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• Foreword / Yuri I. Manin
• Feynman integrals in mathematics and physics / Spencer Bloch
• Feynman integrals and periods in configuration spaces / Özgür Ceyhan and Matilde Marcolli
• Introductory course on l-adic sheaves and their ramification theory on curves / Lars Kindler and Kay Rülling

This is the first volume of the lectures presented at the Clay Mathematics Institute 2014 Summer School, "Periods and Motives: Feynman amplitudes in the 21st century", which took place at the Instituto de Ciencias Matematicas-ICMAT (Institute of Mathematical Sciences) in Madrid, Spain. It covers the presentations by S. Bloch, by M. Marcolli and by L. Kindler and K. Rulling. The main topics of these lectures are Feynman integrals and ramification theory. On the Feynman integrals side, their relation with Hodge structures and heights as well as their monodromy are explained in Bloch's lectures. Two constructions of Feynman integrals on configuration spaces are presented in Ceyhan and Marcolli's notes. On the ramification theory side an introduction to the theory of l -adic sheaves with emphasis on their ramification theory is given. These notes will equip the reader with the necessary background knowledge to read current literature on these subjects.
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• Preface-- Part I. Lectures:
• 1. Notes on coarse median spaces Brian H. Bowditch--
• 2. Semihyperbolicity Martin R. Bridson--
• 3. Acylindrically hyperbolic groups Benjamin Barrett-- Part II. Expository Articles:
• 4. A survey on Morse boundaries and stability Matthew Cordes--
• 5. What is a hierarchically hyperbolic space? Alessandro Sisto-- Part III. Research Articles:
• 6. A counterexample to questions about boundaries, stability, and commensurability Jason Behrstock--
• 7. A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes Indira Chatterji and Alexandre Martin--
• 8. Immutability is not uniformly decidable in hyperbolic groups Daniel Groves and Henry Wilton--
• 9. Sphere systems, standard form, and cores of products of trees Francesca Iezzi--
• 10. Uniform quasiconvexity of the disc graphs in the curve graphs Kate M. Vokes.
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Since the notion was introduced by Gromov in the 1980s, hyperbolicity of groups and spaces has played a significant role in geometric group theory; hyperbolic groups have good geometric properties that allow us to prove strong results. However, many classes of interest in our exploration of the universe of finitely generated groups contain examples that are not hyperbolic. Thus we wish to go 'beyond hyperbolicity' to find good generalisations that nevertheless permit similarly strong results. This book is the ideal resource for researchers wishing to contribute to this rich and active field. The first two parts are devoted to mini-courses and expository articles on coarse median spaces, semihyperbolicity, acylindrical hyperbolicity, Morse boundaries, and hierarchical hyperbolicity. These serve as an introduction for students and a reference for experts. The topics of the surveys (and more) re-appear in the research articles that make up Part III, presenting the latest results beyond hyperbolicity.
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This is the first book to present a detailed discussion of both classical and recent results on the popular Cahn-Hilliard equation and some of its variants. The focus is on mathematical analysis of Cahn-Hilliard models, with an emphasis on thermodynamically relevant logarithmic nonlinear terms, for which several questions are still open. Initially proposed in view of applications to materials science, the Cahn-Hilliard equation is now applied in many other areas, including image processing, biology, ecology, astronomy, and chemistry. In particular, the author addresses applications to image inpainting and tumor growth. Many chapters include open problems and directions for future research. The Cahn?Hilliard Equation: Recent Advances and Applications is intended for graduate students and researchers in applied mathematics, especially those interested in phase separation models and their generalizations and applications to other fields. Materials scientists also will find this text of interest.
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• Introduction Preliminaries Meromorphic continuation of the resolvent Boundary layer operators Dynamical resonance free regions Existence of resonances for the Delta potential Appendix A. Model cases Appendix B. Semiclassical intersecting Lagrangian distributions Appendix C. The semiclassical Melrose-Taylor parametrix Bibliography.
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The author studies high energy resonances for the operators $-\Delta_{\partial\Omega, \delta}:=-\Delta \delta_{\partial\Omega}\otimes V\quad \textrm{and}\quad -\Delta_{\partial\Omega, \delta'}:=-\Delta \delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega\subset{\mathbb{R}}^{d}$ is strictly convex with smooth boundary, $V:L^{2}(\partial\Omega)\to L^{2}(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$.
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• Introduction Potential scattering: Scattering resonances in dimension one Scattering resonances in odd dimensions Geometric scattering: Black box scattering in $\mathbb{R}^n$ Scattering on hyperbolic manifolds Resonances in the semiclassical limit: Resonance-free regions Resonances and trapping Appendices: Notation Spectral theory Fredholm theory Complex analysis Semiclassical analysis Bibliography Index.
