inter species competition and chemorepulsion abstract we study an extension of the lotka volterra competition model in which one of the competing species avoids encounters with rivals by means of chemorepulsion a repulsive reaction to the scent of competitors the evolution of the species densities is described in terms of parabolic equations with cross diffusive and competitive terms the chemical signal used to detect rivals may diffuse in the environment leading to a fully parabolic problem or to a parabolic elliptic system if the diffusion of the chemical is dominant the most difficult case from the viewpoint of well posedness and global existence is the parabolic ode system which corresponds to the case when the chemical signal does not diffuse in the environment we prove global existence of solutions in the three cases for any space dimension for a wide range of parameters and initial conditions for the parabolic elliptic case the moving rectangles method is adapted to prove the converge to a constant steady state for some range of parameters and initial data linear stability analysis indicates that the constant steady state in the fully parabolic case loses stability when the strength of chemorepulsion is too high and that sufficiently low degradation rate of chemorepellent stabilizes the constant steady state

randomized divide and conquer improved path matching and packing algorithms we propose a randomized divide and conquer technique that leads to improved randomized and deterministic algorithms for np hard path matching and packing problems for the parameterized max path problem our randomized algorithm runs in time o 4 k k 2 7 m and polynomial space where m is the number of edges in the input graph improving the previous best randomized algorithm for the problem that runs in time o 5 44 k km and exponential space our randomized algorithms for the parameterized max r d matching and max r set packing problems run in time 4 r 1 k n o 1 and polynomial space improving the previous best algorithms for the problems that run in time 10 88 rk n o 1 and exponential space moreover our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems and they can be extended to solve other matching and packing problems

ergodic theory group representations and rigidity in these lectures we discuss some topics concerning the relationship of ergodic theory representation theory and the structure of lie groups and their discrete subgroups

cartan decomposition subgroups of 2 for g sl 3 r and g so 2 n we give explicit practical conditions that determine whether or not a closed connected subgroup h of g has the property that there exists a compact subset c of g with chc g to do this we fix a cartan decomposition g ka k of g and then carry out an approximate calculation of khk n a for each closed connected subgroup h of g

the instability of spiky steady states for a competing species model with cross diffusion this paper is concerned with the stability instability of a class of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross diffusion system are non linearly unstable

geometrically finite groups patterson sullivan measures and ratner s ridigity theorem the rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated ratner s theorem generalizes to these foliations with respect to the patterson sullivan measure in the spirit of mostow we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups finally we show that the frame flow on geometrically finite hyperbolic manifolds is bernoulli

stability analysis on a type of steady state for the skt competition model with large cross diffusion abstract this paper is concerned with the existence and the stability of steady state solutions for the skt biological competition model with cross diffusion by applying the higher order expansion and some detailed spectral analysis to the limiting system as the cross diffusion rate tends to infinite it is proved that the nontrivial positive steady states with some special bifurcating structure are unstable further the existence and the instability of the corresponding nontrivial positive steady states for the original cross diffusion system are proved by applying perturbation argument finally we show the global existence of solutions for the limiting system and present some numerical simulation on the large time behavior of the solution with more general initial data

analysis of a parabolic cross diffusion population model without self diffusion abstract the global existence of non negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown the cross diffusion terms are allowed to be arbitrarily large whereas the self diffusion terms are assumed to disappear the last assumption complicates the analysis since these terms usually provide h 1 estimates of the solutions the existence proof is based on a positivity preserving backward euler galerkin approximation discrete entropy estimates and l 1 weak compactness arguments furthermore employing the entropy entropy production method we show for special stationary solutions that the transient solution converges exponentially fast to its steady state as a by product we prove that only constant steady states exist if the inter specific competition parameters disappear no matter how strong the cross diffusion constants are

on classification of compact complex solvmanifolds abstract in this paper we complete nakamura s classification of compact complex parallelizable solvmanifolds up to the complex dimension five we find that the holomorphic symplectic ones are either nilpotent or pseudo kahler like i e with a complex solvable lie group as that of a compact complex solvable pseudo kahler space in guan 2010 gu1 we also found that for any even complex dimension all the compact complex pseudo kahler solvmanifolds are hypersymplectic therefore for compact complex solvmanifolds hypersymplectic is as general as pseudo kahler

