structural phase transition in evolving networks a network as a substrate for dynamic processes may have its own dynamics we propose a model for networks which evolve together with diffusing particles through a coupled dynamics and investigate emerging structural property the model consists of an undirected weighted network of fixed mean degree and randomly diffusing particles of fixed density the weight w of an edge increases by the amount of traffics through its connecting nodes or decreases by a constant factor edges are removed with the probability p rew 1 1 w and replaced by new ones having w 0 at random locations we find that the model exhibits a structural phase transition between the homogeneous phase characterized by an exponentially decaying degree distribution and the heterogeneous phase characterized by the presence of hubs the hubs emerge as a consequence of a positive feedback between the particle and the edge dynamics

finiteness of teichmuller curves in non arithmetic rank 1 orbit closures we show that in any non arithmetic rank 1 orbit closure of translation surfaces there are only finitely many teichm uller curves we also show that in any non arithmetic rank 1 orbit closure any completely parabolic surface is veech

the boundary of an affine invariant submanifold we study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area abelian differentials over nodal riemann surfaces modulo zero area components the main result is a formula for the tangent space to the boundary we also prove finiteness results concerning cylinders a partial converse to the cylinder deformation theorem and a result generalizing part of the veech dichotomy

nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations in this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls with a proof that works equally well in a more general equivariant setting for a non equidimensional surjective holomorphic map between compact ball quotients our method applies to show that the set of critical values must be nonempty and of codimension 1 in the equivariant setting the line of arguments extend to holomorphic mappings of maximal rank into the complex projective space or the complex euclidean space yielding in the latter case a lower estimate on the dimension of the singular locus of certain holomorphic maps defined by integrating holomorphic 1 forms in another direction we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume provided that the target complex unit ball is of dimension m 2 giving in particular a new proof that a local biholomorphism between noncompact m ball quotients of finite volume must be a covering map whenever m 2 finally combining our results with hermitian metric rigidity we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball provided that either the lattice is cocompact or the ball is of dimension at least 2

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

small bound for birational automorphism groups of algebraic varieties with an appendix by yujiro kawamata we give an effective upper bound of bir x for the birational automorphism group of an irregular n fold with n 3 of general type in terms of the volume v v x under an albanese smoothness and simplicity condition to be precise rm bir x leq d 3 v 10 an optimum linear bound rm bir x leq frac 1 3 times 42 3 v is obtained for those threefolds with non maximal albanese dimension for all n 3 a bound rm bir x leq d n v 10 is obtained when albx is generically finite alb x is smooth and alb x is simple

fibered algebraic surfaces with low slope the background of this paper is the following conjecture on complex surfaces of general type let s be a smooth projective minimal surface of general type over ti with c 2 throughout the paper

from rational billiards to dynamics on moduli spaces this short expository note gives an elementary introduction to the study of dynamics on certain moduli spaces and in particular the recent breakthrough result of eskin mirzakhani and mohammadi we also discuss the context and applications of this result and connections to other areas of mathematics such as algebraic geometry teichmuller theory and ergodic theory on

square tiled surfaces and rigid curves on moduli spaces abstract we study the algebro geometric aspects of teichmuller curves parameterizing square tiled surfaces with two applications a there exist infinitely many rigid curves on the moduli space of hyperelliptic curves they span the same extremal ray of the cone of moving curves their union is a zariski dense subset hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n pointed rational curves for even n b the limit of slopes of teichmuller curves and the sum of lyapunov exponents for the teichmuller geodesic flow determine each other which yields information about the cone of effective divisors on the moduli space of curves

spatially self organized resilient networks by a distributed cooperative mechanism abstract the robustness of connectivity and the efficiency of paths are incompatible in many real networks we propose a self organization mechanism for incrementally generating onion like networks with positive degree degree correlations whose robustness is nearly optimal as a spatial extension of the generation model based on cooperative copying and adding shortcut we show that the growing networks become more robust and efficient through enhancing the onion like topological structure on a space the reasonable constraint for locating nodes on the perimeter in typical surface growth as a self propagation does not affect these properties of the tolerance and the path length moreover the robustness can be recovered in the random growth damaged by insistent sequential attacks even without any remedial measures

hodge teichmuller planes and finiteness results for teichmuller curves we prove that there are only finitely many algebraically primitive teichmuller curves in the minimal stratum in each prime genus at least 3 the proof is based on the study of certain special planes in the first cohomology of a translation surface which we call hodge teichmuller planes we also show that algebraically primitive teichmuller curves are not dense in any connected component of any stratum in genus at least 3 the closure of the union of all such curves in a fixed stratum is equal to a finite union of affine invariant submanifolds with unlikely properties results of this type hold even without the assumption of algebraic primitivity combined with work of nguyen and the second author a corollary of our results is that there are at most finitely many nonarithmetic teichmuller curves in h 4

singular fibers and kodaira dimensions let f x rightarrow mathbb p 1 be a non isotrivial semi stable family of varieties of dimension m over mathbb p 1 with s singular fibers assume that the smooth fibers f are minimal i e their canonical line bundles are semiample then kappa x le kappa f 1 if kappa x kappa f 1 then s frac 4 m 2 if kappa x ge 0 then s ge frac 4 m 2 in particular if m 1 s 6 and kappa x 0 then the family f is teichmuller

