hodge integrals and degenerate contributions hodge integral techniques are used to compute the degree 1 degenerate contributions of curves of arbitrary genus in the gromov witten theory of 3 folds in the calabi yau case the contributions are compared to related m theoretic calculations in the fano case the contributions suggest new integrality conditions

powers of the euler class abstract let g be the topological group of orientation preserving homeomorphisms of the circle s 1 the classical euler class and its powers generate the cohomology of the classifying space b g in the transition from continuous g to discrete g the behavior of the powers of the euler class is altered significantly there are subgroups k of g for which the powers of the discrete euler class e n h 2 n b g z are non vanishing only up to a certain dimension in this work we determine the threshold dimension for certain groups of homeomorphisms of the circle our results are applied to the based mapping class groups m g and we show e n m g 0 n g but e n m g 0 n g at the threshold dimension n g the discrete euler class has torsion of order 2 g 2 g 1

on the possible multiplicities of the eigenvalues of a hermitian matrix whose graph is a tree we consider the general problem of determining which lists of multiplicities for the eigenvalues occur among hermitian matrices the graph of whose off diagonal entries is a given tree several restrictions are cited and a construction strategy is given together these are sufficient to characterize all lists for each tree in two infinite classes the double paths and generalized stars and to tabulate all lists for trees on fewer than nine vertices such tables should be useful for formulating and dispelling general conjectures

tropical lambda lengths measured laminations and convexity this work uncovers the tropical analogue for measured laminations of the convex hull construction in decorated teichmuller theory namely it is a study in coordinates of geometric degeneration to a point of thurston s boundary for teichmuller space this may offer a paradigm for the extension of the basic cell decomposition of riemann s moduli space to other contexts for general moduli spaces of flat connections on a surface in any case this discussion drastically simplifies aspects of previous related studies as is explained furthermore a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered in the limit finally the tropical analogue of the convex hull construction in minkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent interest

the second homology group of the mapping class group of an orientable surface in i 7 mumford shows that the picard group p ic is isomorphic to h2 f 2 and conjectures the latter is rank one g 3 we prove this below for g 5 another interpretation of this theorem may be obtained by identifying h2 f as bordism classes of fiber bundles f w4 t where t is a closed oriented surface sect 0 when f is closed every such bundle is bordant to f w t a bundle admitting a section s t w the theorem then says that

relations among tautological classes revisited abstract we give a simple generalisation of a theorem of morita 1989 10 11 which leads to a great number of relations among tautological classes on moduli spaces of riemann surfaces

on the cohomological dimension of the moduli space of riemann surfaces the moduli space of riemann surfaces of genus g 2 is up to a finite etale cover a complex manifold and so it makes sense to speak of its dolbeault cohomological dimension the conjecturally optimal bound is g 2 this expectation is verified in low genus and supported by harer s computation of its de rham cohomological dimension and by vanishing results in the tautological intersection ring in this paper we prove that such dimension is at most 2g 2 we also prove an analogous bound for the moduli space of riemann surfaces with marked points the key step is to show that the dolbeault cohomological dimension of each stratum of translation surfaces is at most g in order to do that we produce an exhaustion function whose complex hessian has controlled index in the construction of such a function some basic geometric properties of flat surfaces come into play

evaluating tautological classes using only hurwitz numbers hurwitz numbers count ramified covers of a riemann surface with prescribed monodromy as such they are purely combinatorial objects tautological classes on the other hand are distinguished classes in the intersection ring of the moduli spaces of riemann surfaces of a given genus and are thus geometric localization computations in gromov witten theory provide non obvious relations between the two this paper makes one such computation and shows how it leads to a master relation theorem 0 1 that reduces the ratios of certain interesting tautological classes to the pure combinatorics of hurwitz numbers as a corollary we obtain a purely combinatorial proof of a theorem of bryan and pandharipande expressing in generating function form classical computations by faber looijenga theorem 0 2

the class of the locus of intermediate jacobians of cubic threefolds we study the locus of intermediate jacobians of cubic threefolds within the moduli space mathcal a 5 of complex principally polarized abelian fivefolds and its generalization to arbitrary genus the locus of abelian varieties with a singular odd two torsion point on the theta divisor assuming that this locus has expected codimension g which we show to be true for g 5 and conjecturally for any g we compute the class of this locus and of its closure in the perfect cone toroidal compactification mathcal a g mathrm perf in the chow homology and the tautological ring

