We calculate the cycle class of the Hurwitz divisor D2 on Mg for g = 2k given by the degree k + 1 covers of P 1 with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map from the Hurwitz space H2k,k+1 to the moduli space M2k.

Motivated by the classical Theorems of Picard and Siegel and their generalizations, we define the notion of an essentially large effective divisor and derive some of its arithmetic and function-theoretic consequences.We then investigate necessary and sufficient criteria for divisors to be essentially large.In essence, we prove that on a nonsingular irreducible projective variety X with Pic(X) = Z, every effective divisor with dim X + 2 or more components in general position is essentially large.

Unlike the case for self-similar measures satisfying the open set condition, it has been shown that the m-fold convolution of the uniform Cantor measure on the classical middlethird Cantor set has isolated points in its multifractal spectrum for any m 3. We show that this phenomena of isolated points holds for quite general Cantor measures on Cantor sets that can be far from self-similar.

Building upon works of Hironaka, Bierstone-Milman, Villamayor and Wlodarczyk. We give an a priori estimate for the complexity of the simplified Hironaka algorithm.

To every Gorenstein algebra A of finite vector space dimension greater than 1 over a field F of characteristic zero, and a linear projectionon its maximal ideal m with range equal to the annihilator Ann(m) of m, one can associate a certain algebraic hypersurface S� � m. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for F = C leads to interesting consequences in singularity theory. Also, for F = R such hypersurfaces naturally arise in CR- geometry. Applications of these hypersurfaces to problems in algebra and geometry are particularly striking when the hypersurfaces are affine homogeneous. In the present paper we establish a criterion for the affine homogeneity ofS�. This criterion requires the automorphism group Aut(m) of m to act transitively on the set of hyperplanes in m complementary to Ann(m). As a consequence of this result we obtain the affine homogeneity ofSunder the assumption that the algebra A is graded.

Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact.We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections.Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields.We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant.This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities.Using techniques of projections and elimination one can also recover some well known results.We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables.In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants.

We investigate genericities of reticular Lagrangian maps and reticular Legendrian maps in order to give generic classifications of caustics and wavefronts generated by a hypersurface germ without or with a boundary in a smooth manifold. We also give simpler proofs of main results in T. Tsukada, Reticular Lagrangian Singularities, Asian J. Math., 1 (1997), pp. 572–622 and T. Tsukada, Reticular Legendrian Singularities, Asian J. Math., 5:1 (2001), pp. 109–127.

Nadirashvili presented a beautiful example showing that the Poincare recurrence does not occur near a particular solution to the 2D Euler equation of inviscid incompressible fluids. Unfortunately, Nadirashvili’s setup of the phase space is not appropriate, and details of the proof are missing. This note fixes that.

We study the dynamics of the geodesic flow for a class of non-complete Riemannian metrics on a negatively curved surface M . These finite area surfaces are composed of finitely many “singular” surfaces of revolutions of the form y = x, r > 1 for 0 ≤ x ≤ 1 (thin pieces), together with connecting surfaces of bounded negative curvature (thick pieces). The curvature at every point is negative and bounded away from zero, but is unbounded in the “cusps.” The geodesic flow is non-complete because geodesics corresponding to singular unit tangent vectors pointing into the origin hit the cusps in finite time and then cease to be defined. Such singular unit tangent vectors are dense in the unit tangent bundle. Our main result is ergodicity of the geodesic flow (Theorem 5.1). It immediately follows that these metrics have dense geodesics in the unit tangent bundle SM . We also prove that the closed geodesics are dense in the unit tangent bundle (Theorem 6.1) The motivation for considering these metrics arises from studying the geodesic flow for the Weil-Petersson (WP) metric on two dimensional moduli spaces of Riemann surfaces., e.g., the moduli space for the once punctured torus. In several fundamental ways the geodesic flow we consider provides a good model for the WP geodesic flow. For example, Wolpert showed that the WP metric is also non-complete, has finite area, has negative curvature bounded away from zero, and is unbounded in the ”cusps.” Also, various authors have shown that the cusps can be approximated by singular surface of revolutions. At present, there is only a single technical obstruction (additional estimates on the derivatives of the GF in the cusp) that prevent us from applying this general method to prove ergodicity of the WP geodesic flow. See Section 7. The geometric properties of both types of flows imply that the geodesic flow is a uniformly hyperbolic dynamical system with singularities. The non-completeness causes pathologies in the stable and unstable manifolds, as for many billiard flows. For example, stable and unstable manifolds at a point, if they exist, may intersect a cusp point, and thus have only finite length. One needs extensions of “Pesin theory” to systems with singularities to study these geodesic flows. Another challenging aspect is that these surfaces may be simply connected, making the use of any of the traditional arguments involving boundaries of the covering spaces inapplicable. Studying the dynamics requires the development of new approches. Our strategy to prove ergodicity of the geodesic flow is to first establish non-uniform hyperbolicity, i.e., the Lyapunov exponents are non-zero at almost every point. We then prove that there