Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting recent aspects of this interaction, aimed at a broad audience of both graduate students and researchers. The present volume is based on lectures presented at the summer school on low-dimensional topology. These notes give fresh, concise, and high-level introductions to these developments, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field of low-dimensional topology and to senior researchers wishing to keep up with current developments. The volume begins with notes based on a special lecture by John Milnor about the history of the topology of manifolds. It also contains notes from lectures by Cameron Gordon on the basics of three-manifold topology and surgery problems, Mikhail Khovanov on his homological invariants for knots, John Etnyre on contact geometry, Ron Fintushel and Ron Stern on constructions of exotic four-manifolds, David Gabai on the hyperbolic geometry and the ending lamination theorem, Zoltan Szabo on Heegaard Floer homology for knots and three manifolds, and John Morgan on Hamilton's and Perelman's work on Ricci flow and geometrization.

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting recent aspects of this interaction, aimed at a broad audience of both graduate students and researchers. The present volume is based on lectures presented at the summer school on low-dimensional topology. These notes give fresh, concise, and high-level introductions to these developments, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field of low-dimensional topology and to senior researchers wishing to keep up with current developments. The volume begins with notes based on a special lecture by John Milnor about the history of the topology of manifolds. It also contains notes from lectures by Cameron Gordon on the basics of three-manifold topology and surgery problems, Mikhail Khovanov on his homological invariants for knots, John Etnyre on contact geometry, Ron Fintushel and Ron Stern on constructions of exotic four-manifolds, David Gabai on the hyperbolic geometry and the ending lamination theorem, Zoltan Szabo on Heegaard Floer homology for knots and three manifolds, and John Morgan on Hamilton's and Perelman's work on Ricci flow and geometrization.

Fifty Years Ago : Topology Of Manifolds In The 50's And 60's / John Milnor -- Dehn Surgery And 3-manifolds / Cameron Gordon -- Hyperbolic Geometry And 3-manifold Topology / David Gabai -- Ricci Flow And Thurston's Geometrization Conjecture / John W. Morgan (notes By Max Lipyanskiy) -- Notes On Link Homology / Marta Asaeda, Mikhail Khovanov -- Lecture Notes On Heegaard Floer Homology / Zoltan Szabo -- Contact Geometry In Low Dimensional Topology / John Etnyre -- Six Lectures On Four 4-manifolds / Ronald Fintushel, Ronald J. Stern. Tomasz S. Mrowka, Peter S. Ozsváth, Editors. Includes Bibliographical References (p. 313-315). Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics. The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting aspects of this interaction. This title is based on lectures presented at the summer school on low-dimensional topology.

This book contains written versions of the lectures given at the PCMI Graduate Summer School on the representation theory of Lie groups. The volume begins with lectures by A. Knapp and P. Trapa outlining the state of the subject around the year 1975, specifically, the fundamental results of Harish-Chandra on the general structure of infinite-dimensional representations and the Langlands classification. Additional contributions outline developments in four of the most active areas of research over the past 20 years. The clearly written articles present results to date, as follows: R. Zierau and L. Barchini discuss the construction of representations on Dolbeault cohomology spaces. D. Vogan describes the status of the Kirillov-Kostant ``philosophy of coadjoint orbits'' for unitary representations. K. Vilonen presents recent advances in the Beilinson-Bernstein theory of ``localization''. And Jian-Shu Li covers Howe's theory of ``dual reductive pairs''. Each contributor to the volume presents the topics in a unique, comprehensive, and accessible manner geared toward advanced graduate students and researchers. Students should have completed the standard introductory graduate courses for full comprehension of the work. The book would also serve well as a supplementary text for a course on introductory infinite-dimensional representation theory

The research topic for this IAS/PCMI Summer Session was nonlinear wave phenomena. Mathematicians from the more theoretical areas of PDEs were brought together with those involved in applications. The goal was to share ideas, knowledge, and perspectives. How waves, or 'frequencies', interact in nonlinear phenomena has been a central issue in many of the recent developments in pure and applied analysis. It is believed that wavelet theory - with its simultaneous localization in both physical and frequency space and its lacunarity - is and will be a fundamental new tool in the treatment of the phenomena.Included in this volume are write-ups of the 'general methods and tools' courses held by Jeff Rauch and Ingrid Daubechies. Rauch's article discusses geometric optics as an asymptotic limit of high-frequency phenomena. He shows how nonlinear effects are reflected in the asymptotic theory. In the article "Harmonic Analysis, Wavelets and Applications" by Daubechies and Gilbert the main structure of the wavelet theory is presented.Also included are articles on the more 'specialized' courses that were presented, such as "Nonlinear Schrodinger Equations" by Jean Bourgain and "Waves and Transport" by George Papanicolaou and Leonid Ryzhik. Susan Friedlander provides a written version of her lecture series "Stability and Instability of an Ideal Fluid", given at the Mentoring Program for Women in Mathematics, a preliminary program to the Summer Session. This Summer Session brought together students, fellows, and established mathematicians from all over the globe to share ideas in a vibrant and exciting atmosphere. This book presents the compelling results.