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Scattering resonances generalize bound states/eigenvalues for systems in which energy can scatter to infinity. A typical resonance has a rate of oscillation (just as a bound state does) and a rate of decay. Although the notion is intrinsically dynamical, an elegant mathematical formulation comes from considering meromorphic continuations of Green's functions. The poles of these meromorphic continuations capture physical information by identifying the rate of oscillation with the real part of a pole and the rate of decay with its imaginary part. An example from mathematics is given by the zeros of the Riemann zeta function: they are, essentially, the resonances of the Laplacian on the modular surface. The Riemann hypothesis then states that the decay rates for the modular surface are all either $0$ or $\frac14$. An example from physics is given by quasi-normal modes of black holes which appear in long-time asymptotics of gravitational waves. This book concentrates mostly on the simplest case of scattering by compactly supported potentials but provides pointers to modern literature where more general cases are studied. It also presents a recent approach to the study of resonances on asymptotically hyperbolic manifolds. The last two chapters are devoted to semiclassical methods in the study of resonances.
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This is a proceedings volume of an international conference 'Operator Algebras and Mathematical Physics' held at Tohoku University in August, 2016. This meeting was the 9th MSJ-Seasonal Institute of the Mathematical Society of Japan. Twenty-eight researchers gave lectures on a wide range of topics on operator algebras and their applications to mathematical physics. This volume contains one survey article and 11 research articles based on the lectures given. We recommend this to researchers and graduate students in related fields strongly.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America.
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• Symplectic geometry of unbiasedness and critical points of a potential / Alexey Bondal and Ilya Zhdanovskiy
• A brief survey of FJRW theory / Amanda E. Francis, Tyler J. Jarvis and Nathan Priddis
• Gamma conjecture via mirror symmetry / Sergey Galkin and Hiroshi Iritani
• On a conjecture by N. Takahashi on log mirror symmetry for the projective plane / Michel van Garrel
• Notes on the Hemisphere / Kentaro Hori and Mauricio Romo
• On Dubrovin's Frobenius structures on Hurwitz spaces / Sergei K. Lando
• Calabi-Yau geometry, primitive form and mirror symmetry / Si Li
• TQFT, homological algebra and elements of K. Saito's theory of Primitive form: an attempt of mathematical text written by mathematical physicist / Andrey Losev
• The phase factors in singularity theory / Todor Milanov
• On the homological algebra of relative symplectic geometry / Daniel Pomerleano
• The Landau-Ginzburg/Calabi-Yau correspondence and the mirror quintic / Nathan Priddis
• Quantum invariants via tautological relations / Yefeng Shen
• From a class of Calabi-Yau dg algebras to Frobenius manifolds via primitive forms / Atsushi Takahashi

This is the proceedings of the conference 'Primitive Forms and Related Subjects' held at Kavli IPMU, the University of Tokyo in February 10-14, 2014. The principal aim of the conference was to discuss the current status of rapidly developing subjects related to primitive forms. In particular, Fukaya category, Gromov-Witten and FJRW invariants, mathematical formulation of Landau-Ginzburg models, and mirror symmetry were discussed. The conference had three introductory courses by experts, and 12 lectures on more advanced topics. The present volume contains two survarticles and 11 research articles based on them. Thus the volume is recommended both to researchers who are already experts in the subjects, and also to graduate students who are interested in and want to start study the subjects. Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America.
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• Prologue: listening to the universe
• Mathematics drives away the cloud
• Shining the torch on electricity and magnetism
• Shining the torch on gravity again
• Quantum mathematics
• The long divorce
• Revolution
• Bad company?
• Jokes and magic lead to the string
• Strung together
• Thinking their way to the Millenium
• Diamonds in the rough
• The best possible times.

How math helps us solve the universe's deepest mysteries One of the great insights of science is that the universe has an underlying order. The supreme goal of physicists is to understand this order through laws that describe the behavior of the most basic particles and the forces between them. For centuries, we have searched for these laws by studying the results of experiments. Since the 1970s, however, experiments at the world's most powerful atom-smashers have offered few new clues. So some of the world's leading physicists have looked to a different source of insight: modern mathematics. These physicists are sometimes accused of doing 'fairy-tale physics', unrelated to the real world. But in The Universe Speaks in Numbers, award-winning science writer and biographer Farmelo argues that the physics they are doing is based squarely on the well-established principles of quantum theory and relativity, and part of a tradition dating back to Isaac Newton. With unprecedented access to some of the world's greatest scientific minds, Farmelo offers a vivid, behind-the-scenes account of the blossoming relationship between mathematics and physics and the research that could revolutionize our understanding of reality. A masterful account of the some of the most groundbreaking ideas in physics in the past four decades. The Universe Speaks in Numbers is essential reading for anyone interested in the quest to discover the fundamental laws of nature.
