a necessary one by finding a continuum of relative equilibria in the five body problem which unfortunately includes one negative mass this family persists in similar potential functions including the logarithmic potential used to describe the motion of point vortices in a plane of fluid

elliptic surfaces and linear systems with fat points we investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces for the projectivization of the atiyah bundle over an elliptic curve with a certain polarization we observe that the special case of only one fat point implies the general case of arbitrarily many fat points as well as results concerning other surfaces we conjecture that this special case holds in characteristic 0 but prove that it fails in any positive characteristic

convex central configurations of the 4 body problem with two pairs of equal adjacent masses we study the convex central configurations of the 4 body problem assuming that they have two pairs of equal masses located at two adjacent vertices of a convex quadrilateral under these assumptions we prove that the isosceles trapezoid is the unique central configuration

dziobek configurations of the restricted n 1 body problem with equal masses in this article we count the number of n 1 dimensional central configurations of the restricted n 1 body problem with equal masses for n 4 5 there are 25 56 central configurations for n 3 or n 6 the number of central configurations is 2n n 1

gromov witten invariants in algebraic geometry gromov witten invariants for arbitrary non singular projective vari eties and arbitrary genus are constructed using the techniques from k behrend b fantechi the intrinsic normal cone

the point set method front tracking without connectivity abstract we describe a point set method for extracting the normal curvature and surface area from unordered data points residing on a surface this capability relaxes front tracking s reliance on connectivity between interfacial points and allows front tracking to model topological changes at an interface naturally we use this capability in 2d and 3d front tracking calculations to model coalescence we also describe a simple projection scheme which allows us to suppress parasitic currents in front tracking models

central configurations of the n body problem via equivariant morse theory in this paper we use the equivariant morse theory to give an estimate of the minimal number of central configurations in the n body problem in 3 in the case of equal masses we prove that the planar central configurations are saddle points for the potential energy from this we deduce the presence of non planar central configurations for every n 4

front tracking for shear bands in an antiplane shear model in this paper we describe a numerical algorithm for the study of shear band formation and growth in two dimensional antiplane shear the constitutive model uses a non associative flow rule the numerical scheme is based on a godunov method for updating the velocity while the stress is updated via integration along particle paths the scheme is coupled with a front tracking algorithm for careful evolution of the shear bands the main challenges are the non hyperbolicity of the shear band formation and growth front tracking avoids the catastrophic effects of the loss of hyperbolicity in the godunov type numerical scheme the existence of endpoints for the shear band the tracked front does not separate the computational domain into disconnected regions and the non hyperbolic rate of growth of the shear band we give examples of the success of the algorithm and show convergence tests

bicentric quadrilateral central configurations abstract a bicentric quadrilateral is a tangential cyclic quadrilateral in a tangential quadrilateral the four sides are tangents to an inscribed circle and in a cyclic quadrilateral the four vertices lie on a circumscribed circle in this paper we classify all planar central configurations of the 4 body problem where the four bodies are at the vertices of a bicentric quadrilateral

multidimensional upwind methods for hyperbolic conservation laws abstract we present a class of second order conservative finite difference algorithms for solving numerically time dependent problems for hyperbolic conservation laws in several space variables these methods are upwind and multidimensional in that the numerical fluxes are obtained by solving the characteristic form of the full multidimensional equations at the zone edge and that all fluxes are evaluated and differenced at the same time in particular operator splitting is not used correct behavior at discontinuities is obtained by the use of solutions to the riemann problem and by limiting some of the second order terms numerical results are presented which show that the methods described here yield the same high resolution as the corresponding operator split methods

numerical algorithms for strong discontinuities in elastic plastic solids abstract in this paper the implementation of second order godunov methods for dynamic wave propagation in one dimensional elastic plastic solids is investigated first the lagrangian form of the algorithm is reviewed and then the algorithm is extended to the eulerian frame of reference this extension requires additional evolution equations to handle the history of the material along particle paths both the lagrangian and eulerian versions of the algorithm require appropriately accurate approximations to the solution of riemann problems in order to represent the interaction of waves at cell boundaries two inexpensive approximations to the solution of the riemann problem are constructed and the resulting algorithms are tested against the analytic solution of the riemann problem for longitudinal motion in an elastic plastic bar these approximations to the riemann problem are shown to work well even for strong discontinuities finally the numerical experience gained from the simple longitudinal bar problem is used to design an algorithm for strong shocks predicted by a realistic soil model