<p>This volume, with contributions by leading experts in the field, is a collection of lecture notes of the six minicourses given at the IAS/Park City Summer Mathematics Institute. It introduces advanced graduates and researchers in probability theory to several of the currently active research areas in the field. Each course is self-contained with references and contains basic materials and recent results. Topics include interacting particle systems, percolation theory, analysis on path and loop spaces, and mathematical finance. The volume gives a balanced overview of the current status of probability theory. An extensive bibliography for further study and research is included. This unique collection presents several important areas of current research and a valuable survey reflecting the diversity of the field.</p>

Computational Complexity Theory is the study of how much of a given resource is required to perform the computations that interest us the most. Four decades of fruitful research have produced a rich and subtle theory of the relationship between different resource measures and problems. At the core of the theory are some of the most alluring open problems in mathematics. This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity. The first week gives a general introduction to the field, including descriptions of the basic models, techniques, results and open problems. The second week focuses on lower bounds in concrete models. The final week looks at randomness in computation, with discussions of different notions of pseudorandomness, interactive proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). It is recommended for independent study by graduate students or researchers interested in computational complexity. The volume is recommended for independent study and is suitable for graduate students and researchers interested in computational complexity.

Computational Complexity Theory is the study of how much of a given resource is required to perform the computations that interest us the most. Four decades of fruitful research have produced a rich and subtle theory of the relationship between different resource measures and problems. At the core of the theory are some of the most alluring open problems in mathematics. This book presents three weeks of lectures from the IAS/Park City Mathematics Institute Summer School on computational complexity. The first week gives a general introduction to the field, including descriptions of the basic models, techniques, results and open problems. The second week focuses on lower bounds in concrete models. The final week looks at randomness in computation, with discussions of different notions of pseudorandomness, interactive proof systems and zero knowledge, and probabilistically checkable proofs (PCPs). It is recommended for independent study by graduate students or researchers interested in computational complexity. The volume is recommended for independent study and is suitable for graduate students and researchers interested in computational complexity.

Mapping Class Groups And Moduli Spaces Of Riemann Surfaces Were The Topics Of The Graduate Summer School At The 2011 Ias/park City Mathematics Institute. This Book Presents The Nine Different Lecture Series Comprising The Summer School, Covering A Selection Of Topics Of Current Interest. The Introductory Courses Treat Mapping Class Groups And Teichmüller Theory. The More Advanced Courses Cover Intersection Theory On Moduli Spaces, The Dynamics Of Polygonal Billiards And Moduli Spaces, The Stable Cohomology Of Mapping Class Groups, The Structure Of Torelli Groups, And Arithmetic Mapping Class Groups. The Courses Consist Of A Set Of Intensive Short Lectures Offered By Leaders In The Field, Designed To Introduce Students To Exciting, Current Research In Mathematics. These Lectures Do Not Duplicate Standard Courses Available Elsewhere. The Book Should Be A Valuable Resource For Graduate Students And Researchers Interested In The Topology, Geometry And Dynamics Of Moduli Spaces Of Riemann Surfaces And Related Topics. Titles In This Series Are Co-published With The Institute For Advanced Study/park City Mathematics Institute. Members Of The Mathematical Association Of America (maa) And The National Council Of Teachers Of Mathematics (nctm) Receive A 20% Discount From List Price.

<p>what Distinguishes Differential Geometry In The Last Half Of The Twentieth Century From Its Earlier History Is The Use Of Nonlinear Partial Differential Equations In The Study Of Curved Manifolds, Submanifolds, Mapping Problems, And Function Theory On Manifolds, Among Other Topics. The Differential Equations Appear As Tools And As Objects Of Study, With Analytic And Geometric Advances Fueling Each Other In The Current Explosion Of Progress In This Area Of Geometry In The Last Twenty Years. This Book Contains Lecture Notes Of Minicourses At The Regional Geometry Institute At Park City, Utah, In July 1992. Presented Here Are Surveys Of Breaking Developments In A Number Of Areas Of Nonlinear Partial Differential Equations In Differential Geometry. The Authors Of The Articles Are Not Only Excellent Expositors, But Are Also Leaders In This Field Of Research. All Of The Articles Provide In-depth Treatment Of The Topics And Require Few Prerequisites And Less Background Than Current Research Articles.</p>