parameterized and approximation algorithms for finding two disjoint matchings we first present a randomized fixed parameter algorithm for the np hard problem of deciding if there are two matchings m 1 and m 2 in a given graph g such that m 1 m 2 is a given number k the algorithm runs in o 2 k k m n expected time and can be derandomized to run in o 2 2 k 12 log 2 2 k k n m n time where n respectively m is the number of vertices respectively edges in g we then extend the algorithm to the weighted version of the problem we further present a combinatorial approximation algorithm for the np hard problem of finding two disjoint matchings in a given edge weighted graph g so that their total weight is maximized the algorithm achieves an approximation ratio close to 0 76 and runs in o m n 3 n time where is the inverse ackermann function

global rigidity of holomorphic riemannian metrics on compact complex 3 manifolds we study compact complex 3 manifolds m admitting a locally homogeneous holomorphic riemannian metric g we prove the following i if the killing lie algebra of g has a non trivial semi simple part then it preserves some holomorphic riemannian metric on m with constant sectional curvature ii if the killing lie algebra of g is solvable then up to a finite unramified cover m is a quotient g where is a lattice in g and g is either the complex heisenberg group or the complex sol group

on convex projective manifolds and cusps this study of properly or strictly convex real projective manifolds introduces notions of parabolic horosphere and cusp results include a margulis lemma and in the strictly convex case a thick thin decomposition finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds this is proved using a characterisation of ellipsoids in projective space except in dimension 3 there are only finitely many topological types of strictly convex man ifolds with bounded volume in dimension 4 and higher the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part there is an algebraic characterization of strict convexity in terms of relative hyperbolicity surfaces are ubiquitous throughout mathematics in good measure because of the geometry of riemann surfaces similarly thurston s insights into the geometry of 3 manifolds have led to many developments in diverse areas this paper develops the bridge between real projective geometry and low dimensional topology real projective geometry is a rich subject with many connections in recent years it has been combined with topology in the study of projective structures on manifolds classically it provides a unifying framework as it contains the three constant curvature geometries as subgeometries in dimension 3 it contains the eight thurston geometries up to a subgroup of index 2 in the case of product geometries and there are paths of projective structures that correspond to transitions between different thurston geometries on a fixed manifold moreover there is a link between real projective deformations and complex hyperbolic deformations of a real hyperbolic orbifold see 22 projective geometry therefore offers a general and versatile viewpoint for the study of 3 manifolds another window to projective geometry the symmetric space sl n r so n is isomorphic to the group of projective automorphisms of the convex set in projective space obtained from the open cone of positive definite quadratic forms in n variables this set is properly convex its closure is a compact convex set which is disjoint from some projective hyperplane the boundary of the closure has a rich structure as it consists of semi definite forms and when n 3 contains a dense set of flat 2 discs each corresponding to a family of semi definite forms of rank 2 which may be identified with a copy of the hyperbolic plane from a geometrical point of view there is a crucial distinction between strictly convex domains which contain no straight line segment in the boundary and the more general class of properly convex domains the former behave like manifolds of negative sectional curvature and the latter like arbitrary symmetric spaces however projective manifolds are more general kapovich 34 has shown that there are closed strictly convex 4 manifolds which do not admit a hyperbolic structure the hilbert metric is a complete finsler metric on a properly convex set this is the hyperbolic metric in the klein model when is a round ball a simplex with the hilbert metric is isometric to a normed vector space and appears in a natural geometry on the lie algebra sln a singular version of this metric arises in the study of certain limits of projective structures the hilbert metric has a hausdorff measure and hence a notion of finite volume

topological equivalence of flows on homogeneous spaces and divergence of one parameter subgroups of lie groups soient et des treillis et et des sous groupes a un parametre des groupes de lie connexes g et g si une des conditions a b ou c est vraie alors le theoreme a etablit que si les flots induits sur les espaces homogenes g et g sont topologiquement equivalents alors ils sont topologiquement equivalents par une application affine

