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• 1. The geometry of diffusing and self-attracting particles in a one-dimensional fair-competition regime / Vincent Calvez, José Antonio Carrillo, and Franca Hoffman
• 2. Bubbling blow-up in critical parabolic problems / Manuel del Pino
• 3. Regularity theory for local and nonlocal minimal surfaces: an overview / Matteo Cozzi and Alessio Figalli
• 4. Short tales from nonlinear Calderón-Zygmund theory / Giuseppe Mingione
• 5. The mathematical theories of diffusion: nonlinear and fractional diffusion / Juan Luis Vázquez.

Presenting a selection of topics in the area of nonlocal and nonlinear diffusions, this book places a particular emphasis on new emerging subjects such as nonlocal operators in stationary and evolutionary problems and their applications, swarming models and applications to biology and mathematical physics, and nonlocal variational problems. The authors are some of the most well-known mathematicians in this innovative field, which is presently undergoing rapid development. The intended audience includes experts in elliptic and parabolic equations who are interested in extending their expertise to the nonlinear setting, as well as Ph.D. or postdoctoral students who want to enter into the most promising research topics in the field.
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• Some Introductory Notes on Random Graphs / Fabio Fagnani, Sophie M. Fosson, and Chiara Ravazzi
• Statistical Physics and Network Optimization Problems / Carlo Baldassi, Alfredo Braunstein, Abolfazl Ramezanpour, and Riccardo Zecchina
• Graphical Models and Message-Passing Algorithms: Some Introductory Lectures / Martin J. Wainwright
• Bridging the Gap Between Information Theory and Wireless Networking / P.R. Kumar.

Introducing the reader to the mathematics beyond complex networked systems, these lecture notes investigate graph theory, graphical models, and methods from statistical physics. Complex networked systems play a fundamental role in our society, both in everyday life and in scientific research, with applications ranging from physics and biology to economics and finance. The book is self-contained, and requires only an undergraduate mathematical background.
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• Some themes around first order theories without the independence property.- Lectures on the model theory of real and complex exponentiation.- Lectures on the model theory of valued fields.- Undecidability in number theory.
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Presenting recent developments and applications, the book focuses on four main topics in current model theory: 1) the model theory of valued fields; 2) undecidability in arithmetic; 3) NIP theories; and 4) the model theory of real and complex exponentiation. Young researchers in model theory will particularly benefit from the book, as will more senior researchers in other branches of mathematics.
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• On Bifurcation Currents in Holomorphic Families of Rational maps.- The Complex Monge-Ampere Equation in Kahler Geometry.- Applications of Pluripotential Theory to Algebraic Geometry.- Pluripotential Theory and Monge-Ampere Foliations.
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Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. This volume brings together the lectures held at the 2011 CIME session on "pluripotential theory" in Cetraro, Italy. This CIME course focused on complex Monge-Ampere equations, applications of pluripotential theory to Kahler geometry and algebraic geometry and to holomorphic dynamics. The contributions provide an extensive description of the theory and its very recent developments, starting from basic introductory materials and concluding with open questions in current research.
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• 1 On some recent advances on stabilization for hyperbolic equations.- 2 Notes on the Control of the Liouville Equation.- 3 Some questions of control in fluid mechanics.- 4 Carleman estimates and some applications to control theory.- 5 The Wave Equation: Control and Numerics.
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The term "control theory" refers to the body of results - theoretical, numerical and algorithmic - which have been developed to influence the evolution of the state of a given system in order to meet a prescribed performance criterion. Systems of interest to control theory may be of very different natures. This monograph is concerned with models that can be described by partial differential equations of evolution. It contains five major contributions and is connected to the CIME Course on Control of Partial Differential Equations that took place in Cetraro (CS, Italy), July 19 - 23, 2010. Specifically, it covers the stabilization of evolution equations, control of the Liouville equation, control in fluid mechanics, control and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with application to control. We are confident this work will provide an authoritative reference work for all scientists who are interested in this field, representing at the same time a friendly introduction to, and an updated account of, some of the most active trends in current research.
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• Applications of Boundary Harnack Inequalities for p Harmonic Functions and Related Topics.- Regularity of Supersolutions.- Introduction to random Tug-of-War games and PDEs.- The Problems of the Obstacle in Lower Dimension and for the Fractional Laplacian.
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The issue of regularity has played a central role in the theory of Partial Differential Equations almost since its inception, and despite the tremendous advances made it still remains a very fruitful research field. In particular considerable strides have been made in regularity estimates for degenerate and singular elliptic and parabolic equations over the last several years, and in many unexpected and challenging directions. Because of all these recent results, it seemed high time to create an overview that would highlight emerging trends and issues in this fascinating research topic in a proper and effective way. The course aimed to show the deep connections between these topics and to open new research directions through the contributions of leading experts in all of these fields.