<p>what Distinguishes Differential Geometry In The Last Half Of The Twentieth Century From Its Earlier History Is The Use Of Nonlinear Partial Differential Equations In The Study Of Curved Manifolds, Submanifolds, Mapping Problems, And Function Theory On Manifolds, Among Other Topics. The Differential Equations Appear As Tools And As Objects Of Study, With Analytic And Geometric Advances Fueling Each Other In The Current Explosion Of Progress In This Area Of Geometry In The Last Twenty Years. This Book Contains Lecture Notes Of Minicourses At The Regional Geometry Institute At Park City, Utah, In July 1992. Presented Here Are Surveys Of Breaking Developments In A Number Of Areas Of Nonlinear Partial Differential Equations In Differential Geometry. The Authors Of The Articles Are Not Only Excellent Expositors, But Are Also Leaders In This Field Of Research. All Of The Articles Provide In-depth Treatment Of The Topics And Require Few Prerequisites And Less Background Than Current Research Articles.</p>

Cover Contents Preface Introduction Peter S. Ozsvath and Tomasz S. Mrowka Fifty Years Ago: Topology of Manifolds in the 50's and 60's John Milnor Dehn Surgery and 3-Manifolds Cameron Gordon Hyperbolic Geometry and 3-Manifold Topology David Gabai Ricci Flow and Thurston's Geometrization Conjecture John W. Morgan (notes by Max Lipyanskiy) Notes on Link Homology Marta Asaeda and Mikhail Khovanov Lecture Notes on Heegaard Floer Homology Zoltan Szabo Contact Geometry in Low Dimensional Topology John Etnyre Six Lectures on Four 4-Manifolds Ronald Fintushel and Ronald J. Stern

"Data science is a highly interdisciplinary field, incorporating ideas from applied mathematics, statistics, probability, and computer science, as well as many other areas. This book gives an introduction to the mathematical methods that form the foundations of machine learning and data science, presented by leading experts in computer science, statistics, and applied mathematics. Although the chapters can be read independently, they are designed to be read together as they lay out algorithmic, statistical, and numerical approaches in diverse but complementary ways. This book can be used both as a text for advanced undergraduate and beginning graduate courses, and as a survey for researchers interested in understanding how applied mathematics broadly defined is being used in data science. It will appeal to anyone interested in the interdisciplinary foundations of machine learning and data science."--Site web de l'éditeur

"Data science is a highly interdisciplinary field, incorporating ideas from applied mathematics, statistics, probability, and computer science, as well as many other areas. This book gives an introduction to the mathematical methods that form the foundations of machine learning and data science, presented by leading experts in computer science, statistics, and applied mathematics. Although the chapters can be read independently, they are designed to be read together as they lay out algorithmic, statistical, and numerical approaches in diverse but complementary ways. This book can be used both as a text for advanced undergraduate and beginning graduate courses, and as a survey for researchers interested in understanding how applied mathematics broadly defined is being used in data science. It will appeal to anyone interested in the interdisciplinary foundations of machine learning and data science."--Site web de l'éditeur

A co-publication of the AMS and IAS/Park City Mathematics Institute Random matrix theory has many roots and many branches in mathematics, statistics, physics, computer science, data science, numerical analysis, biology, ecology, engineering, and operations research. This book provides a snippet of this vast domain of study, with a particular focus on the notations of universality and integrability. Universality shows that many systems behave the same way in their large scale limit, while integrability provides a route to describe the nature of those universal limits. Many of the ten contributed chapters address these themes, while others touch on applications of tools and results from random matrix theory. This book is appropriate for graduate students and researchers interested in learning techniques and results in random matrix theory from different perspectives and viewpoints. It also captures a moment in the evolution of the theory, when the previous decade brought major break-throughs, prompting exciting new directions of research.

[2010 Draft version with Solutions, many of which the 2013 release version does not have. A good complement to the 2013 release version.]Main subject categories: • Algebraic geometry • Conics • Cubic curves • Elliptic curves • Affine varieties • Projective varieties • Sheaves • Cohomology2010 Mathematics Subject Classification: • Primary 14–01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometryAlgebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology. The first chapter on conics is appropriate for first-year college students (and many high school students). Chapter 2 leads the reader to an understanding of the basics of cubic curves, while Chapter 3 introduces higher degree curves. Both chapters are appropriate for people who have taken multivariable calculus and linear algebra. Chapters 4 and 5 introduce geometric objects of higher dimension than curves. Abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on. The last chapter is on sheaves and cohomology, providing a hint of current work in algebraic geometry.

Exploring topics from classical and quantum mechanics and field theory, this book is based on lectures presented in the Graduate Summer School at the Regional Geometry Institute in Park City, Utah, in 1991. The chapter by Bryant treats Lie groups and symplectic geometry, examining not only the connection with mechanics but also the application to differential equations and the recent work of the Gromov school. Rabin's discussion of quantum mechanics and field theory is specifically aimed at mathematicians. Alvarez describes the application of supersymmetry to prove the Atiyah-Singer index theorem, touching on ideas that also underlie more complicated applications of supersymmetry. Quinn's account of the topological quantum field theory captures the formal aspects of the path integral and shows how these ideas can influence branches of mathematics which at first glance may not seem connected. Presenting material at a level between that of textbooks and research papers, much of the book would provide excellent material for graduate courses. The book provides an entree into a field that promises to remain exciting and important for years to come.

"Algebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology. The first chapter on conics is appropriate for first-year college students (and many high school students). Chapter 2 leads the reader to an understanding of the basics of cubic curves, while Chapter 3 introduces higher degree curves. Both chapters are appropriate for people who have taken multivariable calculus and linear algebra. Chapters 4 and 5 introduce geometric objects of higher dimension than curves. Abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on. The last chapter is on sheaves and cohomology, providing a hint of current work in algebraic geometry. This book is published in cooperation with IAS/Park City Mathematics Institute."--Provided by publisher

This book is based on lectures given at the Graduate Summer School of the 2015 Park City Mathematics Institute program “Geometry of moduli spaces and representation theory”, and is devoted to several interrelated topics in algebraic geometry, topology of algebraic varieties, and representation theory. Geometric representation theory is a young but fast developing research area at the intersection of these subjects. An early profound achievement was the famous conjecture by Kazhdan–Lusztig about characters of highest weight modules over a complex semi-simple Lie algebra, and its subsequent proof by Beilinson-Bernstein and Brylinski-Kashiwara. Two remarkable features of this proof have inspired much of subsequent development: intricate algebraic data turned out to be encoded in topological invariants of singular geometric spaces, while proving this fact required deep general theorems from algebraic geometry. Another focus of the program was enumerative algebraic geometry. Recent progress showed the role of Lie theoretic structures in problems such as calculation of quantum cohomology, K-theory, etc. Although the motivation and technical background of these constructions is quite different from that of geometric Langlands duality, both theories deal with topological invariants of moduli spaces of maps from a target of complex dimension one. Thus they are at least heuristically related, while several recent works indicate possible strong technical connections. The main goal of this collection of notes is to provide young researchers and experts alike with an introduction to these areas of active research and promote interaction between the two related directions.

Main subject categories: • Algebraic geometry • Conics • Cubic curves • Elliptic curves • Affine varieties • Projective varieties • Sheaves • Cohomology2010 Mathematics Subject Classification: • Primary 14–01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometryAlgebraic Geometry has been at the center of much of mathematics for hundreds of years. It is not an easy field to break into, despite its humble beginnings in the study of circles, ellipses, hyperbolas, and parabolas. This text consists of a series of exercises, plus some background information and explanations, starting with conics and ending with sheaves and cohomology. The first chapter on conics is appropriate for first-year college students (and many high school students). Chapter 2 leads the reader to an understanding of the basics of cubic curves, while Chapter 3 introduces higher degree curves. Both chapters are appropriate for people who have taken multivariable calculus and linear algebra. Chapters 4 and 5 introduce geometric objects of higher dimension than curves. Abstract algebra now plays a critical role, making a first course in abstract algebra necessary from this point on. The last chapter is on sheaves and cohomology, providing a hint of current work in algebraic geometry.

xxix, 410 p. : 23 cm "In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently."--Publisher description Includes bibliographical references (p. 391-399) and index Preface -- 1. Fourier series : some motivation -- 2. Interlude : Analysis concepts -- 3. Pointwise convergence of Fourier series -- 4. Summability methods -- 5. Mean-square convergence of Fourier series -- 6. A tour of discrete Fourier and Haar analysis -- 7. The Fourier transform in paradise -- 8. Beyohd paradise -- 9. From Fourier to wavelets, emphazing Haar -- 10. Zooming properties of wavelets -- 11. Calculating with wavelets -- 12. The hilbert transform -- Appendix. Useful tools

Articles in this volume are based on lectures presented at the Park City summer school on “Mathematics and Materials” in July 2014. The central theme is a description of material behavior that is rooted in statistical mechanics. While many presentations of mathematical problems in materials science begin with continuum mechanics, this volume takes an alternate approach. All the lectures present unique pedagogical introductions to the rich variety of material behavior that emerges from the interplay of geometry and statistical mechanics. The topics include the order-disorder transition in many geometric models of materials including nonlinear elasticity, sphere packings, granular materials, liquid crystals, and the emerging field of synthetic self-assembly. Several lectures touch on discrete geometry (especially packing) and statistical mechanics. The problems discussed in this book have an immediate mathematical appeal and are of increasing importance in applications, but are not as widely known as they should be to mathematicians interested in materials science. The volume will be of interest to graduate students and researchers in analysis and partial differential equations, continuum mechanics, condensed matter physics, discrete geometry, and mathematical physics.

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space.One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kahler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$ - moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.

Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Math Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions. Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Geometric combinatorics describes a wide area of mathematics that is primarily the study of geometric objects and their combinatorial structure. Perhaps the most familiar examples are polytopes and simplicial complexes, but the subject is much broader. This volume is a compilation of expository articles at the interface between combinatorics and geometry, based on a three-week program of lectures at the Institute for Advanced Study/Park City Math Institute (IAS/PCMI) summer program on Geometric Combinatorics. The topics covered include posets, graphs, hyperplane arrangements, discrete Morse theory, and more. These objects are considered from multiple perspectives, such as in enumerative or topological contexts, or in the presence of discrete or continuous group actions. Most of the exposition is aimed at graduate students or researchers learning the material for the first time. Many of the articles include substantial numbers of exercises, and all include numerous examples. The reader is led quickly to the state of the art and current active research by worldwide authorities on their respective subjects. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

"Data science is a highly interdisciplinary field, incorporating ideas from applied mathematics, statistics, probability, and computer science, as well as many other areas. This book gives an introduction to the mathematical methods that form the foundations of machine learning and data science, presented by leading experts in computer science, statistics, and applied mathematics. Although the chapters can be read independently, they are designed to be read together as they lay out algorithmic, statistical, and numerical approaches in diverse but complementary ways. This book can be used both as a text for advanced undergraduate and beginning graduate courses, and as a survey for researchers interested in understanding how applied mathematics broadly defined is being used in data science. It will appeal to anyone interested in the interdisciplinary foundations of machine learning and data science."--Site web de l'éditeur

"The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments"-- Provided by publisher

Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Fractions, Tilings, and Geometry is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. The overall goal of the course is an introduction to non-periodic tilings in two dimensions and space-filling polyhedra. While the course does not address quasicrystals, it provides the underlying mathematics that is used in their study. Because of this goal, the course explores Penrose tilings, the irrationality of the golden ratio, the connections between tessellations and packing problems, and Voronoi diagrams in 2 and 3 dimensions. These topics all connect to precollege mathematics, either as core ideas (irrational numbers) or enrichment for standard topics in geometry (polygons, angles, and constructions). But this book isn't a "course" in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes. These materials provide participants with the opportunity for authentic mathematical discovery--participants build mathematical structures by investigating patterns, use reasoning to test and formalize their ideas, offer and negotiate mathematical definitions, and apply their theories and mathematical machinery to solve problems. Fractions, Tilings, and Geometry is a volume of the book series "IAS/PCMI--The Teacher Program Series" published by the American Mathematical Society. Each volume in this series covers the content of one Summer School Teacher Program year and is independent of the rest

\"This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject. This book is published in cooperation with IAS/Park City Mathematics Institute.\"--Provided by publisher This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject. This book is published in cooperation with...

xi, 187 pages : 26 cm Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Applications of Algebra and Geometry to the Work of Teaching is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. But this book isn't a "course" in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes, and one of the goals of the problem sets is for readers to uncover these themes for themselves. The specific theme developed in Applications of Algebra and Geometry to the Work of Teaching is the use of complex numbers-especially the arithmetic of Gaussian and Eisenstein integers-to investigate some questions that are at the intersection of algebra and geometry, like the classification of Pythagorean triples and the number of representations of an integer as the sum of two squares. Applications of Algebra and Geometry to the Work of Teaching is a volume of the book series IAS/PCMI-The Teacher Program Series published by the American Mathematical Society. Each volume in that series covers the content of one Summer School Teacher Program year and is independent of the rest. -- Provided by publisher Includes bibliographical references Problem Sets -- Facilitator Notes -- Teaching Notes -- Mathematical Overview -- Solutions

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space.One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kahler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$ - moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.

The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. One reason the progress using the Seiberg-Witten invariants was so spectacular was that those studying $SU(2)$-gauge theory had more than ten years' experience with the subject. The tools had been honed, the correct questions formulated, and the basic strategies well understood. The knowledge immediately bore fruit in the technically simpler environment of the Seiberg-Witten theory. Gauge theory long predates Donaldson's applications of the subject to 4-manifold topology, where the central concern was the geometry of the moduli space. One reason for the interest in this study is the connection between the gauge theory moduli spaces of a Kähler manifold and the algebro-geometric moduli space of stable holomorphic bundles over the manifold. The extra geometric richness of the $SU(2)$-moduli spaces may one day be important for purposes beyond the algebraic invariants that have been studied to date. It is for this reason that the results presented in this volume will be essential.

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kahler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in $R^3$, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace-Beltrami operators.

<p>Each summer the IAS/Park City Mathematics Institute Graduate Summer School gathers some of the best researchers and educators in a particular field to present diverse sets of lectures. This volume presents three weeks of lectures given at the Summer School on Quantum Field Theory, Supersymmetry, and Enumerative Geometry, three very active research areas in mathematics and theoretical physics. With this volume, the Park City Mathematics Institute returns to the general topic of the first institute: the interplay between quantum field theory and mathematics. Two major themes at this institute were supersymmetry and algebraic geometry, particularly enumerative geometry. The volume contains two lecture series on methods of enumerative geometry that have their roots in QFT. The first series covers the Schubert calculus and quantum cohomology. The second discusses methods from algebraic geometry for computing Gromov-Witten invariants. There are also three sets of lectures of a more introductory nature: an overview of classical field theory and supersymmetry, an introduction to supermanifolds, and an introduction to general relativity. This volume is recommended for independent study and is suitable for graduate students and researchers interested in geometry and physics.</p>

