categories of fi type a unified approach to generalizing representation stability and character polynomials abstract representation stability is a theory describing a way in which a sequence of representations of different groups is related and essentially contains a finite amount of information starting with church ellenberg farb s theory of fi modules describing sequences of representations of the symmetric groups we now have good theories for describing representations of other collections of groups such as finite general linear groups classical weyl groups and wreath products s n g for a fixed finite group g this paper attempts to uncover the mechanism that makes the various examples work and offers an axiomatic approach that generates the essentials of such a theory character polynomials and free modules that exhibit stabilization we give sufficient conditions on a category c to admit such structure via the notion of categories of fi type this class of categories includes the examples listed above and extends further to new types of categories such as the categorical power fi m whose modules encode sequences of representations of m fold products of symmetric groups the theory is applied in 9 to give homological and arithmetic stability theorems for various moduli spaces e g the moduli space of degree n rational maps p 1 p m

a remarkable periodic solution of the three body problem in the case of equal masses using a variational method we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane the orbit has zero angular momentum and a very rich symmetry pattern its most surprising feature is that the three bodies chase each other around a flxed eight shaped curve setting aside collinear motions the only other known motion along a flxed curve in the inertial plane is the lagrange relative equilibrium in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle our orbit visits in turns every euler conflguration in which one of the bodies sits at the midpoint of the segment deflned by the other two figure 1 numerical computations

asymptotic resonance interaction of modes and subharmonic bifurcation we study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems which mix different normal modes the reference problem is the fermi pasta ulam model a chain of nonlinear oscillators with nearest neighborhood interaction we develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches two sufficient conditions for bifurcation are given one involves only the arithmetic properties of the eigenvalues of the linearized system asymptotic resonance while the other takes into account the nonlinear character of the interaction between normal modes nonlinear coupling both conditions are checked for the fermi pasta ulam problem

coincidences between homological densities predicted by arithmetic abstract motivated by analogies with basic density theorems in analytic number theory we introduce a notion and variations of the homological density of one space in another we use weil s number field function field analogy to predict coincidences for limiting homological densities of various sequences z n d 1 d m x of spaces of 0 cycles on manifolds x the main theorem in this paper is that these topological predictions which seem strange from a purely topological viewpoint are indeed true one obstacle to proving such a theorem is the combinatorial complexity of all possible collisions of points this problem does not arise in the simplest and classical case m n 1 2 of configuration spaces to overcome this obstacle we apply the bjorner wachs theory of lexicographic shellability from algebraic combinatorics

global bifurcation of polygonal relative equilibria for masses vortices and dnls oscillators given a regular polygonal arrangement of identical objects turning around a central object masses vortices or dnls oscillators this paper studies the global bifurcation of relative equilibria in function of a natural parameter central mass central circulation or amplitude of the oscillation the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of a degree theory the symmetries of the bifurcated solutions

global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n body problem abstract given n point masses turning in a plane at a constant speed this paper deals with the global bifurcation of periodic solutions for the masses in that plane and in space as a special case one has a complete study of n identical masses on a regular polygon and a central mass the symmetries of the problem are used in order to find the irreducible representations the linearization and with the help of the orthogonal degree theory all the symmetries of the bifurcating branches

pieri resolutions for classical groups abstract we generalize the constructions of eisenbud floystad and weyman for equivariant minimal free resolutions over the general linear group and we construct equivariant resolutions over the orthogonal and symplectic groups we also conjecture and provide some partial results for the existence of an equivariant analogue of boij soderberg decompositions for betti tables which were proven to exist in the non equivariant setting by eisenbud and schreyer many examples are given

action minimizing orbits in the parallelogram four body problem with equal masses in this paper we prove the existence of a new periodic solution for the planar newtonian four body problem with equal masses on this orbit two mass points travel on one star shaped closed curve while the other two travel on another and have the opposite orientation the configuration of the masses changes from square to collinear periodically and remains a parallelogram for all time

