equivariant stable homotopy and segal s burnside ring conjecture analogous results were proved later for ko in the generality of compact lie groups by atiyah and segal 8 and for kfq the algebraic k theory spectrum associated to the finite field fq by rector 29 using quillen s 26 computation of it kfq in each case the answer involves an appropriate completed representation ring of g and the cohomology theory in question is constructed from the permutative category of finite dimensional vector spaces over a field see 31 if one considers cohomology theories constructed from other permutative categories one expects to find analogous computations in terms of a completed representation ring of g in the given category appropriately defined in particular stable cohomotopy ts is constructed from the category of finite sets 31 and for this category the analogue of the representation ring is a well known object the burnside ring a g 13 a g is a commutative ring with augmentation so one may speak of a g the completed burnside ring moreover there is

the motion of a vortex on a closed surface of constant negative curvature the purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one the algorithm is based on a relation between the laplace beltrami green function and the heat kernel the algorithm is used to compute the motion of a vortex on the bolza surface this is the first determination of the orbits of a vortex on a closed surface of genus greater than one the numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the bolza surface some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc the following theorem is proved a weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium

linking first occurrence polynomials over f2 by steenrod operations abstract it is proved that for every irreducible representation l of the full matrix semigroup mn f 2 the first occurrence of l as a composition factor in the polynomial algebra p f 2 x1 xn is linked by a steenrod operation to the first occurrence of l as a submodule in p this steenrod operation is given explicitly as the image of an admissible monomial in the steenrod squares sqr under the canonical anti automorphism of the mod 2 steenrod algebra a the first occurrences of both kinds are also linked to higher degree occurrences of l by elements of the milnor basis of a

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

the composition factors of fp x1 x2 x3 as a gl 3 p module let gl n p denote the general linear group over the prime field fp and let sd denote the right gl n p module consisting of all homogeneous polynomials of degree d in n variables x1 x2 x over f we wish to determine the simple composition factors of sd unfortunately satisfactory models for all the simple gl n p modules themselves are not known there are p p 1 of these simple modules up to isomorphism and all of them are known to occur in the symmetric algebra s cf sd the general problem is therefore a very difficult one in our present state of knowledge the purpose of this paper is to offer a solution to this problem in the case n 3 this extends the work of david carlisle who calculated in his manchester thesis 2 the occurrences of all the l dimensional modules and developed a method for dealing with the general case let us begin with some general remarks about the problem the case n 1 is trivial as s d detd the dth power of the determinant representation the case n 2 is also easily disposed of for instance by glover 9 the simple modules are the tensor products of the sd for d 1 and d p the module s is not simple for example the pth power sf of the natural module is a proper submodule of 411 0021 8693192 3 00

subalgebras of the steenrod algebra and the action of matrices on truncated polynomial algebras let a t be the subalgebra of the steenrod algebra generated by sqi i 2t we determine which a module indecomposable summands of f2 x1 xn are free over a t by first showing this to be equivalent to the group theoretic question of determining which irreducible modular representations of mn f2 or gln f2 occur as composition factors in the truncated polynomial algebra f2 x1 xn x12 xn2 the second question is then answered

toeplitz operators and quantum mechanics abstract toeplitz operators on the segal bargmann spaces of gaussian measure square integrable entire functions on complex n space c n are studied the c algebra generated by the weyl form of the canonical commutation relations consists precisely of the uniform limits of almost periodic toeplitz operators the question of which toeplitz operators admit a symbol calculus modulo the compact operators is raised and sufficient conditions are given for such a calculus these conditions involve a notion of slow oscillation at infinity

toeplltz operators and the berezin transform on h2 abstract for t a bounded linear operator on the hardy space h2 its berezin transform is the function t on the unit disc defined by t tk k where k z 1 2 1 2 1 z is the normalized reproducing szego kernel when t t gf is a toeplitz operator it turns out that the nontangential boundary values of t gf coincide with gf almost everywhere this allows us to construct extensions and or generalizations of the classical symbol calculus for toeplitz operators to larger operator algebras briefly discussed is also the relation between our results and the recent results of berger coburn and zhu for toeplitz and hankel operators on the bergman space

