fake projective planes a fake projective plane is a complex surface different from but has the same betti numbers as the complex projective plane it is a complex hyperbolic space form and has the smallest euler poincare characteristic among smooth surfaces of general type the first example was constructed by mumford later on two more examples were found by ishida and kato a fourth possible one was recently proposed by keum in a recent joint work with gopal prasad we showed that there are seventeen non empty classes of fake projective planes and there can at most be four more specific classes higher dimensional analogues and examples were also obtained the main purpose of the talk is to explain the joint work with prasad and other related results such as arithmeticity of the lattices involved obtained earlier by klingler and yeung independently

the cusped hyperbolic orbifolds of minimal volume in dimensions less than ten abstract we detect the cusped complete hyperbolic orbifolds of minimal volume in dimensions less than ten our method is geometric and uses results from crystallography and the theory of sphere packings in fact such an n orbifold gives rise to a horoball packing whose orthogonal projection to the cusp boundary yields the densest euclidean lattice packing in dimension n 1

stratifiable maps and topological invariants let x and y be spaces and f x y be a map between them then a large family of problems in topology and geometry involve the attempt to relate the invariants of x and y via f this attempt has been successful often for the case x and y manifolds and f the projection of a fibre bundle for many types of important invariants for example the result of chernhirzebruch serre chs asserts that for y simply connected the signature of x is the product of the signatures of y and of the fibre f 1 y in this paper we undertake the study of maps in which the fiber may vary from point to point the spaces x and y will be any whitney stratified spaces with even codimension strata e g algebraic varieties and f will be a stratified map this means essentially that for v a component of a stratum of y thus v is an open manifold f 1v is a union of strata of x and the restriction of f to f71v is a locally trivial fibre bundle for example subanalytic maps of subanalytic sets can be stratified with respect to whitney

covolumes of nonuniform lattices in pu n 1 this paper studies the covolumes of nonuniform arithmetic lattices in pu n 1 we determine the smallest covolume nonuniform arithmetic lattices for each n the number of minimal covolume lattices for each n and study the growth of the minimal covolume as n varies in particular there is a unique lattice up to isomorphism in pu 9 1 of smallest euler poincare characteristic amongst all nonuniform arithmetic lattices in pu n 1 we also show that for each even n there are arbitrarily large families of nonisomorphic maximal nonuniform lattices in pu n 1 of equal covolume

structural properties of context free sets of graphs generated by vertex replacement abstract we establish that a vertex replacement set of graphs i e a set of graphs generated by a c ednce or equivalently by a separated handle rewriting graph grammar is hyperedge replacement i e is generated by a hyperedge replacement graph grammar iff its graphs do not contain arbitrary large complete bipartite graphs kn n as subgraphs another equivalent condition is that its graphs have a number of edges that is linearly bounded in terms of the number of vertices these properties are decidable by means of an appropriate extension of the theorem by parikh that characterizes the commutative images of context free languages we extend these results to hypergraphs

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

the complexity of the kn n problem for node replacement graph languages the bounded kn n problem is the question whether or not a graph language of a given graph grammar contains arbitrarily large complete bipartite subgraphs kn n in this paper we investigate the complexity of this problem for all relevant classes of node replacement graph grammers our main result states that the bounded kn n problem is nl complete for reduced nonblocking ence graph grammars and for reduced linear nce graph grammars as an application our results settle the complexity of the problems whether or not the graph language of a given confluent boundary or linear graph grammar has bounded tree width and whether or not it is an hr graph language

computing twisted signatures and l classes of non witt spaces in previous joint work with cappell and shaneson we have established an atiyah lusztig meyer type multiplicative characteristic class formula for the twisted signature and more generally the twisted l class of a stratified witt space the present paper shows that these formulae hold even when the stratified space does not satisfy the witt condition it constitutes one of the first applications of signature homology

the l class of non witt spaces characteristic classes for oriented pseudomanifolds can be defined using appropriate self dual complexes of sheaves on non witt spaces self dual complexes compatible to intersection homology are determined by choices of lagrangian structures at the strata of odd codimension we prove that the associated signature and l classes are independent of the choice of lagrangian structures so that singular spaces with odd codimensional strata such as e g certain compactifications of locally symmetric spaces have well defined l classes provided lagrangian structures exist we illustrate the general results with the example of the reductive borel serre compactification of a hilbert modular surface

hodge genera of algebraic varieties ii we study the behavior of hodge genera under algebraic maps we prove thatthemotivic c y genussatisfiesthe stratifiedmultiplicativeproperty whichshows how to compute the invariant of the source of a morphism from its values on varieties arising from the singularities of the map by considering morphisms to a curve we obtain a hodge theoretic version of the riemann hurwitz formula we also study the monodromy contributions to the y genus of a family of compact complex manifolds and prove an atiyah meyer type formula in the algebraic and analytic contexts this formula measures the deviation from multiplicativity of the y genus and expresses the correction terms as higher genera associated to the period map these higher gen era are hodge theoretic extensions of novikov higher signatures to analytic and alge braic settings characteristic class formulae of atiyah meyer type are also obtained by making use of saito s theory of mixed hodge modules

on discontinuous action of monodromy groups on the complex n ball in dm 3 10 3 15 there is defined for each n 3 tuple 1u n 3 of nonintegral real numbers with ei2 i integral a subgroup i acting on a complex n 1 dimensional vector space vj which preserves a hermitian form of signature zi p 1 ei 1 ui 1 dm 2 21 where for any real number r r denotes its fractional part i e 0 0 we will be concerned principally with the case that the signature of the hermitian form is 1 positive n negative in which case the image of v in the projective space pn associated to v is the complex n ball bn also called the n disc we deal almost exclusively with the action of f on bn and pn we call an n 3 tuple u satisfying 0 pi 1 for i 1 n 3 and eiz i 2 a disc n 3 tuple in dm theorem 10 19 it is proved let u be a disc n 3 tuple satisfying condition int for all i a j such that ui uj 1

