Related papers: Boundedness in general type MMP

We give a definition of thickness in $\mathbb{R}^d$ that is useful even for totally disconnected sets, and prove a Gap Lemma type result. We also guarantee an interval of distances in any direction in thick compact sets, relate thick sets…

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in…

We generalize Miyanishi's theory of almost minimal models of log smooth surfaces with reduced boundary to the case of arbitrary log surfaces defined over an algebraically closed field. Given an MMP run of a log surface $(X,D)$ we define and…

Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied.…

We prove the local boundedness for solutions to a class of obstacle problems with non-standard growth conditions. The novelty here is that we are able to establish the local boundedness under a sharp bound on the gap between the growth…

We consider mesh functions which are discrete convex in the sense that their central second order directional derivatives are positive. Analogous to the case of a uniformly bounded sequence of convex functions, we prove that the uniform…

Biggins [Uniform convergence of martingales in the branching random walk. {\em Ann. Probab.}, 20(1):137--151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex…

Nakayama showed that deformation invariance of plurigenera for smooth complex varieties follows from the MMP and Abundance Conjectures. We generalize his result to families of singular pairs over DVRs of positive or mixed characteristic. As…

We study the geometry, topological properties and smoothness of the boundaries of closed $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in \mathbb{R}^2 \, : \, \textrm{dist}(x, E) \leq \varepsilon \}$ of compact planar sets $E \subset…

We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the…

The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the…

The problem we are considering came up in connection with the classification of singularities in positive characteristic. Then it is important that certain invariants like the determinacy can be bounded simultaneously in families of formal…

In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on…

Let $G$ be a closed highly homogeneous subgroup of $S_{\infty}$ not involving circular orderings. We show that the closure of a conjugacy class from $G$ contains a conjugacy class which is comeagre in it. Furthermore, we show that the…

The topology of a separable metrizable space $M$ is \emph{generated} by a family $\mathcal{C}$ of its subsets provided that a set $A\subseteq M$ is closed in $M$ if and only if $A\cap C$ is closed in $C$ for each $C\in \mathcal{C}$. The…

We study the topology of polynomial functions by deforming them generically. We explain how the non-conservation of the total ``quantity'' of singularity in the neighbourhood of infinity is related to the variation of topology in certain…

We consider $K_X$-negative extremal contractions $f\colon X\to (Z,o)$, where $X$ is an algebraic threefold with only $\epsilon$-log terminal Q-factorial singularities and $(Z,o)$ is a two (resp., one)-dimensional germ. The main result is…

For a natural extension of the circular unitary ensemble of order n, we study as n tends to infinity, the asymptotic behavior of the sequence of orthogonal polynomials with respect to the spectral measure. The last term of this sequence is…

In this paper we generalize a key result relating singular limits of certain relative entropies with index in the setting of conformal nets, which has played an important role recently in the mathematical theory of relative entropies in the…

We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width. In this setting we prove that limits may be chosen independently at a countably infinite…