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

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

uniform independence in linear groups we show that for any non virtually solvable finitely generated group of matrices over any field there is an integer m such that given an arbitrary finite generating set for the group one may find two elements a and b that are both products of at most m generators such that a and b are free generators of a free subgroup this uniformity result improves the original statement of the tits alternative

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

isolation equidistribution and orbit closures for the sl 2 r action on moduli space we prove results about orbit closures and equidistribution for the sl 2 r action on the moduli space of compact riemann surfaces which are analogous to the theory of unipotent ows the proofs of the main theorems rely on the measure classication theorem of the rst two authors and a certain isolation property of closed sl 2 r invariant manifolds developed in this paper

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

a technique for efficient realization of wide band fir lti systems this correspondence introduces a technique for efficient realization of wide band finite length impulse response fir linear and time invariant lti systems it divides the overall frequency region into three subregions through lowpass bandpass and highpass filters realized in terms of only one filter the actual function to be approximated is in the low and high frequency regions realized using periodic subsystems in this way one can realize an overall wide band lti function in terms of three low cost subblocks leading to a reduced overall arithmetic complexity as compared to the regular realization a systematic design technique is provided and a detailed example shows multiplication and addition savings of 62 and 48 respectively for a fractional order differentiator with a 96 utilization of the bandwidth another example shows that the savings increase decrease with increased decreased bandwidth

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

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

a posteriori error estimation for the fractional step theta discretization of the incompressible navier stokes equations abstract in this work we derive a goal oriented a posteriori error estimator for the error due to time discretization as time discretization scheme we consider the fractional step theta method that consists of three subsequent steps of the one step theta method in every sub step the full incompressible system has to be solved in contrast to time integrators of operator splitting type the resulting fractional step theta method combines various desirable properties like second order accuracy strong a stability and very little numerical dissipation the derived error estimator is based on a mathematical trick we define an intermediate time discretization scheme based on a petrov galerkin formulation this method is up to a numerical quadrature error equivalent to the theta time stepping scheme the error estimator is assembled as one weighted residual term given by the dual weighted residual method measuring the error between real solution and solution to the petrov galerkin formulation that at no time has to be calculated and one additional residual estimating the discrepancy between actual time stepping scheme used for simulation and the intermediate petrov galerkin formulation

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

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

on uniform exponential growth for linear groups let be a finitely generated group given a finite set of generators s of the word length ls for an element is defined to be the smallest positive integer for which there exist s1 sn s s 1 such that s1 sn for each n n denote by bs n the set of elements in whose word length with respect to s is at most n it follows from the subadditive property of ls that limn bs n 1 n exists which we denote by s a finitely generated group is said to be of exponential growth if s 1 of polynomial growth if for some c 0 and d n bs n c n d for all n 1 and of intermediate growth otherwise for some finite generating set s of observe that the growth type of does not depend on the choice of generating set s if a finitely generated group is linear it is known that is either of polynomial growth in which case is virtually nilpotent or of exponential growth otherwise tit72 mil68 wol68