the stable moduli space of riemann surfaces mumford s conjecture d mumford conjectured in 30 that the rational cohomology of the stable moduli space of riemann surfaces is a polynomial algebra generated by certain classes i of di mension 2i for the purpose of calculating rational cohomology one may replace the stable moduli space of riemann surfaces by b 1 where 1 is the group of isotopy classes of automorphisms of a smooth oriented connected surface of large genus tillmann s insight 41 that the plus construction makes b 1 into an infinite loop space led to a stable homo topy version of mumford s conjecture stronger than the original 22 we prove the stronger version relying on harer s stability theorem 15 vassiliev s theorem concerning spaces of functions with moderate singularities 43 42 and methods from homotopy theory

on the homotopy of the stable mapping class group by considering all surfaces and their mapping class groups at once it is shown that the classifying space of the stable mapping class group after plus construction b has the homotopy type of an infinite loop space the main new tool is a generalized group completion theorem for simplicial categories the first deloop of b coincides with that of miller m induced by the pairs of pants multiplication the classical representation of the mapping class group onto siegel s modular group is shown to induce a map of infinite loop spaces from b to k theory it is then a direct consequence of a theorem by charney and cohen cc that there is a space y such that b im j 1 2 y where im j 1 2 is the image of j localized away from the prime 2

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

the stable mapping class group and q ℂp in t2 it was shown that the classifying space of the stable mapping class groups after plus construction b has an infinite loop space structure this result and the tools developed in bm to analyse transfer maps are used here to show the following splitting theorem let p p e 0 e p 2 be the adams splitting of the p completed suspension spectrum of p then for some infinite loop space w p b p e 0 e p 3 w p where e i denotes the infinite loop space associated to the spectrum e i the homology of e i is known and as a corollary one obtains large families of torsion classes in the homology of the stable mapping class group this splitting also detects all the miller morita mumford classes our results suggest a homotopy theoretic refinement of the mumford conjecture the above p adic splitting uses a certain infinite loop map b p 1 that induces an isomorphims in rational cohomology precisely if the mumford conjecture is true we suggest that might be a homotopy equivalence

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

fptas for optimizing polynomials over the mixed integer points of polytopes in fixed dimension we show the existence of a fully polynomial time approximation scheme fptas for the problem of maximizing a non negative polynomial over mixed integer sets in convex polytopes when the number of variables is fixed moreover using a weaker notion of approximation we show the existence of a fully polynomial time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed integer sets in convex polytopes when the number of variables is fixed

universal moduli spaces of surfaces with flat bundles and cobordism theory for a compact connected lie group g we study the moduli of pairs e w here is a genus g riemann surface and e is a flat g bundle varying both the riemann surface and the flat bundle leads to a moduli space m g parametrizing families riemann surfaces with flat g bundles we show that there is a stable range in which the homology of m g is independent of g the stable range depends on the genus of the surface we then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space rationally the stable cohomology of this moduli space is generated by the mumford morita miller classes and the ring of characteristic classes of principal g bundles h bg equivalently our theorem calculates the homology of the moduli space of semi stable holomorphic bundles on riemann surfaces we then identify the homotopy type of the category of one manifolds and surface cobordisms each equipped with a flat g bundle our methods combine the classical techniques of atiyah and bott with the new techniques coming out of madsen and weiss s proof of mumford s conjecture on the stable cohomology of the moduli space of riemann surfaces

homological stability for moduli spaces of high dimensional manifolds i s galatius was partially supported by nsf grants dms 1105058 and dms 1405001 the european research council erc under the european union s horizon 2020 research and innovation pro gramme grant agreement no 682922 as well as the danish national research foundation through the centre for symmetry and deformation dnrf92 and erc 682992 o randal williams was partially supported by the herchel smith fund and epsrc grant ep m027783 1

mod 2 homology of the stable spin mapping class group using the main result of madsen and weiss mw we compute the mod 2 homology of spin mapping class groups in the stable range in earlier work g we computed the stable mod p homology of the oriented mapping class group and the methods and results here are very similar the forgetful map from the spin mapping class group to the oriented mapping class groups induces a homology isomorphism for odd p but for p 2 it is far from being an isomorphism we include a general discussion of tangential structures on 2 manifolds and their mapping class groups and then specialise to spin structures

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

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

the homology of the mapping class group 1 2 h btg a h mg q in this paper we will show that m g btg and b i l sg get more and more complicated as the genus g tends to infinity more precisely we will prove theorem 1 1 let q z2 z4 z6 denote the polynomial algebra of generators z2n in dimension in n 1 2 3 there are classes y2 y4 y2n with y2n in the 2nth cohomology group h 2n b i sg z such that the homomorphism of algebras sending z2n toy2n

a three valued walsh transform from decimations of helleseth gong sequences the walsh transform of two level autocorrelation sequences has played an important role in the construction of the set of sequences in which any two sequences are orthogonal for p ary sequences there are only two basic classes of two level autocorrelation sequences with no subfield structures for an arbitrary odd prime p one is the class of p ary m sequences and the other is the class of p ary helleseth gong sequences in this paper the walsh transform of a subclass of p ary helleseth gong sequences and the walsh transform of their particular decimations are completely determined and are shown to be three valued

the homotopy type of the cobordism category the embedded cobordism category under study in this paper generalizes the category of conformal surfaces introduced by g segal in s2 in order to formalize the concept of field theories our main result identifies the homotopy type of the classifying space of the embedded d dimensional cobordism category for all d for d 2 our results lead to a new proof of the generalized mumford conjecture somewhat different in spirit from the original one presented in mw