the deligne mumford compactification of the real multiplication locus and teichmuller curves in genus 3 in the moduli space mathcal m g of genus g riemann surfaces consider the locus mathcal r mathcal m mathcal o of riemann surfaces whose jacobians have real multiplication by the order mathcal o in a totally real number field f of degree g if g 3 we compute the closure of mathcal r mathcal m mathcal o in the deligne mumford compactification of mathcal m g and the closure of the locus of eigenforms over mathcal r mathcal m mathcal o in the deligne mumford compactification of the moduli space of holomorphic 1 forms for higher genera we give strong necessary conditions for a stable curve to be in the boundary of mathcal r mathcal m mathcal o boundary strata of mathcal r mathcal m mathcal o are parameterized by configurations of elements of the field f satisfying a strong geometry of numbers type restriction

semisimplicity and rigidity of the kontsevich zorich cocycle we prove that invariant subbundles of the kontsevich zorich cocycle respect the hodge structure in particular we establish a version of deligne semisimplicity in this context this implies that invariant subbundles must vary polynomially on affine manifolds all results apply to tensor powers of the cocycle and this implies that the measurable and real analytic algebraic hulls coincide we also prove that affine manifolds typically parametrize jacobians with non trivial endomorphisms if the field of affine definition is larger than mathbb q then a factor has real multiplication the tools involve curvature properties of the hodge bundles and estimates from random walks in the appendix we explain how methods from ergodic theory imply some of the global consequences of schmid s work on variations of hodge structures we also derive the kontsevich forni formula using differential geometry

canonical class inequality for fibred spaces we establish the canonical class inequality for families of higher dimensional projective manifolds as an application we get a new inequality between the chern numbers of threefolds x with smooth families of minimal surfaces of general type over a curve c 1 3 x 18c 3 x

a characterization of certain shimura curves in the moduli stack of abelian varieties let f x y be a semistable family of complex abelian varieties over a curve y of genus g y and smooth over the com

structure of the orthogonal group on v over l rings abstract we study the orthogonal group on v on a quadratic module v of rank n with an orthogonal basis over a commutative ring r satisfying some condition which holds for the ring of rational integers z the group on v will be shown to be a finite group indeed we factorize on v into a semidirect product of a normal subgroup d and a subroup s where d diagonal e1 e2 en ei 1 and s a direct product of a finite number of symmetric groups also we determine the order of on v further we show that any element in on v is a product of three elements of order 2 moreover we prove that on v is generated by two elements of order 2 and 2n respectively also generated by three elements of order 2

on the generation of finite simple groups by pairs of subgroups 1 12 can be read off from that paper th rs our principal concern here is with projective special linear groups similar methods might yield results for other chevalley groups but the com putational details would be formidable no particular significance should be attached to the bound 4 ill b i2 appearing in these results for large enough 1 a b the bound can certainly be reduced to approximately a i j b and the correct asymptotic bound is probably very much smaller the more serious deficiency in the results is the restriction that j a 1 b 12 in theorem 1 we have proofs that theorem 1 also holds in most of the remaining cases they are cumbersome and require a quite disproportionate amount of effort we have therefore not included them here a proof that every group

variations of hodge structures of a teichmuller curve teichmuller curves are geodesic discs in teichmuller space that project to an algebraic curve in the moduli space mg we show that for all g 2 teichmuller curves map to the locus of real multiplication in the moduli space of abelian varieties observe that mcmullen has shown that precisely for g 2 the locus of real multiplication is stable under the sl2 r action on the tautological bundle mg we also show that teichmuller curves are defined over number fields and we provide a completely algebraic description of teichmuller curves in terms of higgs bundles as a consequence we show that the absolute galois group acts on the set of teichmuller curves

percolation transitions with nonlocal constraint we investigate percolation transitions in a nonlocal network model numerically in this model each node has an exclusive partner and a link is forbidden between two nodes whose r neighbors share any exclusive pair the r neighbor of a node x is defined as a set of at most n r neighbors of x where n is the total number of nodes the parameter r controls the strength of a nonlocal effect the system is found to undergo a percolation transition belonging to the mean field universality class for r 1 2 the system undergoes a peculiar phase transition from a nonpercolating phase to a quasicritical phase where the largest cluster size g scales as g n with 0 74 1 in the marginal case with r 1 2 the model displays a percolation transition that does not belong to the mean field universality class

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles

billiards and teichmuller curves on hilbert modular surfaces this paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of riemann surfaces of genus two these teichmuller curves lie on hilbert modular surfaces parameterizing abelian varieties with real multiplication explicit examples constructed from l shaped polygons give billiard tables with optimal dynamical properties

length problems for automorphisms of modules over local rings abstract let r be a local ring and m a free module of a finite rank over r an element aut r m is said to be simple if 1 fixes a hyperplane of m we shall show that for any aut r m there exist a basis x for m and aut r m such that acts as a permutation on x and 1 is a product of m or less than m simple elements in aut r m where m is the order of the invariant factors of modulo the maximal ideal of r also we shall investigate the problem treated by e w ellers and h ishibashi factorizations of transformations over a valuation ring linear algebra appl 85 1987 17 27 in which they showed that is a product of simple elements and gave an upper bound of the smallest number of such factors of whereas in the present paper we will give lower bounds of in case that r is a local domain moreover we will factorize as a product of symmetries and transvections for some the matrix of which is diagonal