affine structures on abelian lie groups abstract the nagano yagi goldmann theorem states that on the torus t 2 every affine or projective structure is invariant or is constructed on the basis of some goldmann rings nagano t yagi k osaka j math 11 1974 181 it shows the interest to study the invariant affine structure on the torus t 2 or on abelian lie groups recently the works of kim j differential geom 24 1986 373 and dekimpe ongenae p a m s 128 11 2000 3191 precise the number of non equivalent invariant affine structures on an abelian lie group in the case these structures are complete in this paper we propose a study of complete and non complete affine structures on abelian lie groups based on the geometry of the algebraic variety of finite dimensional associative algebras

polynomial structures for nilpotent groups if a polycyclic by finite rank k group admits a faithful affine representation making it acting on rk properly discontinuously and with compact quotient we say that admits an affine structure in 1977 john milnor questioned the existence of affine structures for such groups very recently examples have been obtained showing that even for torsion free finitely generated nilpotent groups n affine structures do not always exist it looks natural to consider affine structures as examples of polynomial structures of degree one we introduce the concept of a canonical type polynomial structure for polycyclic by finite groups using the algebraic framework of the seifert fiber space construction and a nice cohomology vanishing theorem we prove the existence and uniqueness up to conjugation of canonical type polynomial structures for virtually finitely generated nilpotent groups applying this uniqueness to a result going back to mal cev it follows that for torsion free finitely generated nilpotent groups each canonical polynomial structure is expressed in polynomials of limited degree the minimal degree needed for obtaining a polynomial structure will determine the affine defect number we prove that the known counterexamples to milnor s question have the smallest possible affine defect i e affine defect number equal to one

polycyclic by finite groups admit a bounded degree polynomial structure for a polycyclic by finite group gamma of hirsch length h an affine resp polynomial structure is a representation of gamma into rm aff bbb r h resp rm p bbb r h the group of polynomial diffeomorphisms letting gamma act properly discontinuously on bbb r h recently it was shown by counter examples that there exist groups gamma even nilpotent ones which do not admit an affine structure thus giving a negative answer to a long standing question of john milnor we prove that every polycyclic by finite group gamma admits a polynomial structure which moreover appears to be of a special simple type called canonical and as a consequence of this consists entirely of polynomials of a bounded degree the construction of this polynomial structure is a special case of an iterated seifert fiber space construction which can be achieved here because of a very strong and surprising cohomology vanishing theorem

the uniqueness of polynomial crystallographic actions let gamma be a polycyclic by finite group it is proved in 8 that gamma admits a polynomial action of bounded degree on mathbb r n which is properly discontinuous and such that the quotient gamma backslash mathbb r n is compact we prove here that such an action is unique up to conjugation by a polynomial transformation of mathbb r n

autour de la conjecture de l markus sur les varietes affines for any subgroupg of n we introduce some integer discg n called thediscompacity ofg this number measures to what extent the closure ofg is not compact the markus conjecture says that a compact affinely flat unimodular manifold is complete our main result called the discompact theorem is that this conjecture is true under the assumption that the linear holonomy i e the parallel transport has discompacity 1 because discso n 1 1 1 this ensures that a compact flat lorentz manifoldm is geodesically complete hence by a previous result of w goldman and y kamishima gk such am is up to finite covering a solvmanifold this achieves the proof of a bieberbach s theorem for compact lorentz flat manifolds

margulis spacetimes via the arc complex we study strip deformations of convex cocompact hyperbolic surfaces defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface we prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation in an essentially unique way the infinitesimal version of this result gives a parameterization by the arc complex of the moduli space of margulis spacetimes with fixed convex cocompact linear holonomy as an application we provide a new proof of the tameness of such margulis spacetimes m by establishing the crooked plane conjecture which states that m admits a fundamental domain bounded by piecewise linear surfaces called crooked planes the noninfinitesimal version gives an analogous theory for noncompact complete anti de sitter 3 manifolds

