We generalize the notions of F-regular and F-pure rings to those for a pair (R, a) of a ring R and an ideal a ⊂ R with real exponent t > 0, and investigate those properties. Those “F-singularities of pairs” correspond to singularities of pairs of arbitrary codimension in birational geometry. Via this correspondence, we prove Inversion of Adjunction of arbitrary codimension, which states that fo...

We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in th...

The problem of reduction of singularities for germs of codimension one foliations in dimension three has been solved by Cano in [3]. The author divides the proof in two steps. The first one consists in getting pre-simple points and the second one is the passage from pre-simple to simple points. In arbitrary dimension of the ambient space the problem is open. In this paper we solve the second st...

Given a d-web of codimension one on a holomorphic n-dimensional manifold M0 (d > n), we assume that, at any point of M0, the d hyperplanes tangent to the local foliations at a point of M0 are distinct, and that there exists n of them in general position (but we do not require any n of them to be in general position). For such a web, we shall define some specific analytical subset S of M0 which ...

A presentation of a degree d form in n+1 variables as the sum of homogenous elements “essentially” involving n variables is called a codimension one decomposition. Codimension one decompositions are introduced and the related Waring Problem is stated and solved. Natural schemes describing the codimension one decompositions of a generic form are defined. Dimension and degree formulae for these s...

Given a reduced local algebra T over a suitable ring or field k we study the question of whether there are nontrivial algebra surjections R → T which induce isomorphisms Ω R/k ⊗ T → Ω T /k. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions ...

We prove that H( n 2)(SLn Z;Q) = 0, where ( n 2 ) is the cohomological dimension of SLn Z, and similarly for GLn Z. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GLn. These theorems are derived from a presentation of the Steinberg module for SLn Z whose generators are integral apartment classes, generalizing Manin...

Grove and Halperin [3] introduced a notion of taut immersions. Terng and Thorbergsson [5] give a slightly different definition and showed that taut immersions are a simultaneous generalization of taut immersions of manifolds into Euclidean spaces or spheres, and some interesting embeddings constructed by Bott and Samelson [1]. They go on to prove many theorems about such immersions. One particu...

We consider uncertainty quantification in the context of certification, i.e. showing that that the probability of some “failure” event is acceptably small. In this paper, we derive a new method for rigorous uncertainty quantification and conservative certification by combining McDiarmid’s inequality with input domain partitioning and a new concentration-ofmeasure inequality. We show that arbitr...

In this paper, we describe the global structure of webs in codimension one, and study in particular their singularities (the caustic). We define and study different concepts which have no interest locally near regular points, such as the type, the reducibility, the quasi-smoothness, the CIproperty (complete intersection), the dicriticity.... As an example of application, we explain how the alge...