on the noether lefschetz theorem and some remarks on codimension two cycles here of general moduli means that there is a countable union v of subvarieties of the space pn of surfaces of degree d in p3 such that the statement pie s z holds for s p n v noether it would seem stated this theorem but never completely proved it instead he gave a plausibility argument based on the following construction 1 let g be the hilbert scheme of curves of degree n and genus 9 in p3 l gr3 d l ip n the space of surfaces of degree d in p3 and zc t the incidence correspondence s o a c s c z s

holomorphic anomaly equations and the igusa cusp form conjecture let s be a k3 surface and let e be an elliptic curve we solve the reduced gromov witten theory of the calabi yau threefold s times e for all curve classes which are primitive in the k3 factor in particular we deduce the igusa cusp form conjecture the proof relies on new results in the gromov witten theory of elliptic curves and k3 surfaces we show the generating series of gromov witten classes of an elliptic curve are cycle valued quasimodular forms and satisfy a holomorphic anomaly equation the quasimodularity generalizes a result by okounkov and pandharipande and the holomorphic anomaly equation proves a conjecture of milanov ruan and shen we further conjecture quasimodularity and holomorphic anomaly equations for the cycle valued gromov witten theory of every elliptic fibration with section the conjecture generalizes the holomorphic anomaly equations for elliptic calabi yau threefolds predicted by bershadsky cecotti ooguri and vafa we show a modified conjecture holds numerically for the reduced gromov witten theory of k3 surfaces in primitive classes

central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 m 0 and m small in this paper we give a complete description of the families of central configurations of the planar 4 body problem with two pairs of equals masses and two equal masses sufficiently small in particular we give an analytical proof that this particular 4 body problem has exactly 34 different classes of central configurations moreover for this problem we prove the following two conjectures there is a unique convex planar central configuration of the 4 body problem for each ordering of the masses in the boundary of its convex hull which appears in albouy and fu 2007 3 we also prove the conjecture there is a unique convex planar central configuration having two pairs of equal masses located at the adjacent vertices of the configuration and it is an isosceles trapezoid finally the families of central configurations of this 4 body problem are numerically continued to the 4 body problem with four equal masses

central configurations formed by two twisted regular polygons abstract in this paper we study the necessary conditions and sufficient conditions for the central configurations formed by two twisted regular polygons one n regular polygon and one l regular polygon we wish to extend the results of the symmetrical central configurations formed by two twisted n regular polygons however to our own surprise it will be proved that the necessary condition on the central configurations is l n

open gromov witten invariants on elliptic k3 surfaces and wall crossing in this paper we study holomorphic discs in k3 surfaces and defined the open gromov witten invariants using this new invariant we can establish a version of correspondence between tropical discs and holomorphic discs with non trivial invariants we give an example of wall crossing phenomenon of the invariant and expect it satisfies kontsevich soibelman wall crossing formula

a1 curves on log k3 surfaces abstract in this paper we study a 1 curves on log k3 surfaces we classify all genuine log k3 surfaces of type ii which admit countably infinite a 1 curves

finiteness of central configurations of five bodies in the plane we prove there are finitely many isometry classes of planar central configurations also called relative equilibria in the newtonian 5 body problem except perhaps if the 5 tuple of positive masses belongs to a given codimension 2 subvariety of the mass space

on stacked central configurations with n bodies when one body is removed abstract a stacked central configuration is a central configuration of the n body problem for which a proper subset of the bodies is already in a central configuration in this paper we study all the possibilities of stacked central configurations for the n body problem when one body is removed the results have simple and analytic proofs

a higher order godunov method for modeling finite deformation in elastic plastic solids in this paper we develop a first order system of conservation laws for finite deformation in solids describe its characteristic structure and use this analysis to develop a second order numerical method for problems involving finite deformation and plasticity the equations of mass momentum and energy conservation in lagrangian and eulerian frames of reference are combined with kinetic equations of state for the stress and with caloric equations of state for the internal energy as well as with auxiliary equations representing equality of mixed partial derivatives of the deformation gradient particular attention is paid to the influence of a curl constraint on the deformation gradient so that the characteristic speeds transform properly between the two frames of reference next we consider models in rate form for isotropic elastic plastic materials with work hardening and examine the circumstances under which these models lead to hyperbolic systems for the equations of motion in spite of the fact that these models violate thermodynamic principles in such a way that the acoustic tensor becomes nonsymmetric we still find that the characteristic speeds are always real for elastic behavior and essentially always real for plastic response these results allow us to construct a second order godunov method for the computation of three dimensional displacement in a one dimensional material viewed in the lagrangian frame of reference we also describe a technique for the approximate solution of riemann problems in order to determine numerical fluxes in this algorithm finally we present numerical examples of the results of the algorithm