linearly independent vertices and minimum semidefinite rank we study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs we present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph and conjecture that the resulting graph invariant is equal to minimum semidefinite rank rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs we also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex

the johnson homomorphism and its kernel we give a new proof of a celebrated theorem of dennis johnson that asserts that the kernel of the johnson homomorphism on the torelli subgroup of the mapping class group is generated by separating twists in fact we prove a more general result that also applies to subsurface torelli groups using this we extend johnson s calculati on of the rational abelianization of the torelli group not only to the subsurface torelli groups but also to finite index subgroups of the torelli group that contain the kernel of the johnson homomorphism

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

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

local sensitivity diagnostics for bayesian inference we investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k dimensional parameter space we show that several previously suggested diagnostics such as the norm of the frechet derivative diverge at rate n k 2 if the base prior is an interior point in the class of priors under the density ratio topology diagnostics based on divergences exhibit similar asymptotic behavior we show that better asymptotic behavior can be obtained by suitably restricting the classes of priors we also extend the diagnostics to see how various marginals of the prior affect various marginals of the posterior

rational curves on complete intersections in positive characteristic abstract we study properties of rational curves on complete intersections in positive characteristic it has long been known that in characteristic 0 smooth calabi yau and general type varieties are not uniruled in positive characteristic however there are well known counterexamples to this statement we will show that nevertheless a general calabi yau or general type complete intersection in projective space is not uniruled we will also show that the space of complete intersections of degree d 1 d k containing a rational curve has codimension at least i 1 k d i 2 n 2 and give similar results for hypersurfaces containing higher genus curves

the number of distinct eigenvalues for which an index decreases multiplicity abstract for an hermitian matrix a whose graph is a tree t we study the number of eigenvalues of a whose multiplicity decreases when a particular vertex is deleted from t explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter the work is based in part on classical results about multiplicities but also on some new facts including a useful identity this allows us to give strong bounds based on simple facts about the location of the vertex in the tree some facts about matrices whose graphs are not trees are included and the classical diameter bound about the number of distinct eigenvalues for a tree follows

multiplicity lists for symmetric matrices whose graphs have few missing edges abstract we characterize the possible lists of multiplicities occurring among the eigenvalues of real symmetric or hermitian matrices whose graph is one of k n k n less an edge or both possibilities for k n less two edges the lists are quite different from those for trees some construction techniques are developed here and additional results with more missing edges are given including the case of several independent edges

gerby localization z3 hodge integrals and the gw theory of c z3 we exhibit a set of recursive relations that completely determine all equivariant gromov witten invariants of bbb c 3 bbb z 3 we interpret such invariants as bbb z 3 hodge integrals and produce relations among them via atiyah bott localization on moduli spaces of twisted stable maps to gerbes over bbb p 1

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

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

relative virtual localization and vanishing of tautological classes on moduli spaces of curves we prove a localization formula for the moduli space of stable relative maps as an application we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i g 1 genus 0 components as consequences we prove and generalize various conjectures and theorems about various moduli spaces of curves due to getzler ionel faber looijenga pandharipande diaz and others this theorem appears to be the geometric content behind these results the rest is straightforward graph combinatorics the theorem also suggests the importance of the stratification of the moduli space by number of rational components

topological recursive relations in h2g ℳg n we show that any degree at least g monomial in descendant or tautological classes vanishes on g n when g 2 this generalizes a result of looijenga and proves a version of getzler s conjecture the method we use is the study of the relative gromov witten invariants of 1 relative to two points combined with the degeneration formulas of ip1

a new proof of faber s intersection number conjecture we give a new proof of faber s intersection number conjecture concerning the top intersections in the tautological ring of the moduli space of curves mg the proof is based on a very straightforward geometric and combinatorial computation with double ramification cycles