spatial stability of 3d exterior stationary navier stokes flows abstract in this paper we study the stability of stationary solutions w for the navier stokes flows in an exterior domain with zero velocity at infinity with suitable assumptions of w by the works of chen 1993 kozono ogawa 1994 and borchers miyakawa 1995 if u 0 w l r l 3 then one can obtain u t w p o t 3 2 1 r 1 p for 1 r p u t w p o t 3 2 1 r 1 p 1 2 for 1 r p 3 where u x t is a solution of the navier stokes equations with the initial condition u 0 in this paper we will prove that for any 0 3 if x u 0 w belongs to l r then one has x u t w l p o t 3 2 1 r 1 p 2 for p 3 r 3 r

the structure of the tautological ring in genus one we prove getzler s claims about the cohomology of the moduli space of stable curves of genus one that is that the even cohomology ring is spanned by the strata classes and that all relations between these classes follow from the witten dijkgraaf verlinde verlinde wdvv relation and getzler s relation in particular the even cohomology ring is isomorphic to the tautological ring

simple choreographies of the planar newtonian n body problem in the n body problem a simple choreography is a periodic solution where all masses chase each other on a single loop in this paper we prove that for the planar newtonian n body problem with equal masses n 3 there are at least 2n 3 2 n 3 2 different main simple choreographies this confirms a conjecture given by chenciner et al geometry mechanics and dynamics springer new york pp 287 308 2002 all the simple choreoagraphies we prove belong to the linear chain family

stability in the homology of congruence subgroups the homology groups of many natural sequences of groups g n n 1 infty e g general linear groups mapping class groups etc stabilize as n rightarrow infty indeed there is a well known machine for proving such results that goes back to early work of quillen church and farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations they called this phenomena representation stability we prove that the homology groups of congruence subgroups of mathrm gl n r for almost any reasonable ring r satisfy a strong version of representation stability that we call central stability the definition of central stability is very different from church farb s definition of representation stability it is defined via a universal property but we prove that it implies representation stability our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine

variational proof of the existence of the super eight orbit in the four body problem using the variational method chenciner and montgomery ann math 152 881 901 2000 proved the existence of an eight shaped periodic solution of the planar three body problem with equal masses just after the discovery gerver numerically found a similar periodic solution called super eight in the planar four body problem with equal mass in this paper we prove the existence of the super eight orbit by using the variational method the difficulty of the proof is to eliminate the possibility of collisions in order to solve it we apply the scaling technique established by tanaka ann inst h poincare anal non lineaire 10 215 238 1993 proc am math soc 122 275 284 1994 and investigate the asymptotic behavior of a binary collision

standing waves in a counter rotating vortex filament pair abstract the distance among two counter rotating vortex filaments satisfies a beam type of equation according to the model derived in 15 this equation has an explicit solution where two straight filaments travel with constant speed at a constant distance the boundary condition of the filaments is 2 periodic using the distance of the filaments as bifurcating parameter an infinite number of branches of periodic standing waves bifurcate from this initial configuration with constant rational frequency along each branch

e_n cell attachments and a local to global principle for homological stability we define bounded generation for e n algebras in chain complexes and prove that this property is equivalent to homological stability for n ge 2 using this we prove a local to global principle for homological stability which says that if an e n algebra a has homological stability or equivalently the topological chiral homology int mathbb r n a has homology stability then so has the topological chiral homology int m a of any connected non compact manifold m using scanning we reformulate the local to global homological stability principle so that it applies to compact manifolds we also give several applications of our results

multiple periodic solutions for γ symmetric newtonian systems abstract the existence of periodic solutions in symmetric newtonian systems x f x can be effectively studied by means of the o 2 equivariant gradient degree with values in the euler ring u o 2 in this paper we show that in the case of being a finite group the euler ring u o 2 and the related basic degrees are effectively computable using euler ring homomorphisms the burnside ring a o 2 and the reduced o 2 degree with no free parameters we present several examples of newtonian systems with various symmetries for which we show existence of multiple periodic solutions we also provide exact value of the equivariant topological invariant for those problems