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• Transport Rays and Applications to Hamilton-Jacobi Equations.- Functional Inequalities and Dynamics.- Differential, Energetic, and Metric Formulations for Rate-independent Processes.- Optimal Transport and Curvature.
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This volume collects the notes of the CIME course "Nonlinear PDE's and applications" held in Cetraro (Italy) on June 23-28, 2008. It consists of four series of lectures, delivered by Stefano Bianchini (SISSA, Trieste), Eric A. Carlen (Rutgers University), Alexander Mielke (WIAS, Berlin), and Cedric Villani (Ecole Normale Superieure de Lyon). They presented a broad overview of far-reaching findings and exciting new developments concerning, in particular, optimal transport theory, nonlinear evolution equations, functional inequalities, and differential geometry. A sampling of the main topics considered here includes optimal transport, Hamilton-Jacobi equations, Riemannian geometry, and their links with sharp geometric/functional inequalities, variational methods for studying nonlinear evolution equations and their scaling properties, and the metric/energetic theory of gradient flows and of rate-independent evolution problems. The book explores the fundamental connections between all of these topics and points to new research directions in contributions by leading experts in these fields.
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• Braids and Knots.- Topological quantities: calculating winding, writhing, linking, and higher order invariants.- Tangles, Rational Knots and DNA.- The group and Hamiltonian descriptions of hydrodynamical systems.- Singularities in fluid dynamics and their resolution.- Structural complexity and dynamical systems.- Random Knotting: Theorems, Simulations and Applications.
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Helmholtz's seminal paper on vortex motion (1858) marks the beginning of what is now called topological fluid mechanics. After 150 years of work, the field has grown considerably. In the last several decades unexpected developments have given topological fluid mechanics new impetus, benefiting from the impressive progress in knot theory and geometric topology on the one hand, and in mathematical and computational fluid dynamics on the other. This volume contains a wide-ranging collection of up-to-date, valuable research papers written by some of the most eminent experts in the field. Topics range from fundamental aspects of mathematical fluid mechanics, including topological vortex dynamics and magnetohydrodynamics, integrability issues, Hamiltonian structures and singularity formation, to DNA tangles and knotted DNAs in sedimentation. A substantial introductory chapter on knots and links, covering elements of modern braid theory and knot polynomials, as well as more advanced topics in knot classification, provides an invaluable addition to this material.
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• Prologue by the editor (L.L. Bonilla).- Introduction to Image Reconstruction (M. Moscoso).- X-ray Tomography (F. Natterer).- Adjoint Fields and Sensitivities for 3D Electromagnetic Imaging in Isotropic and Anisotropic Media (O. Dorn, H. Bertete-Aguirre, G.C. Papanicolaou).- Polarization-based Optical Imaging (M. Moscoso).- A Brief Review on Point Interactions (G. Dell'Antonia, R. Figar, A. Teta).- Topological derivatives for Shape Reconstruction (A. Carpio, M.L. Rapun).- Time-reversal and the Adjoint Imaging Method with an Application in Telecommunication (O. Dorn).
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Nowadays we are facing numerous and important imaging problems: nondestructive testing of materials, monitoring of industrial processes, enhancement of oil production by efficient reservoir characterization, emerging developments in noninvasive imaging techniques for medical purposes - computerized tomography (CT), magnetic resonance imaging (MRI), positron emission tomography (PET), X-ray and ultrasound tomography, etc. In the CIME Summer School on Imaging (Martina Franca, Italy 2002), leading experts in mathematical techniques and applications presented broad and useful introductions for non-experts and practitioners alike to many aspects of this exciting field. The volume contains part of the above lectures completed and updated by additional contributions on other related topics.
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• Preface by Vincenzo Capasso and Miroslaw Lachowicz.- Jacek Banasiak: Positivity in Natural Sciences.- Vincenzo Capasso and Daniela Morale Rescaling Stochastic Processes: Asymptotics.- Mark A.J. Chaplain: Modelling Aspects of Cancer Growth: Insight from Mathematical and Numerical Analysis and Computational Simulation.- Miroslaw Lachowicz: Links Between Microscopic and Macroscopic Descriptions.- Jacek Miekisz: Evolutionary Game Theory and Population Dynamics.