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• J. Russell and A. Wootton, A lower bound for the number of finitely maximal $C_p$-actions on a compact oriented surface S. A. Broughton, Galois action on regular dessins d'enfant with simple group action D. Swinarski, Equations of Riemann surfaces with automorphisms R. Hidalgo and T. Shaska, On the field of moduli of superelliptic curves L. Beshaj, Minimal integral Weierstrass equations for genus 2 curves L. Beshaj, R. Hidalgo, S. Kruk, A. Malmendier, S. Quispe, and T. Shaska, Rational points in the moduli space of genus two C. Magyar and U. Whitcher, Strong arithmetic mirror symmetry and toric isogenies A. Kumar and M. Kuwata, Inose's construction and elliptic $K$3 surfaces with Mordell-Weil rank 15 revisited C. M. Shor, Higher-order Weierstrass weights of branch points on superelliptic curves E. Previato, Poncelet's porism and projective fibrations A. Levin, Extending Runge's method for integral points D. Joyner and T. Shaska, Self-inversive polynomials, curves, and codes A. Deopurkar and A. Patel, Syzygy divisors on Hurwitz spaces.
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This volume contains the proceedings of the AMS Special Session on Higher Genus Curves and Fibrations in Mathematical Physics and Arithmetic Geometry, held on January 8, 2016, in Seattle, Washington. Algebraic curves and their fibrations have played a major role in both mathematical physics and arithmetic geometry. This volume focuses on the role of higher genus curves; in particular, hyperelliptic and superelliptic curves in algebraic geometry and mathematical physics. The articles in this volume investigate the automorphism groups of curves and superelliptic curves and results regarding integral points on curves and their applications in mirror symmetry. Moreover, geometric subjects are addressed, such as elliptic $K$3 surfaces over the rationals, the birational type of Hurwitz spaces, and links between projective geometry and abelian functions.
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• Elementary Newtonian Mechanics.- The Principles of Canonical Mechanics.- The Mechanics of Rigid Bodies.- Relativistic Mechanics.- Geometric Aspects of Mechanics.- Stability and Chaos.- Continuous Systems.- Exercises.- Solution of Exercises.- Appendix.
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This book covers all topics in mechanics from elementary Newtonian mechanics, the principles of canonical mechanics and rigid body mechanics to relativistic mechanics and nonlinear dynamics. It was among the first textbooks to include dynamical systems and deterministic chaos in due detail. As compared to the previous editions the present 6th edition is updated and revised with more explanations, additional examples and problems with solutions, together with new sections on applications in science. Symmetries and invariance principles, the basic geometric aspects of mechanics as well as elements of continuum mechanics also play an important role. The book will enable the reader to develop general principles from which equations of motion follow, to understand the importance of canonical mechanics and of symmetries as a basis for quantum mechanics, and to get practice in using general theoretical concepts and tools that are essential for all branches of physics. The book contains more than 150 problems with complete solutions, as well as some practical examples which make moderate use of personal computers. This will be appreciated in particular by students using this textbook to accompany lectures on mechanics. The book ends with some historical notes on scientists who made important contributions to the development of mechanics.
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Separation of variables methods for solving partial differential equations are of immense theoretical and practical importance in mathematical physics. They are the most powerful tool known for obtaining explicit solutions of the partial differential equations of mathematical physics. The purpose of this book is to give an up-to-date presentation of the theory of separation of variables and its relation to superintegrability. Collating and presenting it in a unified, updated and a more accessible manner, the results scattered in the literature that the authors have prepared is an invaluable resource for mathematicians and mathematical physicists in particular, as well as science, engineering, geological and biological researchers interested in explicit solutions.

• Preface Acknowledgments Author biography
• 1. Models of Nature
• 2. Randomness
• 3. Bayesian and frequentist approaches to scientific inference
• 4. The principles of inferential statistics
• 5. Parametric inference
• 6. Prior distributions and equiprobable events in the physical sciences
• 7. Conclusionsthe statistical nature of scientific knowledge Appendix AShort review of some basic concepts Appendix BAbbreviations.
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Science often deals with hard-to-see phenomena, and they only stand out and become real when viewed through the lens of complex statistical tools. This book is not a textbook about statistics applied to science - there are already many excellent books to choose from - rather, it tries to give an overview of the basic principles that physical scientists use to analyze their data and bring out the order of Nature from the fog of background noise.