the local gromov witten theory of curves the local gromov witten theory of curves is solved by localization and degeneration methods localization is used for the exact evalu ation of basic integrals in the local gromov witten theory of p 1 a tqft formalism is defined via degeneration to capture higher genus curves together the results provide a compete and effective solution the local gromov witten theory of curves is equivalent to the lo cal donaldson thomas theory of curves the quantum cohomology of the hilbert scheme points of c 2 and the orbifold quantum cohomol ogy the symmetric product of c 2 the results of the paper provide the local gromov witten calculations required for the proofs of these equivalences

the gorenstein conjecture fails for the tautological ring of mathcal overline m _ 2 n we prove that for n equal to at least one of the integers 8 12 16 20 the tautological ring r bullet overline mathcal m 2 n is not gorenstein in fact our n equals the smallest integer such that there is a non tautological cohomology class of even degree on overline mathcal m 2 n by work of graber and pandharipande such a class exists on overline mathcal m 2 20 and we present some evidence indicating that n is in fact 20

a proof of the faber intersection number conjecture we prove the famous faber intersection number conjecture and other more general results by using a recursion formula of n point functions for intersection numbers on moduli spaces of curves we also present some vanishing properties of gromov witten invariants

the classification of punctured torus groups thurston s ending lamination conjecture proposes that a flnitely generated kleinian group is uniquely determined up to isometry by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends we present a proof of this conjecture for punctured torus groups these are free two generator kleinian groups with parabolic commutator which should be thought of as representations of the fundamental group of a punctured torus as a consequence we verify the conjectural topological description of the deformation space of punctured torus groups including bers conjecture that the quasi fuchsian groups are dense in this space and prove a rigidity theorem two punctured torus groups are quasi conformally conjugate if and only if they are topologically conjugate

the hyperbolicity of the sphere complex via surgery paths handel and mosher have proved that the free splitting complex fs for the free group is gromov hyperbolic this is a deep and much sought after result since it establishes fs as a good analogue of the curve complex for surfaces we give a shorter alternative proof of this theorem using surgery paths in hatcher s sphere complex another model for the free splitting complex instead of handel and mosher s fold paths as a byproduct we get that surgery paths are unparameterized quasi geodesics in the sphere complex

a short proof of the λg conjecture without gromov witten theory hurwitz theory and the moduli of curves we give a short and direct proof of the g conjecture the approach is through the ekedahl lando shapiro vainshtein theorem which establishes the polynomiality of hurwitz numbers from which we pick off the lowest degree terms the proof is independent of gromovwitten theory we briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures

generating functions for hurwitz hodge integrals we describe explicit generating functions for a large class of hurwitz hodge integrals these are integrals of tautological classes on moduli spaces of admissible covers a stackily smooth compactification of the hurwitz schemes admissible covers and their tautological classes are interesting mathematical objects on their own but recently they have proved to be a useful tool for the study of the tautological ring of the moduli space of curves and the orbifold gromov witten theory of dm stacks

undirected graphs of hermitian matrices that admit only two distinct eigenvalues abstract we consider the problem of determining those undirected n vertex graphs with a corresponding hermitian matrix that admits only two distinct eigenvalues with multiplicities k and n k after giving some general algebraic characterizations of these dual multiplicity graphs we then prove two major graph theoretic necessary conditions on such graphs construction techniques are then developed and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two

asymptotic global robustness in bayesian decision theory in bayesian decision theory it is known that robustness with respect to the loss and the prior can be improved by adding new observations in this article we study the rate of robustness improvement with respect to the number of observations n three usual measures of posterior global robustness are considered the range of the bayes actions set derived from a class of loss functions the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class we show that the rate of convergence of the first measure of robustness is n while it is n for the other measures under reasonable assumptions on the class of loss functions we begin with the study of two particular cases to illustrate our results

computing top intersections in the tautological ring of mg we derive effective recursion formulae of top intersections in the tautological ring r mathcal m g of the moduli space of curves of genus g 2 as an application we prove a convolution type tautological relation in r g 2 mathcal m g and some interesting bernoulli number identities

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

the κ ring of the moduli of curves of compact type the subalgebra of the tautological ring of the moduli of curves of compact type generated by the classes is studied in all genera relations constructed via the virtual geometry of the moduli of stable quotients are used to obtain minimal sets of generators bases and betti numbers of the rings are computed a universality property relating the higher genus rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps the g formula for hodge integrals arises as the simplest consequence