topological tameness of margulis spacetimes we show that margulis spacetimes without parabolic holonomy elements are topologically tame a margulis spacetime is the quotient of the 3 dimensional minkowski space by a free proper isometric action of the free group of rank geq 2 we will use our particular point of view that the margulis spacetime is a manifold with boundary with an bbb r sf p 3 structure in an essential way the basic tools are a bordification by a closed bbb r sf p 2 surface with free holonomy group and the work of goldman labourie and margulis on geodesics in the margulis spacetimes and 3 manifold topology

classification of simple novikov algebras and their irreducible modules of characteristic 0 abstract in this paper we first present a classification theorem of simple infinite dimensional novikov algebras over an algebraically closed field of characteristic 0 then we classify all the irreducible modules of certain infinite dimensional simple novikov algebras with an idempotent element whose left action is locally finite

archimedean superrigidity of solvables arithmetic groups abstract let g be a connected solvable linear algebraic group over a number fieldk letsbe a finite set of places ofkthat contains all the infinite places and let o s be the ring ofs integers ofk we define a certain closed subgroup formula of g s v s g kvthat contains g o s and prove that g o s is a superrigid lattice in formula by which we mean that finite dimensional representations g o s gln r more or less extend to representations of formula the subgroup formula may be a proper subgroup of g sfor only two reasons first it is well known that g o s is not a lattice in g sif g has nontrivialk characters so one passes to a certain subgroup g s 1 second g o s may fail to be zariski dense in g s 1 in an appropriate sense in this sense the subgroup formula is the zariski closure of g o s in g s 1 furthermore we note that a superrigidity theorem for many nonsolvables arithmetic groups can be proved by combining our main theorem with the margulis superrigidity theorem

on the full holonomy group of lorentzian manifolds the classification of restricted holonomy groups of n dimensional lorentzian manifolds was obtained about ten years ago however up to now not much is known about the structure of the full holonomy group in this paper we study the full holonomy group of lorentzian manifolds with a parallel null line bundle based on the classification of the restricted holonomy groups of such manifolds we prove several structure results about the full holonomy we establish a construction method for manifolds with disconnected holonomy starting from a riemannian manifold and a properly discontinuous group of isometries this leads to a variety of examples most of them being quotients of pp waves with disconnected holonomy including a non flat lorentzian manifold with infinitely generated holonomy group furthermore we classify the full holonomy groups of solvable lorentzian symmetric spaces and of lorentzian manifolds with a parallel null spinor finally we construct examples of globally hyperbolic manifolds with complete spacelike cauchy hypersurfaces disconnected full holonomy and a parallel spinor

algebraic hulls and smooth orbit equivalence let mi ui i 1 2 be two manifolds with quasi invariant measures and let hi c diff mi be connected lie groups if there is a measure class preserving diffeomorphism 0 m1 m2 which is a bijection of hl orbits and h2 orbits then we say that the actions are smoothly orbit equivalent if the hi actions determine foliations g on the manifolds mis then the map 0 is just a diffeomorphism between the foliations these phenomena and the corresponding phenomena arising in situations in which the mi ai are just borel or topological hi spaces and the map 0 is a measure class preserving borel isomorphism or homeomorphism have been studied using a variety of techniques independently by several authors b cfw d1 d2 k pnz w1 w2 z2 z4 most of the results obtained require some strengthening of the hypotheses such as finiteness and invariance of the measures and amenability or semisimplicity in higher rank of the groups acting one of the results that we want to describe in this paper fits in this geometric setting and provides an obstruction in terms of a geometric invariant of the actions to the foliations being diffeomorphic recall that if h acts ergodically and by diffeomorphisms on the n dimensional manifold m the algebraic hull of the h action is the unique up to conjugacy smallest algebraic subgroup l c gl n r such that there exists a measurable h invariant reduction to l of the frame bundle on m on which h acts by automorphisms for an analytic definition see 2 and for the general context see z5 9 2 then the normal algebraic hull of the h action on m will be the projection of l in the direction normal to the orbits

flat compact hermite lorentz manifolds in dimension 4 we give a classification up to finite cover of flat compact complete hermite lorentz manifolds up to complex dimension 4