Analytic And Algebraic Geometers Often Study The Same Geometric Structures But Bring Different Methods To Bear On Them. While This Dual Approach Has Been Spectacularly Successful At Solving Problems, The Language Differences Between Algebra And Analysis Also Represent A Difficulty For Students And Researchers In Geometry, Particularly Complex Geometry. The Pcmi Program Was Designed To Partially Address This Language Gulf, By Presenting Some Of The Active Developments In Algebraic And Analytic Geometry In A Form Suitable For Students On The 'other Side' Of The Analysis-algebra Language Divide. One Focal Point Of The Summer School Was Multiplier Ideals, A Subject Of Wide Current Interest In Both Subjects. The Present Volume Is Based On A Series Of Lectures At The Pcmi Summer School On Analytic And Algebraic Geometry. The Series Is Designed To Give A High-level Introduction To The Advanced Techniques Behind Some Recent Developments In Algebraic And Analytic Geometry. The Lectures Contain Many Illustrative Examples, Detailed Computations, And New Perspectives On The Topics Presented, In Order To Enhance Access Of This Material To Non-specialists.--publisher's Description. Machine Generated Contents Note: An Introduction To Things & Part; / Bo Berndtsson -- Introduction -- Lecture 1 The One-dimensional Case -- 1.1. The & Part;-equation In One Variable -- 1.2. An Alternative Proof Of The Basic Identity -- 1.3. An Application: Inequalities Of Brunn-minkowski Type -- 1.4. Regularity -- A Disclaimer -- Lecture 2 Functional Analytic Interlude -- 2.1. Dual Formulation Of The & Part;-problem -- Lecture 3 The & Part;-equation On A Complex Manifold -- 3.1. Metrics -- 3.2. Norms Of Forms -- 3.3. Line Bundles -- 3.4. Calculation Of The Adjoint And The Basic Identity -- 3.5. The Main Existence Theorem And L2-estimate For Compact Manifolds -- 3.6. Complete Kahler Manifolds -- Lecture 4 The Bergman Kernel -- 4.1. Generalities -- 4.2. Bergman Kernels Associated To Complex Line Bundles -- Lecture 5 Singular Metrics And The Kawamata-viehweg Vanishing Theorem -- 5.1. The Demailly-nadel Vanishing Theorem -- 5.2. The Kodaira Embedding Theorem. 5.3. The Kawamata-viehweg Vanishing Theorem -- Lecture 6 Adjunction And Extension From Divisors -- 6.1. Adjunction And The Currents Defined By Divisors -- 6.2. The Ohsawa-takegoshi Extension Theorem -- Lecture 7 Deformational Invariance Of Plurigenera -- 7.1. Extension Of Pluricanonical Forms -- Bibliography -- Real And Complex Geometry Meet The Cauchy-riemann Equations / John P. D'angelo -- Preface -- Lecture 1 Background Material -- 1. Complex Linear Algebra -- 2. Differential Forms -- 3. Solving The Cauchy-riemann Equations -- Lecture 2 Complex Varieties In Real Hypersurfaces -- 1. Degenerate Critical Points Of Smooth Functions -- 2. Hermitian Symmetry And Polarization -- 3. Holomorphic Decomposition -- 4. Real Analytic Hypersurfaces And Subvarieties -- 5. Complex Varieties, Local Algebra, And Multiplicities -- Lecture 3 Pseudoconvexity, The Levi Form, And Points Of Finite Type -- 1. Euclidean Convexity -- 2. The Levi Form -- 3. Higher Order Commutators -- 4. Points Of Finite Type -- 5. Commutative Algebra. 6. A Return To Finite Type -- 7. The Set Of Finite Type Points Is Open -- Lecture 4 Kohn's Algorithm For Subelliptic Multipliers -- 1. Introduction -- 2. Subelliptic Estimates -- 3. Kohn's Algorithm -- 4. Kohn's Algorithm For Holomorphic And Formal Germs -- 5. Failure Of Effectiveness For Kohn's Algorithm -- 6. Triangular Systems -- 7. Additional Remarks -- Lecture 5 Connections With Partial Differential Equations -- 1. Finite Type Conditions -- 2. Local Regularity For & Part; -- 3. Hypoellipticity, Global Regularity, And Compactness -- 4. An Introduction To L2-estimates -- Lecture 6 Positivity Conditions -- 1. Introduction -- 2. The Classes P & Kappa; -- 3. Intermediate Conditions -- 4. The Global Cauchy-schwarz Inequality -- 5. A Complicated Example -- 6. Stabilization In The Bihomogeneous Polynomial Case -- 7. Squared Norms And Proper Mappings Between Balls -- 8. Holomorphic Line Bundles -- Lecture 7 Some Open Problems -- Bibliography -- Three Variations On A Theme In Complex Analytic Geometry / Dror Varolin. Lecture 0 Basic Notions In Complex Geometry -- 1. Complex Manifolds -- 2. Connections -- 3. Curvature -- 4. Holomorphic Line Bundles -- Lecture 1 The Hormander Theorem -- 1. Functional Analysis -- 2. The Bochner-kodaira Identity -- 3. Manifolds With Boundary -- 4. Density Of Smooth Forms In The Graph Norm -- 5. Hormander's Theorem -- 6. Singular Hermitian Metrics For Line Bundles -- 7. Application: Kodaira Embedding Theorem -- 8. Multiplier Ideal Sheaves And Nadel's Theorems -- 9. Exercises -- Lecture 2 The L2 Extension Theorem -- 1. L2 Extension -- 2. The Deformation Invariance Of Plurigenera -- 3. Pluricanonical Extension On Projective Manifolds -- 4. Exercises -- Lecture 3 The Skoda Division Theorem -- 1. Statement Of The Division Theorem -- 2. Proof Of The Division Theorem -- 3. Global Generation Of Multiplier Ideal Sheaves -- 4. Exercises -- Bibliography -- Structure Theorems For Projective And Kahler Varieties / Jean-pierre Demailly -- 0. Introduction -- 1. Numerically Effective And Pseudo-effective (1,1) Classes. 