Simons Symposia – serie

​​​​This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry.  It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions.  Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families.This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.

The second of three volumes devoted to the study of the trace formula, these proceedings focus on automorphic representations of higher rank groups. Based on research presented at the 2016 Simons Symposium on Geometric Aspects of the Trace Formula that took place in Schloss Elmau, Germany, the volume contains both original research articles and articles that synthesize current knowledge and future directions in the field. The articles discuss topics such as the classification problem of representations of reductive groups, the structure of Langlands and Arthur packets, interactions with geometric representation theory, and conjectures on the global automorphic spectrum. Suitable for both graduate students and researchers, this volume presents the latest research in the field. Readers of the first volume Families of Automorphic Forms and the Trace Formula will find this a natural continuation of the study of the trace formula.

This proceedings volume contains articles related to the research presented at the 2017 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning integral questions and their connections to notions in algebraic topology. This volume features original research articles as well as articles that contain new research and survey some of these recent developments. It is the first of three volumes dedicated to p-adic Hodge theory.

A series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches.

A series of three symposia took place on the topic of trace formulas, each with an accompanying proceedings volume. The present volume is the third and final in this series and focuses on relative trace formulas in relation to special values of L-functions, integral representations, arithmetic cycles, theta correspondence and branching laws. The first volume focused on Arthur’s trace formula, and the second volume focused on methods from algebraic geometry and representation theory. The three proceedings volumes have provided a snapshot of some of the current research, in the hope of stimulating further research on these topics. The collegial format of the symposia allowed a homogeneous set of experts to isolate key difficulties going forward and to collectively assess the feasibility of diverse approaches.

This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research.Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.

This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research.Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.

This proceedings volume contains articles related to the research presented at the 2019 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning non-abelian aspects This volume contains both original research articles as well as articles that contain both new research as well as survey some of these recent developments.

This proceedings volume contains articles related to the research presented at the 2019 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning non-abelian aspects This volume contains both original research articles as well as articles that contain both new research as well as survey some of these recent developments.

This book is a tribute to the memory of Yuri Ivanovich Manin, who passed away on January 7, 2023. Manin was one of the giants of modern mathematics. His work covered a wide range of fields, including logic, number theory, geometry, mathematical physics, theoretical computer science, and linguistics. The contributions collected here are on topics close to his life-long passion: arithmetic and algebraic geometry.

This book is a tribute to the memory of Yuri Ivanovich Manin, who passed away on January 7, 2023. Manin was one of the giants of modern mathematics. His work covered a wide range of fields, including logic, number theory, geometry, mathematical physics, theoretical computer science, and linguistics. The contributions collected here are on topics close to his life-long passion: arithmetic and algebraic geometry.

This book brings together contributions by top-level experts from a wide range of topics in modern Hodge theory, originating in the authors’ participation in the special years on Hodge theory at the Institute of Mathematical Sciences of the Americas (IMSA) in Miami.One of the main themes is the study of moduli spaces and their compactifications. Several articles speak of the singularities occuring in the boundaries of geometrical or Hodge-theoretic compactifications, semistable reduction, the implications of canonical models for model theory in the sense of logic, and fundamental groups of moduli spaces and their associated Torelli groups. Other topics include Mukai lattices, derived moduli spaces, foliations, Higgs bundles and hyperbolicity, the study of pseudoconvexity properties of neighborhoods of infinity, contributions to the theory of degenerations and limiting mixed Hodge structures.This text will provide an indispensable reference for research mathematicians and specialist graduate students, where the modern approaches to moduli spaces are illustrated by their realizations and applications in examples of interest for the interplay between Hodge theory and moduli spaces.

This book grew out of a series of three Simons Symposia in 2019, 2022, and 2024, that focused on the interactions between three types of dynamics—algebraic, complex, and arithmetic—and their connections to the closely related areas of algebraic, complex, and arithmetic geometry. Dynamical systems play a key role both in pure mathematics and its applications. These meetings brought experts together from across these fields for discussions on the most significant recent developments and directions for further research.

This proceedings volume contains articles related to the research presented at the 2022 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory and applications to related fields including number theory and algebraic geometry. This volume contains articles on  some of these new developments in this rapidly evolving field and other research that arose from the symposium.

Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classification of psuedoeffective codimension one distributions.Foliations play a fundamental role in algebraic geometry, for example in the proof of abundance for threefolds and to a solution of the Green-Griffiths conjecture for surfaces of general type with positive Segre class. The purpose of this volume is to foster communication and enable interactions between experts who work on holomorphic foliations and birational geometry, and to bring together leading researchers to demonstrate the powerful connection of ideas, methods, and goals shared by these two areas of study.

Thisvolume grew out of two Simons Symposia on "Nonarchimedean and tropicalgeometry" which took place on the island of St. John in April 2013 and inPuerto Rico in February 2015. Each meeting gathered a small group of expertsworking near the interface between tropical geometry and nonarchimedeananalytic spaces for a series of inspiring and provocative lectures on cuttingedge research, interspersed with lively discussions and collaborative work insmall groups. The articles collected here, which include high-level surveys aswell as original research, mirror the main themes of the two Symposia.Topicscovered in this volume include: Differential forms and currents, andsolutions of Monge-Ampere type differential equations on Berkovich spaces andtheir skeletons; The homotopy types of nonarchimedean analytifications;The existence of "faithful tropicalizations" which encode the topology andgeometry of analytifications;Relations between nonarchimedean analyticspaces and algebraic geometry, including logarithmic schemes, birationalgeometry, and the geometry of algebraic curves;Extended notions oftropical varieties which relate to Huber's theory of adic spaces analogously tothe way that usual tropical varieties relate to Berkovich spaces; and Relationsbetween nonarchimedean geometry and combinatorics, including deep andfascinating connections between matroid theory, tropical geometry, and Hodgetheory.

Featuring the work of twenty-three internationally-recognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, p-adic families, and other recent techniques from harmonic analysis and representation theory. Each peer-reviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in January-February 2014, is the product of intensive research collaboration by the participants over the course of the seven-day workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, p-adic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.

Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classification of psuedoeffective codimension one distributions.Foliations play a fundamental role in algebraic geometry, for example in the proof of abundance for threefolds and to a solution of the Green-Griffiths conjecture for surfaces of general type with positive Segre class. The purpose of this volume is to foster communication and enable interactions between experts who work on holomorphic foliations and birational geometry, and to bring together leading researchers to demonstrate the powerful connection of ideas, methods, and goals shared by these two areas of study.

Thisvolume grew out of two Simons Symposia on "Nonarchimedean and tropicalgeometry" which took place on the island of St. John in April 2013 and inPuerto Rico in February 2015. Each meeting gathered a small group of expertsworking near the interface between tropical geometry and nonarchimedeananalytic spaces for a series of inspiring and provocative lectures on cuttingedge research, interspersed with lively discussions and collaborative work insmall groups. The articles collected here, which include high-level surveys aswell as original research, mirror the main themes of the two Symposia.Topicscovered in this volume include: Differential forms and currents, andsolutions of Monge-Ampere type differential equations on Berkovich spaces andtheir skeletons; The homotopy types of nonarchimedean analytifications;The existence of "faithful tropicalizations" which encode the topology andgeometry of analytifications;Relations between nonarchimedean analyticspaces and algebraic geometry, including logarithmic schemes, birationalgeometry, and the geometry of algebraic curves;Extended notions oftropical varieties which relate to Huber's theory of adic spaces analogously tothe way that usual tropical varieties relate to Berkovich spaces; and Relationsbetween nonarchimedean geometry and combinatorics, including deep andfascinating connections between matroid theory, tropical geometry, and Hodgetheory.

Featuring the work of twenty-three internationally-recognized experts, this volume explores the trace formula, spectra of locally symmetric spaces, p-adic families, and other recent techniques from harmonic analysis and representation theory. Each peer-reviewed submission in this volume, based on the Simons Foundation symposium on families of automorphic forms and the trace formula held in Puerto Rico in January-February 2014, is the product of intensive research collaboration by the participants over the course of the seven-day workshop. The goal of each session in the symposium was to bring together researchers with diverse specialties in order to identify key difficulties as well as fruitful approaches being explored in the field. The respective themes were counting cohomological forms, p-adic trace formulas, Hecke fields, slopes of modular forms, and orbital integrals.

Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.