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

affine manifolds and orbits of algebraic groups this paper is the sequel to the radiance obstruction and parallel forms on a ffne manifolds lyans amer math soc 286 1984 629 649 which introduced a new family of secondary characteristic classes for affine structures on manifolds the present paper utilizes the representation of these classes in lie algebra cohomology and algebraic group cohomology to deduce new results relating the geometric properties of a compact affine manifold mn to the action on rn of the algebraic hull a r of the affine holonomy group r c aff rn a main technical result of the paper is that if m has a nonzero cohomology class represented by a parallel k form then every orbit of a r has dimension k when m is compact then a r acts transitively provided that m is complete or has parallel volume the converse holds when r is nilpotent a 4 dimensional subgroup of aff r3 is exhibited which does not contain the holonomy group of any compact affine 3 manifold when m has solvable holonomy and is complete then m must have parallel volume conversely if m has parallel volume and is of the homotopy type of a solvmanifold then m is complete if m is a compact homogeneous affine manifold or if m possesses a rational riemannian metric then it is shown that the conditions of parallel volume and completeness are equivalent this paper is the sequel to our previous paper 22 in that paper we exploited certain characteristic classes exterior powers of the radiance obstruction to obtain relationships between various properties of affine manifolds our results supported the conjecture first made by l markus 36 that a compact affine manifold is complete if and only if it has parallel volume we shall refer to this as the main conjecture let m be a compact affine manifold with developing map dev m e and affine holonomy representation h 7r h 7r r c aff e here 1r is the group of deck transformations of the universal cover m e is the vector space rn and aff e is the group of affine automorphisms of e the map dev is a locally affine immersion which is equivariant respecting h the linear holonomy homb morphism is the composition a 1r gl e of h with the natural homomorphism l aff e gl e the obstruction to r fixing a point in e is a 1 dimensional cohomology class cm e h1 m ex with coefficients in e twisted by a it comes from a universal class in the group cohomology h1 aff e el which contains the crossed homomorphism u aff e e g g 0 received by the editors december 12 1984 and in revised form may 29 1985 1980 mathematics subject classification 1985 revision primary 57r99 53c05 secondary 53c10 55r25 1 research partially supported by fellowships and grants from the national science foundation a 1986 american mathematical society 0002 9947 86 1 00 25 per page

three dimensional affine crystallographic groups those groups r which act properly discontinuously and aillnely on ii with compact fundamental domain are classified first it is shown that such a group f contains a solvable subgroup of finite index thus establishing a conjecture of auslander in dimension three then unimodular simply transitive altine actions on ir are classified this leads to a classification of attine crystallographic groups acting on ir3 a characterization of which abstract groups admit such an action is given moreover it is proved that every isomorphism between virtually solvable atline crystallographic groups respectively simply transitive afline groups is induced by conjugation by a polynomial automorphism of the affrne space a characterization is given of which closed 3 manifolds can be represented as quotients of ir by groups of afftne transformations a closed 3 manifold m admits a complete atline structure if and only if m has a finite covering homeomorphic or homotopy equivaient to a 2 torus bundle over the circle

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

on the zariski closure of the linear part of a properly discontinuous group of affine transformations let be a subgroup of the group of affine transformations of the affine space r 2n 1 suppose acts properly discontinuously on r 2n 1 the paper deals with the question which subgroups of gl 2n 1 r occur as zariski closure of the linear part of such a group the two main results of the paper say that so n 1 n does occur as of such a group if n is odd but does not if n is even

proper affine actions and geodesic flows of hyperbolic surfaces let to c 0 2 1 be a schottky group and let h2 to be the corresponding hyperbolic surface let s denote the space of unit length geodesic currents on s the cohomology group if1 to v parametrizes equivalence classes of affine deformations tu of to acting on an irreducible representation v of 0 2 1 we define a continuous biaffine map i s x hl to v u which is linear on the vector space h1 fo v an affine deformation tu acts properly if and only if x u 7 0 for all jl e s consequently the set of proper affine actions whose linear part is a schottky group identifies with a bundle of open convex cones in hl to v over the fricke teichmiiller space of e

dynamics in dumbbell domains i continuity of the set of equilibria we analyze the dynamics of a reaction diffusion equation with homogeneous neumann boundary conditions in a dumbbell domain we provide an appropriate functional setting to treat this problem and as a first step we show in this paper the continuity of the set of equilibria and of its linear unstable manifolds