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

linear and quadratic ranges in representation stability abstract we prove two general results concerning spectral sequences of fi modules these results can be used to significantly improve stable ranges in a large portion of the stability theorems for fi modules currently in the literature we work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range previously the best stable ranges known in these examples were exponential up to an additive constant our work on congruence subgroups verifies a conjecture of djament

interpolation between hp spaces the complex method the corollary follows immediately from theorem 1 and fefferman s theorem see 111 that bmo is the dual space of h before proceeding to further results we make some comments on the history of theorem 1 unfortunately the theorem h lm e lp l p 1 0 has been folklore for many years and is stated e g in 2 221 this is due to a small but inopportune typographical error in 111 it is stated on page 157 of that paper that h lp lq l q 1 8 e p 1 p co

new periodic solutions for 2n body problems in r3 for newtonian 2n body problems with equal masses in r3 we prove the existence of new noncollision periodic solution such that 2n bodies move on two different closed curves

representation stability for families of linear subspace arrangements abstract church ellenberg farb 8 used the language of fi cha to identify certain sequences of hyperplane arrangements with s n actions that satisfy cohomological representation stability here we vastly extend their results and define when a collection of arrangements is finitely generated using this notion we get stability results to general linear subspace arrangements not necessarily of hyperplanes a wide class of group actions replacing fi by a general category c we show that the cohomology of such collections of arrangements satisfies a strong form of representation stability with many concrete applications for example this implies that their betti numbers are always given by certain polynomials for this purpose we use the theory of representation stability for quite general classes of groups developed in a 17 we apply this theory to get classical cohomological stability of quotients of linear subspace arrangements with coefficients in certain constructible sheaves

symmetry groups and non planar collisionless action minimizing solutions of the three body problem in three dimensional space periodic and quasi periodic solutions of the n body problem can be found as minimizers of the lagrangian action functional restricted to suitable spaces of symmetric paths the main purpose of this paper is to develop a systematic approach to the equivariant minimization for the three body problem in three dimensional space first we give a finite complete list of symmetry groups fitting to the minimization of the action with the property that any other symmetry group can be reduced to be isomorphic to one of these representatives a second step is to prove that the resulting local and global symmetric action minimizers are always collisionless when they are not already bound to collisions furthermore we prove some results which address the question of whether minimizers are planar or non planar as a consequence of our theory we will give general criteria for a symmetry group to yield planar or homographic minimizers either homographic or not as in the chenciner montgomery eight solution on the other hand we will provide a rigorous proof of the existence of some interesting one parameter families of periodic and quasi periodic non planar orbits these include the choreographic marchal s p12 family with equal masses together with a less symmetric choreographic family which anyway probably coincides with the p12 family

binary decompositions for planar n body problems and symmetric periodic solutions a binary decomposition for a system of n masses is a way of treating the system as big n atop 2 big binaries with the total action exactly the same as that of the original system by considering binary decompositions we are able to provide effective lower bound estimates for the action of collision paths in several spaces of symmetric loops as applications we use our estimates to prove the existence of some new classes of symmetric periodic solutions for the n body problem

symmetric trajectories for the 2n body problem with equal masses we consider the problem of 2n bodies of equal masses in mathbb r 3 for the newtonian like weak force potential r and we prove the existence of a family of collision free nonplanar and nonhomographic symmetric solutions that are periodic modulo rotations in addition the rotation number with respect to the vertical axis ranges in a suitable interval these solutions have the hip hop symmetry a generalization of that introduced in 19 for the case of many bodies and taking account of a topological constraint the argument exploits the variational structure of the problem and is based on the minimization of lagrangian action on a given class of paths

entire minimal parabolic trajectories the planar anisotropic kepler problem we continue the variational approach to parabolic trajectories introduced in our previous paper barutello et al entire parabolic trajectories as minimal phase transitions arxiv 1105 3358v1 2011 which sees parabolic orbits as minimal phase transitions we deepen and complete the analysis in the planar case for homogeneous singular potentials we characterize all parabolic orbits connecting two minimal central configurations as free time morse minimizers in a given homotopy class of paths these may occur for at most one value of the homogeneity exponent in addition we link this threshold of existence of parabolic trajectories with the absence of collisions for all the minimizers of fixed end problems and also with the existence of action minimizing periodic trajectories with nontrivial homotopy type

existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses poincare made the first attempt in 1896 on applying variational calculus to the three body problem and observed that collision orbits do not necessarily have higher values of action than classical solutions little progress had been made on resolving this difficulty until a recent breakthrough by chenciner and montgomery afterward variational methods were successfully applied to the jv body problem to construct new classes of solutions in order to avoid collisions the problem is confined to symmetric path spaces and all new planar solutions were constructed under the assumption that some masses are equal a question for the variational approach on planar problems naturally arises are minimizing methods useful only when some masses are identical this article addresses this question for the three body problem for various choices of masses it is proved that there exist infinitely many solutions with a certain topological type called retrograde orbits that minimize the action functional on certain path spaces cases covered in our work include triple stars in retrograde motions double stars with one outer planet and some double stars with one planet orbiting around one primary mass our results largely complement the classical results by the poincare continuation method and conley s geometric approach

transitive decomposition of symmetry groups for the n body problem periodic and quasi periodic solutions of the n body problem are critical points of the action functional constrained to the sobolev space of symmetric loops variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions such as the rotating circle property here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis by a decomposition into irreducible transitive components as examples we show approximate trajectories of some of the resulting symmetric minimizers

motivic random variables and representation stability ii hypersurface sections abstract we prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity we show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed grothendieck ring of varieties and deduce from this the stabilization of the hodge euler characteristic of natural families of local systems constructed from the vanishing cohomology we prove explicit formulas for the stable values using a probabilistic interpretation along with the natural analogs in point counting over finite fields we explain how these results provide new geometric examples of a weak version of representation stability for symmetric symplectic and orthogonal groups this interpretation of representation stability was studied in the prequel 20 for configuration spaces

on central configurations this paper concerns an old problem which arose in celestial mechanics and has since taken on a life of its own the probem of central configurations of the n body problem

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

on the existence of collisionless equivariant minimizers for the classical n body problem we show that the minimization of the lagrangian action functional on suitable classes of symmetric loops yields collisionless periodic orbits of the n body problem provided that some simple conditions on the symmetry group are satisfied more precisely we give a fairly general condition on symmetry groups g of the loop space for the n body problem with potential of homogeneous degree with 0 which ensures that the restriction of the lagrangian action mathcal a to the space g of g equivariant loops is coercive and its minimizers are collisionless without any strong force assumption in proving that local minima of g are free of collisions we develop an averaging technique based on marchal s idea of replacing some of the point masses with suitable shapes see 10 as an application several new orbits can be found with some appropriate choice of g furthermore the result can be used to give a simplified and unitary proof of the existence of many already known minimizing periodic orbits

on the interaction of nearly parallel vortex filaments the system for the interaction of nearly parallel vortex filaments deduced by klein majda and damodaran is considered local existence for the associated initial value problem with data which are small perturbations of perfect parallel filaments is proved assumptions on the unperturbed configurations and on the initial parameters which guarantee the existence of global regular solutions no finite time collapse are deduced

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

a new branch of mountain pass solutions for the choreographical 3 body problem we prove the existence of a new branch of solutions of mountain pass type for the periodic 3 body problem with choreographical constraint at first we describe the variational structure of the action functional associated to the choreographical three body problem in mathbb r 3 in the second part using a bisection algorithm we provide a numerical non rigorous solution of mountain pass type for this problem in a rotating frame with angular velocity 1 5 the last step consists in the rigorous computer assisted proof of the existence of a full branch of solutions for the problem starting from the mountain pass solution detected numerically

