Related papers: Mather discrepancy and the arc spaces

Stolarsky's invariance principle quantifies the deviation of a subset of a metric space from the uniform distribution. Classically derived for spherical sets, it has been recently studied in a number of other situations, revealing a general…

For a nonsingular matrix $A$, we propose the form $Tr(^t\!A A^{-1})$, the trace of the product of its transpose and inverse, as a new invariant under congruence of nonsingular matrices.

There are two well known tasks, related to Newton polyhedra: to study invariants of singularities in terms of their Newton polyhedra, and to describe Newton polyhedra of resultants and discriminants. We introduce so called resultantal…

In this paper, we show that Shokurov's conjectures on the ACC for $a$-lc thresholds and the ACC for minimal log discrepancies are equivalent in the interval $[0,1)$. That is, the conjecture on ACC for $a$-lc thresholds holds for every…

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$.…

This paper formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepacy is computed by a divisor corresponding to a…

Given a lattice in an isocrystal, Mazur's inequality states that the Newton point of the isocrystal is less than or equal to the invariant measuring the relative position of the lattice and its transform under Frobenius. Conversely, it is…

We introduce an approach of Riemann--Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that its minimal log…

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic…

We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case…

We prove that the ACC conjecture for minimal log discrepancies holds for threefolds in $[1-\delta,+\infty)$, where $\delta>0$ only depends on the coefficient set. We also study Reid's general elephant for pairs, and show Shokurov's…

This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the determinantal variety over $K$.…

We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories…

We introduce and study invariants of singularities in positive characteristic called F-thresholds. They give an analogue of the jumping coefficients of multiplier ideals in characteristic zero. We discuss the connection between the…

For a fixed symmetric matrix A and symmetric perturbation E we develop purely deterministic bounds on how invariant subspaces of A and A+E can differ when measured by a suitable "row-wise" metric rather than via traditional measures of…

This paper identifies a new class of shape invariant models. These models are based on extensions of conventional quantum mechanics that satisfy a string-motivated minimal length uncertainty relation. An important feature of our…

For a fixed pair and fixed exponents, we prove the discreteness of log discrepancies over all log canonical triples formed by attaching a product of ideals with given exponents.

Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties…

Finding the distance to singularity for a matrix is a ubiquitous problem in numerical linear algebra, and is elegantly solved by the Eckart-Young-Mirsky theorem. Its structured variant naturally emerges when one considers structured…

We establish a new set of pointwise inequalities that order curvature invariants across various Petrov and Segre types of spacetimes. In arbitrary spacetime dimension, we systematically analyze inequalities among contractions of the Ricci…