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• Adiabatic limits of co-associative Kovalev-Lefschetz fibrations.- Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories.- Spectral sequences for cyclic homology.- Derived varieties of complexes and Kostant's theorem for gl(mjn).- Higher symmetry and gapped phases of gauge theories.- Constructing Buildings and Harmonic Maps.- Cohomological Hall algebras, semicanonical bases and Donaldson-Thomas invariants for 2-dimensional Calabi-Yau categories.- Fukaya A1-structures associated to Lefschetz fibrations. II.
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This volume is a tribute to Maxim Kontsevich, one of the most original and influential mathematicians of our time. Maxim's vision has inspired major developments in many areas of mathematics, ranging all the way from probability theory to motives over finite fields, and has brought forth a paradigm shift at the interface of modern geometry and mathematical physics. Many of his papers have opened completely new directions of research and led to the solutions of many classical problems. This book collects papers by leading experts currently engaged in research on topics close to Maxim's heart. Contributors: S. Donaldson A. Goncharov D. Kaledin M. Kapranov A. Kapustin L. Katzarkov A. Noll P. Pandit S. Pimenov J. Ren P. Seidel C. Simpson Y. Soibelman R. Thorngren.
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• Fourier series and the discrete fourier transform
• Introduction
• Formulation of fourier series
• Fourier coefficients and their properties
• Convolution and parseval's equality
• Fejér means of fourier series : uniqueness of the fourier series
• The riemann-lebesgue lemma
• The fourier series of a square-integrable function : the riesz-fischer theorem
• Besov and holder spaces
• Absolute convergence : bernstein and peetre theorems
• Dirichlet kernel : pointwise and uniform convergence
• Formulation of the discrete fourier transform and its properties
• Connection between the discrete fourier transform and the fourier transform
• Some applications of the discrete fourier transform
• Applications to solving some mod el equations
• The one-dimensional heat equation
• The one-dimensional wave equation
• The laplace equation in a rectangle and in a disk
• Fourier transform and distributions
• Introduction
• The fourier transform in schwartz space
• The fourier transform in LP(RN), 1 < ̲p < ̲2
• Tempered distributions
• Convolutions in S and S¹
• Sobolev spaces
• Sobolev spaces on bounded domains
• Homogeneous distributions
• Fundamental solution of the helmholtz operator
• Estimates for the laplacian and hamiltonian
• Operator theory and integral equations
• Introduction
• Inner product spaces and hilbert spaces
• Symmetric operators in hilbert spaces
• John von neumann's spectral theorem
• Spectra of self-adjoint operators
• Quadratic forms : friedrichs extension
• Elliptic differential operators
• Spectral functions
• The schrodinger operator
• The magnetic schrodinger operator
• Integral operators with weak singularities : integral equations of the first and second kinds
• Volterra and singular integral equations
• Approximate methods
• Partial differential equations
• Introduction
• Local existence theory
• The laplace operator
• The dirichlet and neumann problems
• Layer potentials
• Elliptic boundary value problems
• The direct scattering problem for the helmholtz equation
• Some inverse scattering problems for the schrodinger operator
• The heat operator
• The wave operator
• References
• Index.

This text serves as an introduction to the modern theory of analysis and differential equations with applications in mathematical physics and engineering sciences. Having outgrown from a series of half-semester courses given at University of Oulu, this book consists of four self-contained parts. The first part, Fourier Series and the Discrete Fourier Transform, is devoted to the classical one-dimensional trigonometric Fourier series with some applications to PDEs and signal processing. The second part, Fourier Transform and Distributions, is concerned with distribution theory of L. Schwartz and its applications to the Schroedinger and magnetic Schroedinger operations. The third part, Operator Theory and Integral Equations, is devoted mostly to the self-adjoint but unbounded operators in Hilbert spaces and their applications to integral equations in such spaces. The fourth and final part, Introduction to Partial Differential Equations, serves as an introduction to modern methods for classical theory of partial differential equations. Complete with nearly 250 exercises throughout, this text is intended for graduate level students and researchers in the mathematical sciences and engineering.
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• * Background material: Assorted notions from differential geometry* Differential forms* Curves and surfaces in homogeneous spaces via the method of moving frames: Homogeneous spaces* Curves and surfaces in Euclidean space* Curves and surfaces in Minkowski space* Curves and surfaces in equi-affine space* Curves and surfaces in projective space* Applications of moving frames: Minimal surfaces in $\mathbb{E}^3$ and $\mathbb{A}^3$* Pseudospherical surfaces in Backlund's theorem* Two classical theorems* Beyond the flat case: Moving frames on Riemannian manifolds: Curves and surfaces in elliptic and hyperbolic spaces* The nonhomogeneous case: Moving frames on Riemannian manifolds* Bibliography* Index.
