cubic curves and totally geodesic subvarieties of moduli space in this paper we present the rst example of a primitive totally geodesic subvariety f m g n with dim f 1 the variety we consider is a surfacef m 1 3 dened using the projective geometry of

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

connected components of prym eigenform loci in genus three this paper is devoted to the classification of connected components of prym eigenform loci in the strata mathcal h 2 2 mathrm odd and mathcal h 1 1 2 of the bundle of abelian differentials omega mathcal m 3 over mathcal m 3 these loci discovered by mcmullen duke math j 133 569 590 2006 are mathrm gl 2 mathbb r invariant submanifolds of complex dimension 3 of omega mathcal m g that project to the locus of riemann surfaces whose jacobian variety has a factor admitting real multiplication by some quadratic order mathcal o d these algebraic varieties are not necessarily irreducible the main result we show is that for each discriminant d the corresponding locus has one component if d equiv 0 4 text mod 8 two components if d equiv 1 text mod 8 and is empty if d equiv 5 text mod 8 our result contrasts with the case of prym eigenform loci in the strata mathcal h 1 1 in genus 2 studied by mcmullen ann math 2 165 2 397 456 2007 which is connected for every discriminant d

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

on the effective cone of m g n abstract for every g 2 and n g 1 we exhibit infinitely many extremal effective divisors in m g n coming from the strata of abelian differentials

erratum teichmuller curves in moduli space eisenstein series and an application to triangular billiards invent math 97 553 583 1989

completely periodic directions and orbit closures of many pseudo anosov teichmueller discs in q 1 1 1 1 in this paper we investigate the closure of a large class of teichmuller discs in the stratum mathcal q 1 1 1 1 or equivalently in a rm gl 2 mathbb r invariant locus mathcal l of translation surfaces of genus three we describe a systematic way to prove that the rm gl 2 mathbb r orbit closure of a translation surface in mathcal l is the whole locus mathcal l the strategy of the proof is an analysis of completely periodic directions on such a surface and an iterated application of ratner s theorem to unipotent subgroups acting on an adequate splitting this analysis applies for example to all teichmuller discs obtained by the thurston veech s construction with a trace field of degree three which are moreover obviously not veech we produce an infinite series of such examples and show moreover that the favourable splitting situation does not arise everywhere on mathcal l contrary to the situation in genus two we also study completely periodic directions on translation surfaces in mathcal l for instance we prove that completely periodic directions are dense on surfaces obtained by the thurston veech s construction

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

weak mixing directions in non arithmetic veech surfaces we show that the billiard in a regular polygon is weak mixing in almost every invariant surface except in the trivial cases which give rise to lattices in the plane triangle square and hexagon more generally we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non arithmetic veech surface and show that the hausdorff dimension of the set of non weak mixing directions is not full we also provide a necessary condition verified for instance by the veech surface corresponding to the billiard in the pentagon for the set of non weak mixing directions to have positive hausdorff dimension

connected components of the moduli spaces of abelian differentials with prescribed singularities consider the moduli space of pairs c where c is a smooth compact complex curve of a given genus and is a holomorphic 1 form on c with a given list of multiplicities of zeroes we describe connected components of this space this classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons and in the study of geometry of translation surfaces

algebraic curves uniformized by congruence subgroups of triangle groups we construct certain subgroups of hyperbolic triangle groups which we call congruence subgroups these groups include the classical congruence subgroups of sl2 z hecke triangle groups and 19 families of arithmetic triangle groups associated to shimura curves we determine the eld of moduli of the curves associated to these groups and thereby realize the groups psl2 fq and pgl2 fq regularly as galois groups

dynamics of sl2 ℝ over moduli space in genus two this paper classifies orbit closures and invariant measures for the natural action of sl2 r on um2 the bundle of holomorphic 1 forms over the moduli space of riemann surfaces of genus two

caratheodory s metrics on teichmuller spaces and l shaped pillowcases one of the most important results in teichmuller theory is royden s theorem which says that the teichmuller and kobayashi metrics agree on the teichmuller space of a given closed riemann surface the problem that remained open is whether the caratheodory metric agrees with the teichmuller metric as well in this article we prove that these two metrics disagree on each t g the teichmuller space of a closed surface of genus g 2 the main step is to establish a criterion to decide when the teichmuller and caratheodory metrics agree on the teichmuller disk corresponding to a rational jenkins strebel differential first we construct a holomorphic embedding h k tg n corresponding to the criterion says that the two metrics agree on this disk if and only if a certain function h k h can be extended to a holomorphic function t g n h we then show by explicit computation that this is not the case for quadratic differentials arising from l shaped pillowcases

teichmuller curves triangle groups and lyapunov exponents we construct a teichmuller curve uniformized by a fuchsian triangle group commensurable to m n for every m n in most cases for example when m 6 n and m or n is odd the uniformizing group is equal to the triangle group m n our construction includes the teichmuller curves constructed by veech and ward as special cases the construction essentially relies on properties of hypergeometric differential operators for small m we find billiard tables that generate these teich muller curves we interpret some of the so called lyapunov exponents of the kontsevich zorich cocycle as normalized degrees of a natural line bundle on a teichmuller curve we determine the lyapunov exponents for the teichmuller curves we construct

prym varieties and teichmuller curves this paper gives a uniform construction of infinitely many primitive teichmuller curves v mg for g 2 3 and 4

