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In a previous paper \cite{SV}, the authors studied the isolated singular fibers that can occur in algebraic fibrations of certain genus two fibrations. There the goal was to determine their monodromy factorizations with the goal of…

We determine the one point genus zero correlators of compactly supported forms of a subcritical Stein filling whose first Chern class vanishes. This is a step towards determining the full potential function of the filling. As an…

Graded rings provide a natural algebraic framework for encoding symmetry via decompositions into homogeneous components indexed by a group, together with multiplication rules reflecting the group operation. Among graded rings, strongly…

Let $K$ be a field of characteristic $p>0$, $A=K[[Y]]$ be a power series ring in one variable and $Q(A)$ be the field of fraction of $A$. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e.,…

In this article we study smooth families of stratified bundles in positive characteristic and the variation of their monodromy group.Our aim is, in particular, to strengthen the weak form of the positive equicharacteristic $p$-curvature…

In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves…

Given a well-ordered semi-group $\Gamma$ with a minimal system of generators of ordinal type at most $\omega n$ and of rational rank $r$, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation…

We study the vanishing neighbourhood of non-isolated singularities of functions on singular spaces by associating a general linear function. We use the carrousel monodromy in order to show how to get a better control over the attaching of…

For a real polynomial $f$ we present explicit zero-free angular sectors in the complex plane, symmetric with respect to the real axis, with angles depending only on the degree of $f$, and vertices expressed in terms of the coefficients of…

As a step toward uncovering the relation between the weak and the strong coupling regimes of the $\mathcal{N}=4$ super Yang-Mills theory beyond the specral level, we have developed in a previous paper [arXiv:1410.8533] a novel group…

Let $X$ be a generic curve of genus $g$ defined over an algebraically closed field $k$ of characteristic $p\geq 0$. We show that for $n$ sufficiently large there exists a tame rational map $f:X\to \PP^1_k$ with monodromy group $A_n$. This…

This paper proposes the use of $F$-split and globally $F$-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over…

This paper is a survey of the authors' recent results on "abc-surfaces" and the monodromy of their natural Lefschetz fibrations and projections to P^1 x P^1, see (arXiv:0910.2142). The results being surveyed explore various fundamental…

In this paper we obtain a system of flat coordinates on the monodromy manifold of each of the Painlev\'e equations. This allows us to quantise such manifolds. We produce a quantum confluence procedure between cubics in such a way that…

We study D-branes on smooth noncompact toric Calabi-Yau manifolds that are resolutions of abelian orbifold singularities. Such a space has a distinguished basis {S_i} for the compactly supported K-theory. Using local mirror symmetry we…

We study degenerate singular points of planar vector fields inside a (degenerated) flow-box. These kind of singularities are called fake saddles and their linear parts are always zero. We characterize fake saddles with non-zero second order…

We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product…

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular…

The Banana manifold $X_{\text{Ban}}$ is a compact Calabi-Yau threefold constructed as the conifold resolution of the fiber product of a generic rational elliptic surface with itself, first studied by Bryan. We compute Katz's genus 0…

In this brief note we explore the space of genus one and elliptic fibrations within CY manifolds, their organizing principles, and how they relate to the set of all CY manifolds. We provide examples of genus one fibered manifolds that…