1.a. Pseudo-effective Line Bundles And Metrics With Minimal Singularities -- 1.b. Nef Line Bundles -- 1.c. Description Of The Positive Cones -- 1.d. The Kawamata-viehweg Vanishing Theorem -- 1.e. A Uniform Global Generation Property Due To Y.t. Siu -- 1.f. Hard Lefschetz Theorem With Multiplier Ideal Sheaves -- 2. Holomorphic Morse Inequalities -- 3. Approximation Of Closed Positive (1,1)-currents By Divisors -- 3.a. Local Approximation Theorem Through Bergman Kernels -- 3.b. Global Approximation Of Closed (1,1)-currents On A Compact Complex Manifold -- 3.c. Global Approximation By Divisors -- 3.d. Singularity Exponents And Log Canonical Thresholds -- 4. Subadditivity Of Multiplier Ideals And Fujita's Approximate Zariski Decomposition Theorem -- 5. Numerical Characterization Of The Kahler Cone -- 5.a. Positive Classes In Intermediate (p, P) Bidegrees -- 5.b. Numerically Positive Classes Of Type (1,1) -- 5.c. Deformations Of Compact Kahler Manifolds -- 6. Structure Of The Pseudo-effective Cone And Mobile Intersection Theory. 6.a. Classes Of Mobile Curves And Of Mobile (n -- 1, N -- 1)-currents -- 6.b. Zariski Decomposition And Mobile Intersections -- 6.c. The Orthogonality Estimate -- 6.d. Dual Of The Pseudo-effective Cone -- 7. Super-canonical Metrics And Abundance -- 7.a. Construction Of Super-canonical Metrics -- 7.b. Invariance Of Plurigenera And Positivity Of Curvature Of Super-canonical Metrics -- 7.c. Tsuji's Strategy For Studying Abundance -- 8. Siu's Analytic Approach And Paun's Non Vanishing Theorem -- Bibliography -- Lecture Notes On Rational Polytopes And Finite Generation / Mihai Paun -- 0. Introduction -- 1. Basic Definitions And Notations -- 2. Proof Of (i) -- 2.1. The Case Nd({kx + Yto + A}) = 0 -- 2.2. The X Method For Sequences -- 2.3. The Induced Polytope And Its Properties -- 3. Proof Of (ii) -- 3.1. The First Step -- 3.2. Iteration Scheme -- References -- Introduction To Resolution Of Singularities / Mircea Mustata -- Lecture 1 Resolutions And Principalizations -- 1.1. The Main Theorems. 1.2. Strengthenings Of Theorem 1.3 -- 1.3. Historical Comments -- Lecture 2 Marked Ideals -- 2.1. Marked Ideals -- 2.2. Derived Ideals -- Lecture 3 Hypersurfaces Of Maximal Contact And Coefficient Ideals -- 3.1. Hypersurfaces Of Maximal Contact -- 3.2. The Coefficient Ideal -- Lecture 4 Homogenized Ideals -- 4.1. Basics Of Homogenized Ideals -- 4.2. Comparing Hypersurfaces Of Maximal Contact: Formal Equivalence -- 4.3. Comparing Hypersurfaces Of Maximal Contact: Etale Equivalence -- Lecture 5 Proof Of Principalization -- 5.1. The Statements -- 5.2. Part I: The Maximal Order Case -- 5.3. Part Ii: The General Case -- 5.4. Proof Of Principalization -- Bibliography -- A Short Course On Multiplier Ideals / Robert Lazarsfeld -- Introduction -- Lecture 1 Construction And Examples Of Multiplier Ideals -- Definition Of Multiplier Ideals -- Monomial Ideals -- Invariants Defined By Multiplier Ideals -- Lecture 2 Vanishing Theorems For Multiplier Ideals -- The Kawamata-viehweg-nadel Vanishing Theorem -- Singularities Of Plane Curves And Projective Hypersurfaces -- Singularities Of Theta Divisors -- Uniform Global Generation. Lecture 3 Local Properties Of Multiplier Ideals -- Adjoint Ideals And The Restriction Theorem -- The Subadditivity Theorem -- Skoda's Theorem -- Lecture 4 Asymptotic Constructions -- Asymptotic Multiplier Ideals -- Variants -- Etale Multiplicativity Of Plurigenera -- A Comparison Theorem For Symbolic Powers -- Lecture 5 Extension Theorems And Deformation Invariance Of Plurigenera -- Bibliography -- Exercises In The Birational Geometry Of Algebraic Varieties / Janos Kollar -- 1. Birational Classification Of Algebraic Surfaces -- 2. Naive Minimal Models -- 3. The Cone Of Curves -- 4. Singularities -- 5. Flips -- 6. Minimal Models -- Bibliography -- Higher Dimensional Minimal Model Program For Varieties Of Log General Type / Christopher D. Hacon -- Introduction -- Lecture 1 Pl-flips -- Lecture 2 Multiplier Ideal Sheaves -- Asymptotic Multiplier Ideal Sheaves -- Extending Pluricanonical Forms -- 3. Finite Generation Of The Restricted Algebra -- Rationality Of The Restricted Algebra -- Proof Of (1.10) -- Lecture 4 The Minimal Model Program With Scaling -- Solutions To The Exercises -- Bibliography -- Lectures On Flips And Minimal Models / Mircea Mustata. Lecture 1 Extension Theorems -- 1.1. Multiplier And Adjoint Ideals -- 1.2. Proof Of The Main Lemma -- Lecture 2 Existence Of Flips I -- 2.1. The Setup -- 2.2. Adjoint Algebras -- 2.3. The Hacon-mckernan Extension Theorem -- 2.4. The Restricted Algebra As An Adjoint Algebra -- Lecture 3 Existence Of Flips Ii -- Lecture 4 Notes On Birkar-cascini-hacon-mckernan -- 4.1. Comparison Of 3 Mmp's -- 4.2. Mmp With Scaling -- 4.3. Mmp With Scaling Near [& Delta;] -- 4.4. Bending It Like Bchm -- 4.5. Finiteness Of Models. Jeffery Mcneal, Mircea Mustata, Editors. Includes Bibliographical References (p. 583)

