Related papers: Introduction to the Minimal Model Program and the …
The purpose of this note is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We apply then this result in order to give a partial proof of a conjecture raised by Bocci,…
Generalizing work of Smith and Hara, we give a new characterization of log-terminal singularities for finitely generated algebras over $\mathbb C$, in terms of purity properties of ultraproducts of characteristic $p$ Frobenii. The first…
Lecture notes of a course on birational geometry (taught at College de France, Winter 2011, with the support of Fondation Sciences Math\'ematiques de Paris). Topics covered: introduction into the subject, contractions and extremal rays,…
This is the first of a series of papers studying real algebraic threefolds using the minimal model program. The main results are outlined in Part II. The present part I. contains the necessary preliminary work concerning terminal…
Moser's Bernstein theorem \cite{moser61} says that an entire minimal graph of codimension 1 with bounded slope must be a hyperplane. An analogous result for arbitrary codimension is not true, by an example of Lawson-Osserman. Here, we show…
Suppose $R\rightarrow S$ is a faithfully flat ring map. The theory of twisted forms lets one compute, given an $R$-module $M$, how many isomorphism classes of $R$-modules $M^{\prime}$ satisfy $S\otimes_R M\cong S\otimes_R M^{\prime}$. This…
We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…
We consider four related problems. (1) Obtaining dimension estimates for the set of exceptional vantage points for the pinned Falconer distance problem. (2) Nonlinear projection theorems, in the spirit of Kaufman, Bourgain, and Shmerkin.…
This paper is concerned at the minimization fundamental gap problem for a class of two-dimensional degenerate sub-elliptic operators. We establish existence results for weak solutions, Sobolev embedding theorem and spectral theory of…
The aim of this text is to provide an elementary and self-contained exposition of Gromov's argument on topological overlap (the presentation is based on Gromov's work, as well as two follow-up papers of Matousek and Wagner, and of…
We consider two-dimensional $\mathcal{N}=(0,2)$ sigma models with the CP(1) target space. A minimal model of this type has one left-handed fermion. Nonminimal extensions contain, in addition, $N_f$ right-handed fermions. Our task is to…
In this paper, we propose a new representation of the minimal form factors in integrable quantum field theories. These are solutions of the two-particle form factor equations, which have no poles on the physical sheet. Their expression…
We establish the existence and uniqueness of rational conformal maps of minimal degree $n+1$ for opening up $n$ arcs. In earlier results, the degree was exponential in $n$. We also discuss two related problems. (a) We establish existence of…
In this paper, we prove the cone theorem and the contraction theorem for pairs $(X, B)$, where $X$ is a normal variety and $B$ is an effective $\mathbb R$-divisor on $X$ such that $K_X+B$ is $\mathbb R$-Cartier.
In this paper, we develop a theory of diminished multiplier ideals on singular varieties which was introduced by Hacon, and developed by Lehmann. We prove a result regarding the termination of certain type of flips with scaling of an ample…
We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.
In this article I investigate the phenomenon of minimum models of second-order set theories, focusing on Kelley--Morse set theory $\mathsf{KM}$, G\"odel--Bernays set theory $\mathsf{GB}$, and $\mathsf{GB}$ augmented with the principle of…
The Mori's memory function approach to spin dynamics in doped antiferromagnetic insulator combined with the assumption of temperature independent static spin correlations and constant collective mode damping leads to w/T scaling in a broad…
Monadic stability and the more general monadic dependence (or NIP) are tameness conditions for classes of logical structures, studied in the 80's in Shelah's classification program in model theory. They recently emerged in algorithmic and…
The aim of this note is to prove that almost-minimizers of the perimeter are Reifenberg flat, for a very weak notion of minimality. The main observation is that smallness of the excess at some scale implies smallness of the excess at all…