the symplectic sum formula for gromov witten invariants in the symplectic category there is a connect sum operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds this paper establishes a formula for the gromov witten invariants of a symplectic sum z x y in terms of the relative gw invariants of x and y several applications to enumerative geometry are given gromov witten invariants are counts of holomorphic maps into symplectic manifolds to define them on a symplectic manifold x w one introduces an almost complex structure j compatible with the symplectic form w and

rational curves in holomorphic symplectic varieties and gromov witten invariants abstract we use gromov witten theory to study rational curves in holomorphic symplectic varieties we present a numerical criterion for the existence of uniruled divisors swept out by rational curves in the primitive curve class of a very general holomorphic symplectic variety of k 3 n type we also classify all rational curves in the primitive curve class of the fano variety of lines in a very general cubic 4 fold and prove the irreducibility of the corresponding moduli space our proofs rely on gromov witten calculations by the first author and in the fano case on a geometric construction of voisin in the fano case a second proof via classical geometry is sketched

singular continuation of planar central configurations with clusters of bodies abstract planar central configurations of point masses that have one or more clusters of bodies are created by analytic continuation the singularity of the gravitational interaction is removed from the continuation equations by algebraic means the continuation splits a single body into several and the initial mass of the single body can be nonzero necessary conditions are derived for these continuations that may be useful in addressing the question of finiteness of central configurations

density of rational curves on k3 surfaces using the dynamics of self rational maps of elliptic k3 surfaces together with deformation theory we prove that the union of rational curves is dense on a very general k3 surface and that the union of elliptic curves is dense in the 1st jet space of a very general k3 surface both in the strong topology

central configurations with many small masses abstract by using the method of analytical continuation we find the exact numbers of central configurations for some open sets of n positive masses for any choice of n it turns out that the numbers increase dramatically as n increases e g for some open set of 18 positive masses some 2 08766 1020 classes of distinctive central configurations are found in the mean time we obtained some results about the hausdorff measure for the set of n positive masses where degenerate central configuration arises

generic finiteness for dziobek configurations the goal of this paper is to show that for almost all choices of n masses mi there are only finitely many central configurations of the newtonian n body problem for which the bodies span a space of dimension n 2 such a central configuration is called a dziobek configuration the result applies in particular to two dimensional configurations of four bodies and threedimensional configurations of five bodies

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

some notes on four body co circular central configurations abstract there are two symmetric families of four body co circular central configurations namely the kite and isosceles trapezoid using mutual distances as coordinates we prove that if the four body central configuration is an isosceles trapezoid then the diagonals of the isosceles trapezoid cannot be perpendicular to each other furthermore we show that for any four body co circular central configuration the diagonals of the quadrilateral cannot be perpendicular except that the configuration is a kite

projective degenerations of k3 surfaces gaussian maps and fano threefolds in this article we exhibit certain projective degenerations of smoothk3 surfaces of degree 2g 2 in g whose picard group is generated by the hyperplane class to a union of two rational normal scrolls and also to a union of planes as a consequence we prove that the general hyperplane section of suchk3 surfaces has a corank one gaussian map ifg 11 org 13 we also prove that the general such hyperplane section lies on a uniquek3 surface up to projectivities finally we present a new approach to the classification of prime fano threefolds of index one which does not rely on the existence of a line

genus zero gopakumar vafa type invariants for calabi yau 4 folds abstract in analogy with the gopakumar vafa conjecture on cy 3 folds klemm and pandharipande defined gv type invariants on calabi yau 4 folds using gromov witten theory and conjectured their integrality in this paper we propose a sheaf theoretic interpretation of their genus zero invariants using donaldson thomas theory on cy 4 folds more specifically we conjecture genus zero gv type invariants are d t 4 invariants for one dimensional stable sheaves on cy 4 folds some examples are computed for both compact and non compact cy 4 folds to support our conjectures we also propose an equivariant version of the conjectures for local curves and verify them in certain cases

constructing rational curves on k3 surfaces we develop a mixed characteristic version of the morimukai technique for producing rational curves on k3 surfaces we reduce modulo p produce rational curves on the resulting k3 surface over a finite field and lift to characteristic zero as an application we prove that all complex k3 surfaces with picard group generated by a class of degree two have an infinite number of rational curves