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

modular representations of gl 3 fp symmetric squares and mod p cohomology of gl 3 z abstract we prove certain cases of nonabelian reciprocity between mod p galois representations and hecke eigenclasses in the mod p cohomology of gl 3 z using the symmetric square liftings from gl 2 and congruences on hecke eigenvalues we need to develop some modular representation theory for gl 3 z p in some cases we also need to invoke certain computer calculations to finish the proof

nonlinear stability of a latitudinal ring of point vortices on a nonrotating sphere we study the nonlinear stability of relative equilibria of configurations of identical point vortices on the surface of a sphere in particular we study how the stability changes as a function of the colatitude theta and of the number of vortices n by using the integrals of motion we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest then after a sufficient criterion due to dirichlet the stability ranges are the theta intervals for which the hessian of the hamiltonian evaluated at the equilibrium configuration is positive or negative definite we find that the stability intervals coincide with those for linear stability determined by polvani and dritschel j fluid mech 255 1993 pp 35 64 for n 3 we recover the result previously established by pekarsky and marsden j math phys 39 1998 pp 5894 5907

foncteurs polynomiaux stricts et modules instables sur l algebre de steenrod resume soient u f et p respectivement les categories des modules instables sur l algebre de steenrod modulo p sans bockstein des representations generiques des groupes lineaires generaux h w henn j lannes and l schwartz 1993 1 n j kuhn 1994 2 et des foncteurs polynomiaux stricts a la friedlander and suslin 1997 3 un foncteur f u f est construit dans h w henn j lannes and l schwartz 1993 1 utilisant le foncteur de lannes en degre 0 on construit dans cet article un foncteur m p u exact et commutant au produit tensoriel et a la torsion de frobenius qui constitute un relevement du foncteur oubli o p f m f o

weyl modules and the mod 2 steenrod algebra abstract this paper continues our study of the action of the mod 2 steenrod algebra a 2 on the polynomial algebra p n f 2 t 1 t n we obtain further partial results on the hit problem of f p peterson which asks for a minimal generating set for p n as an a 2 module we also study the structure of the quotient by the hit elements as a graded representation of the finite general linear group g n gl n f 2 i e as a module over the finite group algebra f 2 g n these results were obtained in previous work of the authors for the special case of the steinberg module for g n by extending the scalars to f 2 the algebraic closure of f 2 we obtain commuting actions of a 2 and g n on p n f 2 t 1 t n while this makes no essential difference to the representation theory of g n or to the hit problem it allows us to treat the action of g n on p n as the restriction of that of the algebraic group g n gl n f 2 in particular we make use of tilting modules for g n to show that for every irreducible representation l of g n a minimal set of a 2 generators of p n must contain a copy of the corresponding dual weyl module

statistics of point vortex turbulence in non neutral flows and in flows with translational and rotational symmetries a theory esler and ashbee in j fluid mech 779 275 308 2015 describing the statistics of n freely evolving point vortices in a bounded two dimensional domain is extended first the case of a non neutral vortex gas is addressed and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non central chi squared random variables the details of which depend only on the shape of the domain equations for the equilibrium energy spectrum and other statistical quantities follow the validity of which are verified against direct numerical simulations of the equations of motion second domains with additional conserved quantities associated with a symmetry e g circle periodic channel are investigated and it is shown that the treatment of the non neutral case can be modified to account for the additional constraint

deligne lusztig characters of the finite general linear groups and eigenvectors of lannes t functor abstract the eigenvalue conjecture of lionel schwartz states that the induced action of lannes t functor on the grothendieck ring of reduced injective unstable modules over the mod p steenrod algebra is diagonalizable over q with eigenvalues the powers of p while some proofs have been given for the conjecture none of them could shed light on the problem of determining the eigenvectors of t the aim of this paper is to compute at the same time eigenvalues and eigenvectors of t by making use of the deligne lusztig characters of the finite general linear groups based on this computation some remarks and conjectures about the multiplicative structure of the grothendieck ring and the poincare series of eigenvectors will be also given

lannes t functor on summands of h b z p s let h be the mod p cohomology of the classifying space b z p thought of as an object in the category u of unstable modules over the steenrod algebra lannes constructed a functor t u u which is left adjoint to the functor a a h in this paper we evaluate t on the indecomposable u summands of h s the tensor product of s copies of h our formula involves the composition factors of certain tensor products of irreducible representations of the semigroup ring f p m s s z p

the morava k theories of some classifying spaces let p be a finite abelian p group with classifying space bp we compute in representation theoretic terms the morava k theories of the stable wedge summands of bp in particular we obtain a simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup a minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup m 1 z p 1 main results for a fixed prime p there exists a sequence of cohomology theories k 1 k 2 with k s periodic with period 2 ps 1 these are the morava k theories 13 and generalize ordinary complex k theory in the sense that k 1 is one of the p 1 isomorphic summands of k z p recently it has been becoming clear that they play a central role in homotopy theory see for example the work of m hopkins and j smith 6 k s has further computational virtues k s x is always a free module over the coefficient ring k s and there is a kunneth isomorphism a major outstanding problem is to find good models for the spaces representing these theories related to this is the question of finding for finite groups g a group theoretic description for the rings k s bg analogous to atiyah s isomorphism 2 r g k bg the author together with j harris recently analyzed stable wedge decompositions of classifying spaces of abelian p groups 5 in this paper we compute in representation theoretic terms the morava k theories of the resulting wedge summands as a consequence we obtain a very simple and purely group theoretic description of the rank of k s bg for any finite group g with an abelian p sylow subgroup to state our results we establish some notation it is convenient to let k o be rational cohomology we let k s x always denote the reduced sth morava k theory of x if x is a space or spectrum we let k5 x equal the rank of k s x as a k s module and by abuse of notation we let ks g ks bg for a finite group g bg is the union of bg with a disjoint basepoint received bv the editors may 27 1986 1980 marhemat7es subject cl lssific ltion primary 55n22 secondary 20g05 research partially supported by the sloan foundation and the nsf t1987 american mathematical society 0002 9947 87 1 00 25 per page