The overall theme of the 2009 IAS/PCMI Graduate Summer School was connections between special values of $L$-functions and arithmetic, especially the Birch and Swinnerton-Dyer Conjecture and Stark's Conjecture. These conjectures are introduced and discussed in depth, and progress made over the last 30 years is described. This volume contains the written versions of the graduate courses delivered at the summer school. It would be a suitable text for advanced graduate topics courses on the Birch and Swinnerton-Dyer Conjecture and/or Stark's Conjecture. The book will also serve as a reference volume for experts in the field. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

This volume contains lectures presented at the third Regional Geometry Institute at Park City. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second and third year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics.

This book is based on an undergraduate course taught at the IAS/Park City Mathematics Institute (Utah) on linear and nonlinear waves. The first part of the text overviews the concept of a wave, describes one-dimensional waves using functions of two variables, provides an introduction to partial differential equations, and discusses computer-aided visualization techniques. The second part of the book discusses traveling waves, leading to a description of solitary waves and soliton solutions of the Klein-Gordon and Korteweg-deVries equations. The wave equation is derived to model the small vibrations of a taut string, and solutions are constructed via d'Alembert's formula and Fourier series. The last part of the book discusses waves arising from conservation laws. After deriving and discussing the scalar conservation law, its solution is described using the method of characteristics, leading to the formation of shock and rarefaction waves. Applications of these concepts are then given for models of traffic flow.

<p>This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject. This book is published in cooperation with IAS/Park City Mathematics Institute.</p>

This volume contains lectures presented at the third Regional Geometry Institute at Park City. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second and third year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics.

Exploring topics from classical and quantum mechanics and field theory, this book is based on lectures presented in the Graduate Summer School at the Regional Geometry Institute in Park City, Utah, in 1991. The chapter by Bryant treats Lie groups and symplectic geometry, examining not only the connection with mechanics but also the application to differential equations and the recent work of the Gromov school. Rabin's discussion of quantum mechanics and field theory is specifically aimed at mathematicians. Alvarez describes the application of supersymmetry to prove the Atiyah-Singer index theorem, touching on ideas that also underlie more complicated applications of supersymmetry. Quinn's account of the topological quantum field theory captures the formal aspects of the path integral and shows how these ideas can influence branches of mathematics which at first glance may not seem connected. Presenting material at a level between that of textbooks and research papers, much of the book would provide excellent material for graduate courses. The book provides an entree into a field that promises to remain exciting and important for years to come.