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

algebraic cycles and mumford griffiths invariants let x be a projective algebraic manifold and let ch r x be the chow group of algebraic cycles of codimension r on x modulo rational equivalence working with a candidate bloch beilinson filtration f 0 on ch r x q due to the second author we construct a space of arithmetic hodge theoretic invariants j r x and corresponding map r x gr f ch r x q j r x and determine conditions on x for which the kernel and image of r x are uncountably large

existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi homogeneous n body problems and the curved n body problem we prove that if for relative equilibrium solutions of a generalisation of quasi homogeneous n body problems the masses and rotation are given then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero we furthermore prove that the set of such relative equilibria is compact and prove related results for n body problems in spaces of constant gaussian curvature

the symmetric central configurations of the 4 body problem with masses m1 m2 m3 m4 we characterize the planar central configurations of the 4 body problem with masses m 1 m 2 m 3 m 4 which have an axis of symmetry it is known that this problem has exactly two classes of convex central configurations one with the shape of a rhombus and the other with the shape of an isosceles trapezoid we show that this 4 body problem also has exactly two classes of concave central configurations with the shape of a kite this proof is assisted by computer

counting rational curves on text k3 surfaces the aim of these notes is to explain the remarkable formula found by yau and zaslow y z to express the number of rational curves on a k3 surface projective k3 surfaces fall into countably many families fg g 1 a surface in fg admits a gdimensional linear system of curves of genus g a naive count of constants suggests that such a system will contain a positive number say n g of rational highly singular curves the formula is

a conjectural generating function for numbers of curves on surfaces i give a conjectural generating function for the numbers of nodal curves in a linear system of dimension on an algebraic surface it reproduces the results of vainsencher v for the case le 6 and kleiman piene k p for the case le 8 the numbers of curves are expressed in terms of five universal power series three of which i give explicitly as quasimodular forms this gives in particular the numbers of curves of arbitrary genus on a k3 surface and an abelian surface in terms of quasimodular forms generalizing the formula of yau zaslow for rational curves on k3 surfaces the coefficients of the other two power series can be determined by comparing with the recursive formulas of caporaso harris for the severi degrees in 2 we verify the conjecture for genus 2 curves on an abelian surface we also discuss a link of this problem with hilbert schemes of points

rational curves on k3 surfaces we show that projective k3 surfaces with odd picard rank contain infinitely many rational curves our proof extends the bogomolov hassett tschinkel approach i e uses moduli spaces of stable maps and reduction to positive characteristic

macmahon s sum of divisors functions chebyshev polynomials and quasi modular forms we investigate a relationship between macmahon s generalized sum of divisors functions and chebyshev polynomials of the rst kind this determines a recurrence relation to compute these functions as well as proving a conjecture of macmahon about their general form by relating them to quasimodular forms these functions arise as solutions to a curve counting problem on abelian surfaces

rational curves and points on k3 surfaces we study the distribution of algebraic points on k3 surfaces 1 introduction let k be a field and k a fixed algebraic closure of k we are interested in connections between geometric properties of algebraic varieties and their arithmetic properties over k over its finite extensions k k or over k here we study certain varieties of intermediate type namely k3 surfaces and their higher dimensional generalizations calabi yau varieties to motivate the following discussion let 5 be a k3 surface over k in positive characteristic 5 may be unirational and covered by rational curves examples are supersingular k3 surfaces over fields of characteristic two or the surface

angular momenta of relative equilibrium motions and real moment map geometry chenciner and jimenez perez mosc math j 13 4 621 630 2013 showed that the range of the spectra of the angular momenta of all the rigid motions of a fixed central configuration in a general euclidean space form a convex polytope in this note we explain how this result follows from a general convexity theorem of o shea and sjamaar in real moment map geometry math ann 31 415 457 2000 finally we provide a representation theoretic description of the pushforward of the normalized measure under the real moment map for riemannian symmetric pairs