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The aim of this volume that presents lectures given at a joint CIME and Banach Center Summer School, is to offer a broad presentation of a class of updated methods providing a mathematical framework for the development of a hierarchy of models of complex systems in the natural sciences, with a special attention to biology and medicine. Mastering complexity implies sharing different tools requiring much higher level of communication between different mathematical and scientific schools, for solving classes of problems of the same nature. Today more than ever, one of the most important challenges derives from the need to bridge parts of a system evolving at different time and space scales, especially with respect to computational afford ability. As a result the content has a rather general character; the main role is played by stochastic processes, positive semi groups, asymptotic analysis, kinetic theory, continuum theory, and game theory.
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• Preface by Paolo Nistri and Gianna Stefani.- Andrei A. Agrachev: Geometry of Optimal Control Problems and Hamiltonian Systems.- A. Stephen Morse: Lecture notes on Logically Switched Dynamical Systems.- Eduardo D. Sontag: Input to State Stability: Basic Concepts and Results.- Hector J. Sussmann: Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints.- Vadim I. Utkin: Sliding Mode Control: Mathematical Tools, Design and Applications.
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• Preface by E. Casadio Tarabusi, A. D'Agnolo, M. Picardello.- M. Cowling: Applications of representation theory to harmonic analysis of Lie groups (and vice versa).- E. Frenkel: Ramifications of the geometric Langlands Program.- M. Kashiwara: Equivariant derived category and representation of real semisimple Lie groups.- A.Valette: Amenability and Margulis super-rigidity.- D. A. Vogan, Jr: Unitary Representations and Complex Analysis.- N. R. Wallach: Quantum computing and entanglement for mathematicians.
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• Some Methods of Infinite Dimensional Analysis in Hydrodynamics: An Introduction (Sergio Albeverio, Benedetta Ferrario).- An Introduction to 3D Stochastic Fluid Dynamics (Franco Flandoli).- Mathematical Results Related to the Navier-Stokes System (Yakov G. Sinai).
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Of the three lecture courses making up the CIME summer school on Fluid Dynamics at Cetraro in 2005 reflected in this volume, the first, due to Sergio Albeverio describes deterministic and stochastic models of hydrodynamics. In the second course, Franco Flandoli starts from 3D Navier-Stokes equations and ends with turbulence. Finally, Yakov Sinai, in the 3rd course, describes some rigorous mathematical results for multidimensional Navier-Stokes systems and some recent results on the one-dimensional Burgers equation with random forcing.
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• P. Marcati: Preface.- A. Bressan: BV Solutions to Hyperbolic Systems by Vanishing Viscosity.- D. Serre: Discrete Shock Profiles: Existence and Stability.- M. Williams: Stability of Multidimensional Viscous Shocks.- K. Zumbrun: Planar Stability Criteria for Viscous Shock Waves of Systems with Real Viscosity.- A. Bressan: Tutorial on Center Manifold Theorem.
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The present Cime volume includes 4 lecture courses by Bressan, Serre, Zumbrun and Williams and a Tutorial by Bressan on the Center Manifold Theorem. Bressan's notes start with an extensive review of hyperbolic conservation laws. Then he introduces the vanishing viscosity approach and explains clearly the building blocks of the theory in particular the crucial role of the decomposition by travelling waves. Serre focuses on existence and stability for discrete shock profiles, he reviews the existence both in the rational and in the irrational cases and gives a concise introduction to the use of spectral methods for stability analysis. Finally the lectures by Williams and Zumbrun deal with the stability of multidimensional fronts. Williams' lectures describe the stability of multidimensional viscous shocks. Zumbrun discusses planar stability for viscous shocks with a realistic physical viscosity, and necessary and sufficient conditions for nonlinear stability.
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• Preface.- J. B. Friedlander: Producing Prime Numbers via Sieve Methods.- D. R. Heath-Brown: Counting Rational Points on Algebraic Varieties.- H. Iwaniec: Conversations on the Exceptional Character.- J. Kaczorowski: Axiomatic Theory of L-Functions: the Selberg Class.
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The four contributions collected in this volume deal with several advanced results in analytic number theory. Friedlander's paper contains some recent achievements of sieve theory leading to asymptotic formulae for the number of primes represented by suitable polynomials. Heath-Brown's lecture notes mainly deal with counting integer solutions to Diophantine equations, using among other tools several results from algebraic geometry and from the geometry of numbers. Iwaniec's paper gives a broad picture of the theory of Siegel's zeros and of exceptional characters of L-functions, and gives a new proof of Linnik's theorem on the least prime in an arithmetic progression. Kaczorowski's article presents an up-to-date survey of the axiomatic theory of L-functions introduced by Selberg, with a detailed exposition of several recent results.