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

periodic points on veech surfaces and the mordell weil group over a teichmuller curve periodic points are points on veech surfaces whose orbit under the group of affine diffeomorphisms is finite we characterize those points as being torsion points if the veech surfaces is suitably mapped to its jacobian or an appropriate factor thereof for a primitive veech surface in genus two we show that the only periodic points are the weierstras points and the singularities

deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod ulus 1 the functions we consider are the corresponding eigen functions in theorem 1 1 we prove that the limit inferior of the ergodic sums n x 0 x n 1 n n is bounded for every point x in the phase space in theorem 1 2 we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length under additional assumptions we prove that ergodic integrals satisfy the central limit theorem theorem 1 3 theorem 1 9

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

braid groups and hodge theory this paper gives an account of the unitary representations of the braid group that arise via the hodge theory of cyclic branched coverings of mathbb p 1 highlighting their connections with ergodic theory complex reflection groups moduli spaces of 1 forms and open problems in surface topology

splitting mixed hodge structures over affine invariant manifolds we prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties the result is deduced from a generalization of a theorem of moller namely we prove that the image of a certain twisted abel jacobi map lands in the torsion of a factor of the jacobians this statement can be viewed as a splitting of certain mixed hodge structures

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

characterizations of lattice surfaces we answer a question of vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles we show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality we deduce several other characterizations of the lattice property

the gauss bonnet theorem for cone manifolds and volumes of moduli spaces this paper generalizes the gauss bonnet formula to a class of stratied spaces called riemannian cone manifolds as an application we compute the volumes of the moduli spacesm0 n with respect to the complex hyperbolic metrics introduced by picard deligne mostow and thurston

spectral decomposition of square tiled surfaces we consider n copies of a square s0 and define selfadjoint extensions of the euclidean laplacian acting on left mathcal c 0 infty s 0 right n by choosing some boundary conditions that are parametrized by two unitary matrices h and v acting on mathbb c n denoting by sp n h v the spectrum of such an operator we derive conditions on h and v so that the following spectral decomposition holds rm sp delta n h v bigcup 1 leq i leq k rm sp delta n i h i v i quad rm with sum 1 k n i n if h and v are permutation matrices this gives a spectral decomposition of the spectrum of the square tiled surface defined by the corresponding permutations we apply this to derive examples related to isospectrality and to high multiplicity

foliations of hilbert modular surfaces the hilbert modular surface xd is the moduli space of abelian varieties a with real multiplication by a quadratic order of discriminant d 1 the locus where a is a product of elliptic curves determines a finite union of algebraic curves xd 1 xd in this paper we show the lamination xd 1 extends to an essentially unique foliation fd of xd by complex geodesics the geometry of fd is related to teichm uller theory holomorphic motions polygonal billiards and lattes rational maps we show every leaf of fd is either closed or dense and compute its holonomy we also introduce refinements tn i of the classical modular curves on xd leading to an explicit description of xd 1

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

a new look at type 2 fuzzy sets and type 2 fuzzy logic systems we propose the concept of a conditional fuzzy set and prove that a type 2 fuzzy set is equivalent to a conditional fuzzy set we show that both the conditional fuzzy sets and the type 2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other whereas in the type 2 fuzzy set framework the secondary variable depends on the primary variable by definition it is this dependence between the primary and secondary variables that makes the type 2 fuzzy sets a complex subject while the conditional fuzzy sets do not have this built in dependence and thus are much easier to analyze with the fuzzy relation formulation powerful tools in fuzzy set theory such as zadeh s compositional rule of inference can be used to obtain the marginal fuzzy sets of the type 2 and conditional fuzzy sets transforming the type 2 problems back to the conventional type 1 domain with the help of the marginal fuzzy set concept we show that a type 2 fuzzy logic system can be designed in the same way as designing a type 1 fuzzy logic system

schiffer variations and abelian differentials abstract deformations of compact riemann surfaces are considered using a cech cohomology sliding overlaps approach cocycles are calculated for conformal cutting and regluing deformations at zeros of abelian differentials deformations fixing the periods of a differential and deformations splitting zeros are considered a second order deformation expansion is presented for the riemann period matrix a complete deformation expansion is presented for abelian differentials schiffer s kernel function approach for deformations of a green s function is followed

prym covers theta functions and kobayashi curves in hilbert modular surfaces algebraic curves in hilbert modular surfaces that are totally geodesic for the kobayashi metric have very interesting geometric and arithmetic properties e g they are rigid there are very few methods known to construct such algebraic geodesics that we call kobayashi curves we give an explicit way of constructing kobayashi curves using determinants of derivatives of theta functions this construction also allows to calculate the euler characteristics of the teich m uller curves constructed by mcmullen using prym covers

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

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