variational constructions for some satellite orbits in periodic gravitational force fields in a gravitational force field a bounded orbit of an infinitesimal point mass is called a satellite orbit the purpose of this paper is to establish a variational theory for the existence of some relative periodic satellite orbits in periodic gravitational force fields the gravitation field can be generated by a relative equilibrium or a uniformly rotating asteroid our approach is to regard the aggregate of primaries together with the small satellite as a restricted full two body problem and then look for direct and retrograde relative periodic satellite orbits by direct methods of calculus of variations regularity of the action minimizing satellite orbit is obtained by providing satisfactory lower bound estimates for the distance between the satellite and the mass center an upper bound estimate for this distance is also provided as a contrast to classical existence proofs by continuation from infinity

homological stability for configuration spaces of manifolds let cn m be the configuration space of n distinct ordered points in m we prove that if m is any connected orientable manifold closed or open the homology groups hi cn m are representation stable in the sense of church and farb arxiv 1008 1368 applying this to the trivial representation we obtain as a corollary that the unordered configuration space bn m satisfies classical homological stability for each i hi bn m hi bn 1 m for n i this improves on results of mcduff segal and others for open manifolds applied to closed manifolds this provides natural examples where rational homological stability holds even though integral homological stability fails

bifurcation of periodic solutions from a ring configuration in the vortex and filament problems this paper gives an analysis of the movement of n 1 almost parallel filaments or vortices starting from a polygonal equilibrium of n vortices with equal circulation and one vortex at the center of the polygon we find bifurcation of periodic solutions the bifurcation result makes use of the orthogonal degree in order to prove the global bifurcation of periodic solutions depending on the circulation of the central vortex in the case of the filament problem these solutions are periodic traveling waves

fi modules and stability for representations of symmetric groups in this paper we introduce and develop the theory of fi modules we apply this theory to obtain new theorems about the cohomology of the configuration space of n distinct ordered points on an arbitrary connected oriented manifold the diagonal coinvariant algebra on r sets of n variables the cohomology and tautological ring of the moduli space of n pointed curves the space of polynomials on rank varieties of n n matrices the subalgebra of the cohomology of the genus n torelli group generated by h1 and more the symmetric group sn acts on each of these vector spaces in most cases almost nothing is known about the characters of these representations or even their dimensions we prove that in each fixed degree the character is given for n large enough by a polynomial in the cycle counting functions that is independent of n in particular the dimension is eventually a polynomial in n in this framework representation stability in the sense of church farb for a sequence of sn representations is converted to a finite generation property for a single fi module

representation theory and homological stability abstract we introduce the idea of representation stability and several variations for a sequence of representations v n of groups g n a central application of the new viewpoint we introduce here is the importation of representation theory into the study of homological stability this makes it possible to extend classical theorems of homological stability to a much broader variety of examples representation stability also provides a framework in which to find and to predict patterns from classical representation theory littlewood richardson and murnaghan rules stability of schur functors to cohomology of groups pure braid torelli and congruence groups to lie algebras and their homology to the equivariant cohomology of flag and schubert varieties to combinatorics the n 1 n 1 conjecture the majority of this paper is devoted to exposing this phenomenon through examples in doing this we obtain applications theorems and conjectures beyond the discovery of new phenomena the viewpoint of representation stability can be useful in solving problems outside the theory in addition to the applications given in this paper it is applied by church ellenberg farb in preparation 20 to counting problems in number theory and finite group theory representation stability is also used by church 2012 19 to give broad generalizations and new proofs of classical homological stability theorems for configuration spaces on oriented manifolds

removing collision singularities from action minimizers for the n body problem with free boundaries for the planar and spatial n body problems it has been proved by marchal and chenciner that solutions for the minimizing problem with fixed ends are free from interior collisions this important result has been extended by ferrario terracini to newtonian type problems and equivariant problems it has also been used to construct many symmetric solutions for the n body problem

on the minimizing triple collision orbits in the planar newtonian three body problem in this paper we study the minimizing triple collision orbits in the planar newtonian three body problem with arbitrary masses we show that for a given non collinear initial configuration the minimizing triple collision orbit is collision free until a simultaneous collision and its limiting configuration is the lagrangian configuration with the same orientation as the initial configuration for the collinear initial configuration under a certain technical assumption there exist two minimizing orbits the limiting configurations of these orbits are the two opposite lagrangian configurations