Point-Counting and the Zilber–Pink Conjecture
Point-Counting and the Zilber–Pink Conjecture

Author: Jonathan Pila

Publisher: Cambridge University Press

Published: 2022-06-09

Total Pages: 268

ISBN-13: 1009301926

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Point-counting results for sets in real Euclidean space have found remarkable applications to diophantine geometry, enabling significant progress on the André–Oort and Zilber–Pink conjectures. The results combine ideas close to transcendence theory with the strong tameness properties of sets that are definable in an o-minimal structure, and thus the material treated connects ideas in model theory, transcendence theory, and arithmetic. This book describes the counting results and their applications along with their model-theoretic and transcendence connections. Core results are presented in detail to demonstrate the flexibility of the method, while wider developments are described in order to illustrate the breadth of the diophantine conjectures and to highlight key arithmetical ingredients. The underlying ideas are elementary and most of the book can be read with only a basic familiarity with number theory and complex algebraic geometry. It serves as an introduction for postgraduate students and researchers to the main ideas, results, problems, and themes of current research in this area.

Families of Varieties of General Type
Families of Varieties of General Type

Author: János Kollár

Publisher: Cambridge University Press

Published: 2023-04-30

Total Pages: 491

ISBN-13: 1009346105

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The first complete treatment of the moduli theory of varieties of general type, laying foundations for future research.

Fractional Sobolev Spaces and Inequalities
Fractional Sobolev Spaces and Inequalities

Author: D. E. Edmunds

Publisher: Cambridge University Press

Published: 2022-10-31

Total Pages: 169

ISBN-13: 1009254634

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Provides an account of fractional Sobolev spaces emphasising applications to famous inequalities. Ideal for graduates and researchers.

Variations on a Theme of Borel
Variations on a Theme of Borel

Author: Shmuel Weinberger

Publisher: Cambridge University Press

Published: 2022-11-30

Total Pages: 365

ISBN-13: 1107142598

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Explains, using examples, the central role of the fundamental group in the geometry, global analysis, and topology of manifolds.

Large Deviations for Markov Chains
Large Deviations for Markov Chains

Author: Alejandro D. de Acosta

Publisher:

Published: 2022-10-12

Total Pages: 264

ISBN-13: 1009063359

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This book studies the large deviations for empirical measures and vector-valued additive functionals of Markov chains with general state space. Under suitable recurrence conditions, the ergodic theorem for additive functionals of a Markov chain asserts the almost sure convergence of the averages of a real or vector-valued function of the chain to the mean of the function with respect to the invariant distribution. In the case of empirical measures, the ergodic theorem states the almost sure convergence in a suitable sense to the invariant distribution. The large deviation theorems provide precise asymptotic estimates at logarithmic level of the probabilities of deviating from the preponderant behavior asserted by the ergodic theorems.

O-Minimality and Diophantine Geometry
O-Minimality and Diophantine Geometry

Author: G. O. Jones

Publisher: Cambridge University Press

Published: 2015-08-20

Total Pages: 235

ISBN-13: 1316301060

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This collection of articles, originating from a short course held at the University of Manchester, explores the ideas behind Pila's proof of the Andre–Oort conjecture for products of modular curves. The basic strategy has three main ingredients: the Pila–Wilkie theorem, bounds on Galois orbits, and functional transcendence results. All of these topics are covered in this volume, making it ideal for researchers wishing to keep up to date with the latest developments in the field. Original papers are combined with background articles in both the number theoretic and model theoretic aspects of the subject. These include Martin Orr's survey of abelian varieties, Christopher Daw's introduction to Shimura varieties, and Jacob Tsimerman's proof via o-minimality of Ax's theorem on the functional case of Schanuel's conjecture.

Some Problems of Unlikely Intersections in Arithmetic and Geometry
Some Problems of Unlikely Intersections in Arithmetic and Geometry

Author: Umberto Zannier

Publisher: Princeton University Press

Published: 2012-03-25

Total Pages: 175

ISBN-13: 1400842719

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This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).

Around the Zilber-Pink Conjecture
Around the Zilber-Pink Conjecture

Author: Philipp Habegger

Publisher:

Published: 2017

Total Pages: 0

ISBN-13: 9782856298565

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"Following Faltings and Vojta's work proving the Mordell-Lang conjecture for abelian varieties and Raynaud's work proving the Manin-Mumford conjecture, many new diophantine questions appeared, often described as problems of unlikely intersections. The arithmetic of moduli spaces of abelian varieties and, more generally, Shimura varieties has been parallel-developed around the central André-Oort conjecture. These two themes can be placed in a common frame--the Zilber-Pink conjecture. This volume is an introduction to these problems and to the various techniques used: geometry, height theory, reductive groups and Hodge theory, Shimura varieties, and model theory via the notion of o-minimal structure."--Publisher.

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)
Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Author: Boyan Sirakov

Publisher: World Scientific

Published: 2019-02-27

Total Pages: 5393

ISBN-13: 9813272899

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The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.