instability of spiky steady states for s k t biological competing model with cross diffusion abstract this paper is concerned with the stability instability of a type of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable when the diffusion rate is small enough and the positive spiky steady states for the original cross diffusion system are unstable when the cross diffusion rate for one species is large enough and the diffusion rate of another species is small

stability results for a reaction diffusion problem with mixed boundary conditions and applications to some symmetric cases abstract in this article we consider a one dimensional reaction diffusion problem with mixed boundary conditions we provide conditions for the existence or nonexistence of stable nonconstant solutions whose derivative vanishes at some point as an application we obtain similar results for problems with dirichlet boundary conditions posed in some symmetric domains an n dimensional ball surfaces of revolution and model manifolds

mobius disjointness for homogeneous dynamics we prove sarnak s mobius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected lie groups namely we show that if g is any such group g a lattice and u g an ad unipotent element then for every x g and every function f continuous on the 1 point compactification of g the sequence f xun cannot correlate with the mobius function on average

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

minimal coexistence configurations for multispecies systems abstract we deal with strongly competing multispecies systems of lotka volterra type with homogeneous neumann boundary conditions in dumbbell like domains under suitable non degeneracy assumptions we show that as the competition rate grows indefinitely the system reaches a state of coexistence of all the species in spatial segregation furthermore the limit configuration is a local minimizer for the associated free energy

measurable quotients of unipotent translations on homogeneous spaces let u be a nilpotent unipotent subgroup of a lie group g and let f be a closed subgroup of g marina ratner showed that every ergodic u invariant probability measure on the homogeneous space f g is of a simple algebraic form we use this fundamental new result to show that every measurable quotient of the u action on j g is of a simple algebraic form roughly speaking any quotient is a double coset space a g k

the radiance obstruction and parallel forms on affine manifolds a manifold m is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine when they are locally linear m is called radiant the obstruction to radiance is a one dimensional class cm with coefficients in the flat tangent bundle of m exterior powers of cm give information on the existence of parallel forms on m especially parallel volume forms as applications various kinds of restrictions are found on the holonomy and topology of compact affine manifolds introduction an affine manifold m is a manifold with a distinguished maximal atlas of charts all of whose coordinate changes are locally affine on such a manifold there is an intrinsic notion of a parallel tensor one whose components in any affine chart are constants more generally there is the notion of a polynomial tensor field of given degree in 1962 l markus conjectured in mk that a compact orientable affine n dimensional manifold has parallel volume form if and only if it is complete meaning that its universal covering is affinely isomorphic to euclidean n space rn the problem of constructing a parallel volume form determines an n dimensional twisted real cohomology class originally studied by j smillie sm2 in this paper we express this class as the nth exterior power of a twisted one dimensional real class which we call the radiance obstruction cm by computing cm in various cohomology theories cech singular de rham and others we are able to exploit cm in several ways to yield more information on the structure of affine manifolds some of these results will appear in a subsequent paper gh3 the basic tool used is a formula proved in 2 6 which expresses the cohomology class of a parallel exterior k form in terms of the k th exterior power of the radiance obstruction akcvf min 2 7 the special case of a parallel volume form on a compact n dimensional manifold m is examined the existence of such a form implies that ancm 0 in 2 8 we show that the affine holonomy group f of such an m cannot preserve a proper affine subspace in gh3 this result will be improved by showing that r preserves no proper semialgebraic subset of rn in 2 10 we show that there tend to be plenty of parallel forms on a compact affine manifold with nilpotent affine holonomy received by the editors may 26 1983 1980 mathematics subject classification primary 57r99 53c05 secondary 53c10 55r25 4c1984 amencan mathematical society 0002 9947 84 1 00 25 per page

on the uniqueness of some affinely flat infra nilmanifolds i p irn irk u p irn irk since pi irn irk pi irn ir k each element of i 1 ir irk is completely determined by k polynomials of total degree at most i in n variables let g be a group and suppose we have a morphism 4 g g l k ir and a homomorphism p g aff ir we can provide a g module structure for p ir f k in the following way for g e g and pepi ir irk we define

