Related papers: Relative and logarithmic resolution of singulariti…

These notes aim to provide a classical approach to solving some conformable differential equations based on prior knowledge of how to solve ordinary differential equations. That is, using the methods of separation of variables, homogeneous…

Let X be a singular affine normal variety with coordinate ring R and assume that there is an R-order admitting a stability structure such that the scheme of relevant semistable representations is smooth, then we construct a partial…

These lecture notes focus on some numerical linear algebra algorithms in scientific computing. We assume that students are familiar with elementary linear algebra concepts such as vector spaces, systems of equations, matrices, norms,…

We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and nonconvex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental…

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from…

By generalizing the notion of linearization, a concept originally arising from microlocal analysis and symbolic calculus, to diffeological spaces, we make a first proposal setting for optimization problems in this category. We show how…

This paper surveys, in the first place, some basic facts from the classification theory of normal complex singularities, including details for the low dimensions 2 and 3. Next, it describes how the toric singularities are located within the…

We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which…

In [Kat94b], Kato defined his notion of a log regular scheme and studied the local behavior of such schemes. A toric variety equipped with its canonical logarithmic structure is log regular. And, these schemes allow one to generalize toric…

In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local…

The algebraic method of singular reduction is applied for non regular group action on manifolds which provides singular symplectic spaces. The problem of deformation quantization of the singular surfaces is the focus. For some examples of…

This paper formulates an elementary algorithm for resolution of singularities in a neighborhood of a singular point over a field of characteristic zero. The algorithm is composed of finite sequences of Newton polyhedra and monomial…

We introduce the notion of a relative log scheme with boundary: a morphism of log schemes together with a (log schematically) dense open immersion of its source into a third log scheme. The sheaf of relative log differentials naturally…

We discuss Hironaka's theorem on resolution of singularities in charactetistic 0 as well as more recent progress, both on simplifying and improving Hironaka's method of proof and on new results and directions on families of varieties,…

These are notes from a mini-course about the main results of arXiv:2206.03438: I explain how, using suitable valued fields, one obtains a natural notion of canonical stratifications (of e.g. algebraic subsets of $\mathbb{R}^n$). I also…

It is shown that, for any reduced algebraic variety in characteristic zero, one can resolve all but simple normal crossings (snc) singularities by a finite sequence of blowings-up with smooth centres which, at every step, avoids points…

On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line…

In this article we use our constructions from "Enlargements of Categories" (Theory and Applications of Categories, 14:357-398) to lay down some foundations for the application of A. Robinson's nonstandard methods to modern Algebraic…

This work unifies the analysis of various randomized methods for solving linear and nonlinear inverse problems by framing the problem in a stochastic optimization setting. By doing so, we show that many randomized methods are variants of a…