相关论文: On boundedness of singularities and minimal log di…

We show that log canonical thresholds of fixed dimension are standardized. More precisely, we show that any sequence of log canonical thresholds in fixed dimension $d$ accumulates in a way which is i) either similar to how standard and…

We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded…

We deduce results on the dimension and connectedness of degeneracy loci of maps of finite modules $f:M\to N$ over a local noetherian ring $(A,{\mathfrak m})$. We show for instance that the expected determinantal bounds on the dimension of…

The future progress of semi-device independent quantum information science depends crucially on our ability to bound the strength of the nonlocal correlations achievable with finite dimensional quantum resources. In this work, we…

In 2013, Koldobsky posed the problem to find a constant $d_n$, depending only on the dimension $n$, such that for any origin-symmetric convex body $K\subset\mathbb{R}^n$ there exists an $(n-1)$-dimensional linear subspace…

In this paper, an algebraic theory for local rings of finite embedding dimension is developed. Several extensions of (Krull) dimension are proposed, which are then used to generalize singularity notions from commutative algebra. Finally,…

In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for…

We construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size…

We answer the following long-standing question of Kolchin: given a system of algebraic-differential equations $\Sigma(x_1,\dots,x_n)=0$ in $m$ derivatives over a differential field of characteristic zero, is there a computable bound, that…

We consider the following conjecture: on a klt germ (X,x), for every finite set I there is a positive integer N with the property that for every R-ideal J on X with exponents in I, there is a divisor E over X that computes the minimal log…

I prove, answering a question of Zilber, that if $M$ is an algebraic variety of dimension strictly greater than one and $(M, \ldots)$ is a strongly minimal structure with atomic relations definable in the Zariski language on $M$, then $M$…

In this paper we study singularities in arbitrary characteristic. We propose Finite Determination Conjecture for Mather-Jacobian minimal log discrepancies in terms of jet schemes of a singularity. The conjecture is equivalent to the…

This is the major revision. The main purpose of this paper is to prove that minimal discrepancies of $n$-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension…

We construct partitions of $\mathbb{R}^n$ into three sets $\{\mathscr{X}(1),\mathscr{X}(2),\mathscr{X}(3)\}$ that locally minimize interfacial area among compactly supported volume preserving variations and that blow down at infinity to…

We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a…

In this paper, we prove that klt singularities are invariant under deformations if the generic fiber is $\mathbb{Q}$-Gorenstein. We also obtain a similar result for slc singularities. These are generalizations of results of Esnault-Viehweg…

We show that the number of marked minimal models of an n-dimensional smooth complex projective variety of general type can be bounded in terms of its volume, and, if n=3, also in terms of its Betti numbers. For an n-dimensional projective…

For a toric log variety with standard coefficients, we show that the minimal log discrepancy at a closed invariant point bounds the Cartier index of a neighbourhood.

We prove the local boundedness of the solutions to degenerate second order partial differential equations of Kolmogorov type with measurable coefficients in divergence form, under minimal integrability assumption on the lower order…

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that…