local rigidity of affine actions of higher rank groups and lattices let j be a semisimple lie group with all simple factors of real rank at least two let r j be a lattice we prove a very general local rigidity result about actions of j or r this shows that almost all so called standard actions are locally rigid as a special case we see that any action of by toral automorphisms is locally rigid more generally given a manifold m on which acts isometrically and a torus t n on which it acts by automorphisms we show that the diagonal action on t n m is locally rigid this paper is the culmination of a series of papers and depends heavily on our work in two recent articles the reader willing to accept the main results of those papers as black boxes should be able to read the present paper without referring to them

spatial chaotic structure of attractors of reaction diffusion systems the dynamics described by a system of reaction diffusion equations with a nonlinear potential exhibits complicated spatial patterns these patterns emerge from preservation of homotopy classes of solutions with bounded energies chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree this group corresponds to level sets of the potential the estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem

on the nonexistence of closed timelike geodesics in flat lorentz 2 step nilmanifolds the main purpose of this paper is to prove that there are no closed timelike geodesics in a compact or noncompact flat lorentz 2 step nilmanifold n where n is a simply connected 2 step nilpotent lie group with a flat left invariant lorentz metric and a discrete subgroup of n acting on n by left translations for this purpose we shall first show that if n is a 2 step nilpotent lie group endowed with a flat left invariant lorentz metric g then the restriction of g to the center z of n is degenerate we shall then determine all 2 step nilpotent lie groups that can admit a flat left invariant lorentz metric we show that they are trivial central extensions of the three dimensional heisenberg lie group h 3 if n g is one such group we prove that no timelike geodesic in n g can be translated by an element of n by the way we rediscover that the heisenberg lie group h 2k 1 admits a flat left invariant lorentz metric if and only if k 1

delay partial differential equations with some large diffusion it is often the case in infinite dimensional dynamical systems that a certain unbounded operator depends upon a parameter v its eigenvalues and eigenfunctions will also depend upon v in many cases see for example l 7 there is a finite number of eigenvalues of the unbounded operator that stay bounded as the parameter varies the remaining eigenvalues blow up at a certain rate in this case one expects the dynamics to be dictated by a system of ordinary differential equations this is indeed the case in several applications and its proof strongly uses the invariant manifold theory in the cases considered in this paper we study the dynamics of infinite dimensional dynamical systems for which a countable number of eigenvalues of the unbounded operator converges to the eigenvalues of an unbounded operator in a lower dimensional domain whereas the remaining blow up this is the case for example in parabolic equations in thin domains studied by hale and raugel s we will not be able to address the general case of thin domains but only the case of a thin straight channel however the property that allows this theory to work is present in the general case that is there is a sequence of eigenvalues that stay bounded and a sequence of eigenvalues that blow up see also arrieta 9 the theory of invariant manifolds will no longer work in this case since it is centered around the existence of backward solutions which cannot be expected in this situation the main tools in this work are a converse theorem on existence of compact attractors and an eigenvalue problem a physical interpretation of a parabolic problem in a thin straight channel is that there is a very high diffusion in one of the directions from which our intuition would say that in that direction spatial changes will not be observed the same interpretation can be given to a damped wave problem in a thin straight channel the wave speed can be interpreted as being very large in one direction this would intuitively imply that there would be no spatial changes for the amplitude of the wave in that direction we also consider the problem called shadow systems where the diffusion is very large for some of the substances in the reaction and not so large for the remaining substances in this

on isospectral locally symmetric spaces and a theorem of von neumann 1 introduction call two lattices f1 and f2 in a locally compact group g isospectral if the representations of g on l2 g fi and l2 g f2 are unitarily equivalent here we show that isospectral lattices are quite common in semisimple real algebraic groups g we construct them using a variation of a technique of sunada su as a consequence we obtain new examples of locally symmetric spaces with isospectral laplacians in particular we get such examples with higher rank previously the only higher rank isospectral manifolds known were locally reducible these were vigneras s examples of quotients of products of hyperbolic spaces i v on the other hand brooks found irreducible higher rank locally symmetric spaces with isospectral potentials i br i however the underlying manifolds in his examples happened to be isometric this was due to mostow rigidity since the fundamental groups were isomorphic in fact this work was largely motivated by the question of whether the rigidity properties of lattices in higher rank force an isospectral rigidity