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• A. Giorgilli: Preface.- G. Benettin: Physical Applications of Nekhoroshev Theorem and Exponential Estimates.- J. Henrard: The Adiabatic Invariant Theory and Applications.- S. Kuksin: Lectures on Hamiltonian Methods in Nonlinear PDEs.
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This volume compiles three series of lectures on applications of the theory of Hamiltonian systems, contributed by some of the specialists in the field. The aim is to describe the state of the art for some interesting problems, such as the Hamiltonian theory for infinite-dimensional Hamiltonian systems, including KAM theory, the recent extensions of the theory of adiabatic invariants, and the phenomena related to stability over exponentially long times of Nekhoroshev's theory. The books may serve as an excellent basis for young researchers, who will find here a complete and accurate exposition of recent original results and many hints for further investigation.
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• Preface.- Kerry Back: Incomplete and Asymmetric Information in Asset Pricing Theory.- Tomasz R. Bielecki, Monique Jeanblanc, Marek Rutkowski: Modeling and Valuation of Credit Risk.- Christian Hipp: Stochastic Control with Application in Insurance.- Shige Peng: Nonlinear Expectations, Nonlinear Evaluations and Risk Measures.- Walter Schachermayer: Utility Maximisation in Incomplete Markets.
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This volume includes the five lecture courses given at the CIME-EMS School on "Stochastic Methods in Finance" held in Bressanone/Brixen, Italy 2003. It deals with innovative methods, mainly from stochastic analysis, that play a fundamental role in the mathematical modelling of finance and insurance: the theory of stochastic processes, optimal and stochastic control, stochastic differential equations, convex analysis and duality theory. Five topics are treated in detail: Utility maximization in incomplete markets; the theory of nonlinear expectations and its relationship with the theory of risk measures in a dynamic setting; credit risk modelling; the interplay between finance and insurance; incomplete information in the context of economic equilibrium and insider trading.
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• Preface.- S.-N. Chow: Lattice Dynamical Systems.- R. Conti, M. Galeotti: Totally bounded cubic systems in R2.- R. Johnson, F. Mantellini: Non-Autonomous Differential Equations.- J. Mallet-Paret: Traveling Waves in Spatially Discrete Dynamical Systems of Diffuse Type.- R.D. Nussbaum: Limiting Profiles For Solutions of Differential-Delay Equations.
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The C.I.M.E. session on Dynamical Systems, held in Cetraro (Italy), June 19-26, 2000, focused on the latest developments in several important areas in dynamical systems, with full development and historical context. The lectures of Chow and Mallet-Paret focus on the area of lattice differential systems, the lectures of Conto and Galleotti treat the classical problem of classification of orbits for two-dimensional autonomous systems with polynomial right sides, the lectures of Nussbaum focus on applications of fixed point theorems to the problem of limiting profiles for the solutions of singular perturbations of delay differential equations, and the lectures of Johnson and Mantellini deal with the existence of periodic and quasi-periodic orbits to non-autonomous systems. The volume will be of interest to researchers and graduate students working in these areas.
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• Preface.- A. Cohen: Theoretical Applied and Computational Aspects of Nonlinear Approximation.- W. Dahmen: Multiscale and Wavelet Methods for Operator Equations.- J. H. Bramble: Multilevel Methods in Finite Elements.
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This volume aims to disseminate a number of new ideas that have emerged in the last few years in the field of numerical simulation, all bearing the common denominator of the "multiscale" or "multilevel" paradigm. This covers the presence of multiple relevant "scales" in a physical phenomenon; the detection and representation of "structures", localized in space or in frequency, in the solution of a mathematical model; the decomposition of a function into "details" that can be organized and accessed in decreasing order of importance; and the iterative solution of systems of linear algebraic equations using "multilevel" decompositions of finite dimensional spaces.
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• William H. Meeks III: Minimal Surfaces in Flat Three Dimensional Spaces.- Joaquin Perez, Antonio Ros: Properly embedded minimal surfaces with finite total curvature.- Harold Rosenberg: Bryant Surfaces.
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In the second half of the twentieth century the global theory of minimal surface in flat space had an unexpected and rapid blossoming. Some of the classical problems were solved and new classes of minimal surfaces found. Minimal surfaces are now studied from several different viewpoints using methods and techniques from analysis (real and complex), topology and geometry. In this lecture course, Meeks, Ros and Rosenberg, three of the main architects of the modern edifice, present some of the more recent methods and developments of the theory. The topics include moduli, asymptotic geometry and surfaces of constant mean curvature in the hyperbolic space.
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