arithmetic fake projective spaces and arithmetic fake grassmannians in a recent paper we have classified fake projective planes natural higher dimensional generalization of these surfaces are arithmetic fake bf p n 1 c and arithmetic fake bf gr m n in this paper we show that arithmetic fake bf p n 1 c can exist only if n 3 5 and an arithmetic fake bf gr m n can exist with n3 odd only if n 5 here we construct four distinct arithmetic fake bf p 4 c and four distinct fake arithmetic bf gr 2 5 furthermore we use certain results and computations of py to exhibit five irreducible arithmetic fake bf p 2 c times bf p 2 c all these are connected smooth complex projective shimura varieties

the bounded degree problem for ence graph grammars the complexity of the bounded degree problem is analyzed for graph languages generated by ence graph grammars in particular the bounded degree problem is shown to be undecidable for ence graph grammars dexptime complete for confluent boundary ence graph grammars pspace complete for linear ence graph grammars and p complete for non blocking ence graph grammars in our main theorem we show that the bounded degree problem is nl complete for reduced non blocking ence graph grammars many of the shown results carry over to other types of graph grammars

a linear time algorithm for finding tree decompositions of small treewidth in this paper we give for constant k a linear time algorithm that given a graph g v e determines whether the treewidth of g is at most k and if so finds a tree decomposition of g with treewidth at most k a consequence is that every minor closed class of graphs that does not contain all planar graphs has a linear time recognition algorithm another consequence is that a similar result holds when we look instead for path decompositions with pathwidth at most some constant k

a comparison of boundary graph grammars and context free hypergraph grammars context free hypergraph grammars and boundary graph grammars of bounded nonterminal degree have the same power both for generating sets of graphs and for generating sets of hypergraphs arbitrary boundary graph grammars have more graph generating power than context free hypergraph grammars but they have the same hypergraph generating power to obtain these results several normal forms for boundary graph grammars are given it is also shown that the class of boundary graph languages is closed under the operation of edge contraction where the label of the edge indicates whether or not the edge should be contracted

computing twisted signatures and l classes of stratified spaces this paper establishes an atiyah type characteristic class formula for both the twisted signature and the twisted l classes of a singular space the result in its present form applies to coefficient systems that satisfy a transversality condition near the singularities this condition is frequently automatically satisfied e g on supernormal spaces or for local systems arising in certain geometric mapping situations we obtain applications to stratified maps by combining our formula with the cappell shaneson signature formula

an axiomatic definition of context free rewriting and its application to nlc graph grammars abstract an abstract notion of context free grammar is introduced it deals with abstract objects that can be words trees graphs or other combinatorial objects it is applied to nlc graph grammars introduced by rozenberg and janssens the monadic second order theory of a context free nlc set of graphs is decidable

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

the leech lattice and complex hyperbolic reflections we construct a natural sequence of finite covolume reflection groups acting on the complex hyperbolic spaces h 13 h 9 and h 5 and show that the 9 dimensional example coincides with the largest of the groups of mostow 11 our reflection groups arise as automorphism groups of certain lorentzian lattices over the eisenstein integers and we obtain our largest example by using the complex leech lattice in a manner inspired by conway 5 we also construct finite covolume reflection groups on the quaternionic hyperbolic spaces h 7 h 5 and h 3 again using the leech lattice and apply results of borcherds 4 to obtain automorphic forms for our groups

discrete subgroups acting transitively on vertices of a bruhat tits building we describe all the discrete subgroups of ad g0 f o aut f which act transitively on the set of vertices of b b f g0 the bruhat tits building of a pair f g0 of a characteristic zero non archimedean local eld and a simply connected absolutely almost simple f group if b is of dimension at least 4 in fact we classify all the maximal such subgroups we show that there are exactly eleven families of such subgroups and explicitly construct them moreover we show that four of these families act simply transitively on the vertices in particular we show that there is no such actions if either the dimension of the building is larger than 7 f is not isomorphic to qp for some prime p or the building is associated to sln d0 where d0 is a non commutative division algebra along the way we also give a new proof of siegel klingen theorem on the rationality of certain dedekind zeta functions and l functions 1

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

linear graph grammars power and complexity abstract a graph grammar is linear if it generates graphs with at most one nonterminal node linear graph grammars can simulate nonterminal bounded graph grammars which generate graphs with a bounded number of nonterminal nodes and derivation bounded graph grammars if a linear graph language contains connected graphs of bounded degree only then it is in nspace log n these results are shown for graph grammars with neighbourhood controlled embedding and with dynamic edge relabeling ence grammars

rene thom s work on geometric homology and bordism by the early 1950 s algebraic topology had reached great heights with serre s thesis and the calculations in the seminar of henri cartan of the cohomology of spaces with one nonzero homotopy group in terms of steenrod operations there was also the appearance of the new characteristic classes of vector bundles pontryagin classes z coefficients and chern weil classes coefficients in r joining those of stiefel whitney z mod 2 coefficients rene thom absorbed all this structure made vigorous use of it and added a geometric perspective that combined to revolutionize topology manifold theory and algebraic geometry for the unexpected and fertile results in bordism closed manifolds mod boundaries of manifolds of the 1954 paper 1 thom received the fields medal at edinburgh in 1958 many more applications of thom s ideas came even later