the fundamental group of a compact flat lorentz space form is virtually polycyclic a flat lorentz space form is a geodesically complete lorentzian manifold of zero curvature it is well known see auslander markus 3 that such a space m may be represented as a quotient r where r is an dimensional minkowski space n equals the dimension of m and is a group of lorentz isometries acting properly discontinuously and freely on r in particular the universal covering of m is isometric to r and the fundamental group m is isomorphic to theorem let m be a compact flat lorentz space form then m is virtually poly cyclic recall that a group is virtually polycyclic if it can be built by iterated extensions from finitely many finite groups and cyclic groups this result affirms a conjecture of milnor 13 in a special case for discussion of this conjecture as well as another special case we refer to fried goldman 8 the importance of this result is that it reduces the classification of compact flat lorentz space forms to fairly elementary problems concerning lie algebras and lattices in solvable lie groups we hope to pursue this classification in a future publication for a description of this reduction and the classification in dimension 3 see fried goldman 8 in dimension 4 the classification is worked out in fried 7 one immediate consequence of the structure theory developed in fried goldman 8 1 and kamishima 12 is the following corollary let m be the compact flat lorentz space form then m has a finite covering which is diffeomorphic to a solvmanifold the outline of this paper is as follows in the first section we collect some basic facts about the group e n 1 1 of isometries of dimensional minkowski space in the second section we prove the theorem in the special

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

on the existence of closed timelike geodesics in compact spacetimes in 19 tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact cauchy surfaces necessarily contains a closed timelike geodesic the restriction to compact spacetimes with just regular globally hyperbolic coverings i e the cauchy surfaces are not required to be compact is still an open question here we shall answer this question negatively by providing examples of compact flat lorentz space forms without closed timelike geodesics and shall give some criterion for the existence of such geodesics more generally we will show that in a compact spacetime having a regular globally hyperbolic covering each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic whether or not a compact flat spacetime contains closed nonspacelike geodesics is as far as we know an open question we shall answer this question affirmatively we shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

invariant measures for algebraic actions zariski dense subgroups and kazhdan s property t let k be any locally compact non discrete field we show that finite invariant measures for k algebraic actions are obtained only via actions of compact groups this extends both borel s density and fixed point theorems over local fields for semisimple solvable groups resp we then prove that for k algebraic actions finitely additive finite invariant measures are obtained only via actions of amenable groups this gives a new criterion for zariski density of subgroups and is shown to have representation theoretic applications the main one is to kazhdan s property t for algebraic groups which we investigate and strengthen

superrigidity of lattices in solvable lie groups let be a closed cocompact subgroup of a simply connected solvable lie groupg such that adg has the same zariski closure as adg if gln is any finite dimensional representation of we show that virtually extends to a representation ofg by combining this with work of margulis on lattices in semisimple groups we obtain a similar result for lattices in many groups that are neither solvable nor semisimple furthermore we show that if is isomorphic to a closed cocompact subgroup of another simply connected solvable lie groupg then any isomorphism from to extends to a crossed isomorphism fromg tog in the same vein we prove a more concrete form of mostow s theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic

minimal models formality and hard lefschetz properties of solvmanifolds with local systems for a simply connected solvable lie group g with a cocompact discrete subgroup gamma we consider the space of differential forms on the solvmanifold g gamma with values in a certain flat bundle so that this space has a structure of a differential graded algebra dga we construct sullivan s minimal model of this dga this result is an extension of nomizu s theorem for ordinary coefficients in the nilpotent case by using this result we refine hasegawa s result of formality of nilmanifolds and benson gordon s result of hard lefschetz properties of nilmanifolds

affine transformations and parallel lightlike vector fields on compact lorentzian 3 manifolds we describe the compact lorentzian 3 manifolds admitting a parallel lightlike vector field the classification of compact lorentzian 3 manifolds admitting non isometric affine diffeomorphisms follows together with the complete description of these morphisms such a lorentzian manifold is in some sense an equivariant deformation of a flat one