the sarason sub symbol and the recovery of the symbol of densely defined toeplitz operators over the hardy space abstract while the symbol map for the collection of bounded toeplitz operators is well studied there has been little work on a symbol map for densely defined toeplitz operators in this work a family of candidate symbols the sarason sub symbols is introduced as a means of reproducing the symbol of a densely defined toeplitz operator this leads to a partial answer to a question posed by donald sarason in 2008 in the bounded case the toeplitzness of an operator can be classified in terms of its sarason sub symbols this justifies the investigation into the application of the sarason sub symbols on densely defined operators it is shown that analytic closed densely defined toeplitz operators are completely determined by their sarason sub symbols and it is shown for a broader class of operators that they extend closed densely defined toeplitz operators of multiplication type

vortex dynamics in complex domains on a spherical surface we consider the motion of both point vortices and uniform vortex patches in arbitrary possibly multiply connected regions bounded by impenetrable walls on the surface of a sphere by exploiting knowledge of the functional form of the relevant green s function in a pre image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface we first generalize kirchhoff routh theory for point vortex motion in the plane to point vortex motion on a spherical shell next we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls we describe a novel numerical scheme whereby this motion can be computed some illustrative calculations are shown

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

on the p adic completions of nonnilpotent spaces this paper deals with the p adic completion f p x developed by bousfield kan for a space x and prime p a space x is called f p good when the map x f p x is a mod p homology equivalence and called f p bad otherwise general examples of f p good spaces are established beyond the usual nilpotent or virtually nilpotent ones these include the polycyclic by finite spaces however the wedge of a circle with a sphere of positive dimension is shown to be f p bad this provides the first example of an f p bad space of finite type and implies that the p profinite completion of a free group on two generators must have nontrivial higher mod p homology as a discrete group

a proof of schwartz s conjecture about the eigenvalues of lannes t functor abstract this paper gives an algebraic proof of a conjecture due to lionel schwartz which asserts that the linear operator induced by lannes t functor on the grothendieck group k n red generated by indecomposable summands of the mod p cohomology of a rank n elementary abelian p group is diagonalizable over q with eigenvalues 1 p p n and with multiplicities p n p n 1 p n 1 p n 2 p 1 1 respectively using work of harris and shank we first reduce this to an algebraic question involving the grothendieck ring g 0 m n p of modules over the semigroup ring f p end f p n showing that the induced action of t on k n red corresponds to the multiplication by an explicit element in the second step we establish the separability of the algebra c g 0 m n p from which the diagonalizability and the computation of the eigenvalues and their multiplicities follow easily the arguments use ingredients from the theory of brauer characters of finite groups

the distribution function inequality for a finite sum of finite products of toeplitz operators abstract a generalized area function associated with a finite sum of finite products of toeplitz operators is introduced a distribution function inequality is established for the generalized area function by using the distribution function inequality we characterize when a finite sum of finite products of toeplitz operators on the hardy space is a compact perturbation of a toeplitz operator

point vortex dynamics a classical mathematics playground the idealization of a two dimensional ideal flow as a collection of point vortices embedded in otherwise irrotational flow yields a surprisingly large number of mathematical insights and connects to a large number of areas of classical mathematics several examples are given including the integrability of the three vortex problem the interplay of relative equilibria of identical vortices and the roots of certain polynomials addition formulas for the cotangent and the weierstras function projective geometry and other topics the hope and intent of the article is to garner further participation in the exploration of this intriguing dynamical system from the mathematical physics community

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze

toeplitz operators on the segal bargmann space in this paper we give a complete characterization of those functions on 2n dimensional euclidean space for which the berezin toeplitz quantizations admit a symbol calculus modulo the compact operators the functions in question are characterized by a condition of small oscillation at infinity

the structure ofī λnin generic representation theory abstract this paper pursues a study of the category f of functors between f 2 vector spaces it is proved that the objects i n occur in short exact sequences k n i n k n 1 where each k n is a non finite functor which has the property that every proper sub functor is finite moreover every sub functor of i n either contains k n or is finite