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The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others. This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames on Riemannian manifolds. The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance regarding the use of the computer algebra system MapleTM to perform many of the computations involved in the exercises.
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• Introduction.- Models for Double and Triple Porosity.- Double Porosity and Voids.- Comparison of Porosity and Voids Theories.- Uniqueness and Stability by Energy Methods.- Uniqueness Without Definiteness Conditions.- Continuous Dependence in Multi-Porosity Elasticity.- Waves in Double Porosity Elasticity.- Acceleration Waves in Double Voids.- Double Porosity and Second Sound.
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This book is devoted to describing theories for porous media where such pores have an inbuilt macro structure and a micro structure. For example, a double porosity material has pores on a macro scale, but additionally there are cracks or fissures in the solid skeleton. The actual body is allowed to deform and thus the underlying theory is one of elasticity. Various different descriptions are reviewed. Chapter 1 introduces the classical linear theory of elastodynamics together with uniqueness and continuous dependence results. Chapters 2 and 3 review developments of theories for double and triple porosity using a pressure-displacement structure and also using voids-displacement. Chapter 4 compares various aspects of the pressure-displacement and voids-displacement theories via uniqueness studies and wave motion analysis. Mathematical analyses of double and triple porosity materials are included concentrating on uniqueness and stability studies in chapters 5 to 7. In chapters 8 and 9 the emphasis is on wave motion in double porosity materials with special attention paid to nonlinear waves. The final chapter embraces a novel area where an elastic body with a double porosity structure is analyzed, but the thermodynamics allows for heat to travel as a wave rather than simply by diffusion. This book will be of value to mathematicians, theoretical engineers and other practitioners who are interested in double or triple porosity elasticity and its relevance to many diverse applications.
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• Preface to CIME lecture notes / John M. Ball, Eduard Feireisl, Felix Otto
• Liquid crystals and their defects / John M. Ball
• Mathematical thermodynamics of viscous fluids / Eduard Feireisl
• Rigorous bounds on scaling laws in fluid dynamics / Felix Otto, Steffen Pottel, and Camilla Nobili.

The main goal of this book is to provide an overview of the state of the art in the mathematical modeling of complex fluids, with particular emphasis on its thermodynamical aspects. The central topics of the text, the modeling, analysis and numerical simulation of complex fluids, are of great interest and importance both for the understanding of various aspects of fluid dynamics and for its applications to special real-world problems. New emerging trends in the subject are highlighted with the intent to inspire and motivate young researchers and PhD students.
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"The Monge-Ampère equation is one of the most important partial differential equations, appearing in many problems in analysis and geometry. This monograph is a comprehensive introduction to the existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix wherein one can find precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, nonlinear analysis, and PDEs). This book is intended for graduate students and researchers interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study and also as a reference." -- Back cover.

• Classical Deformation Theory. Noncommutative Algebras and Simple Modules. Noncommutative Deformation Theory. The Noncommutative Phase Space. A Cosmological Toy Model. Moduli of Endomorphisms of Rank Three.
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Noncommutative Deformation Theory is aimed at mathematicians and physicists studying the local structure of moduli spaces in algebraic geometry. This book introduces a general theory of noncommutative deformations, with applications to the study of moduli spaces of representations of associative algebras and to quantum theory in physics. An essential part of this theory is the study of obstructions of liftings of representations using generalised (matric) Massey products. Suitable for researchers in algebraic geometry and mathematical physics interested in the workings of noncommutative algebraic geometry, it may also be useful for advanced graduate students in these fields.
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• IntroductionNotation and preliminariesKnown existence and regularity resultsClasses of solutionsEstimates in Gaussian spacesThe generalized frequency functionExistence and homogeneity of blowupsHomogeneous global solutionsOptimal regularity of solutionsClassification of free boundary pointsFree boundary: Regular setFree boundary: Singular setWeiss and Monneau type monotonicity formulasStructure of the singular set}Appendix A. Estimates in Gaussian spaces: ProofsAppendix B. Parabolic Whitney's extension theoremBibliography.
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The authors give a comprehensive treatment of the parabolic Signorini problem based on a generalization of Almgren's monotonicity of the frequency. This includes the proof of the optimal regularity of solutions, classification of free boundary points, the regularity of the regular set and the structure of the singular set.
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