Related papers: Rational points on analytic varieties
For many years I have taught an advanced statistical inference course for master's students using the text of Casella and Berger (2002). The book gives a comprehensive treatment of the core topics at a level that avoids measure theory while…
This is an expanded version of the notes for the lectures given by the author at RIMS in the summer of 1999 to give a detailed account of the proof for the (weak) factorization theorem of birational maps by…
This is an expanded version of my plenary lecture at the 8th European Congress of Mathematics in Portoro\v{z} on 23 June 2021. The main part of the paper is a survey of recent applications of complex-analytic techniques to the theory of…
We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with…
This is the expanded version of my talk at the workshop "Groups of Automorphisms in Birational and Affine Geometry", October 29--November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the…
We present a selection of basic results on Borel reducibility of Borel ideals and equivalence relations, especially those with comparably short proofs. The focal point are reducibility/irreducibility results related to some special…
This paper is an extended version of four lectures at PIMS in Vancouver given June 27 - 30, 2016. The primary goal of these lectures was to publicize the author's recent efforts to extend to representations of linear algebraic groups the…
In this note a characterization of anallytically Riesz operators is given. This work completes the article [1].
These are detailed notes on a variant of the non-abelian cohomology developed by Minhyong Kim in arXiv:0409456 (published in Invent. Math.) to study rational points of varieties over number fields. The current variant is used in…
Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry…
These expository notes discuss the arithmetic of rationally connected varieties. Detailed proofs of theorems of Koll\'ar, of Koll\'ar and Szab\'o and of Esnault about rationally connected varieties over finite fields and local fields are…
We prove Manin's conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study…
CONTENTS: 1 Introduction 2 Analytic Manifolds and Analytic Continuation of Metrics 3 Walker's Spacetimes and their Maximal Extension 4 Global Structure of de Sitter and Reissner-Nordstr\"om-de Sitter Cosmos 4.1 Special Cases 4.2 Collapsing…
A number of topics in analysis are discussed, with emphasis on basic principles. There is some overlap with "Elements of linear and real analysis" (arXiv:math/0108030), with numerous changes in content and presentation since then.
In a recent paper of Blomer, Bourgain, Radziwi{\l}{\l} and Rudnick, the authors proved the existence of small gaps between eigenvalues of the Laplacian in a rectangular billiard with sides $\pi$ and $\pi/\sqrt\alpha$, i.e. numbers of the…
This note replaces two earlier preprints (1101.3737 by Koll\'ar) and (1211.6681 by Nowak). It studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one…
We introduce invariant rings for forms (homogeneous polynomials) and for d points on the projective space, from the point of view of representation theory. We discuss several examples, addressing some computational issues. We introduce the…
These are the lecture notes of a set of lectures delivered at the 1995 Trieste summer school in June. I review some recent work on duality in four dimensional Maxwell theory on arbitrary four manifolds, as well as a new set of topological…
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12--16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski…
The purpose of this article is to explain the Pila-Zannier strategy for proving the Andr\'e-Oort conjecture. First, however, we will provide a brief introduction to the theory of Shimura varieties.