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• Elimination theory Numerical algebraic geometry Geometric modeling Rigidity theory Chemical reaction networks Illustration credits Bibliography Index.
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Systems of polynomial equations can be used to model an astonishing variety of phenomena. This book explores the geometry and algebra of such systems and includes numerous applications. The book begins with elimination theory from Newton to the twenty-first century and then discusses the interaction between algebraic geometry and numerical computations, a subject now called numerical algebraic geometry. The final three chapters discuss applications to geometric modeling, rigidity theory, and chemical reaction networks in detail. Each chapter ends with a section written by a leading expert. Examples in the book include oil wells, HIV infection, phylogenetic models, four-bar mechanisms, border rank, font design, Stewart-Gough platforms, rigidity of edge graphs, Gaussian graphical models, geometric constraint systems, and enzymatic cascades. The reader will encounter geometric objects such as Bezier patches, Cayley-Menger varieties, and toric varieties; and algebraic objects such as resultants, Rees algebras, approximation complexes, matroids, and toric ideals. Two important subthemes that appear in multiple chapters are toric varieties and algebraic statistics. The book also discusses the history of elimination theory, including its near elimination in the middle of the twentieth century. The main goal is to inspire the reader to learn about the topics covered in the book. With this in mind, the book has an extensive bibliography containing over 350 books and papers.
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• Introduction The example of the GUE Wigner random matrices Beta-ensembles Discrete beta-ensembles Continuous beta-models: The several cut case Several matrix-ensembles Universality for beta-models Bibliography Index.
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Probability theory is based on the notion of independence. The celebrated law of large numbers and the central limit theorem describe the asymptotics of the sum of independent variables. However, there are many models of strongly correlated random variables: for instance, the eigenvalues of random matrices or the tiles in random tilings. Classical tools of probability theory are useless to study such models. These lecture notes describe a general strategy to study the fluctuations of strongly interacting random variables. This strategy is based on the asymptotic analysis of Dyson-Schwinger (or loop) equations: the author will show how these equations are derived, how to obtain the concentration of measure estimates required to study these equations asymptotically, and how to deduce from this analysis the global fluctuations of the model. The author will apply this strategy in different settings: eigenvalues of random matrices, matrix models with one or several cuts, random tilings, and several matrices models.
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• Introduction A dynamical systems approach Initial value spaces Foliated initial value spaces Cremona mappings Asymptotic analysis Lax pairs Riemann-Hilbert problems Foliations and vector bundles Projective spaces Reflection groups Lists of discrete-Painleve equations Asymptotics of discrete equations Bibliography Index.
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Discrete Painleve equations are nonlinear difference equations, which arise from translations on crystallographic lattices. The deceptive simplicity of this statement hides immensely rich mathematical properties, connecting dynamical systems, algebraic geometry, Coxeter groups, topology, special functions theory, and mathematical physics. This book necessarily starts with introductory material to give the reader an accessible entry point to this vast subject matter. It is based on lectures that the author presented as principal lecturer at a Conference Board of Mathematical Sciences and National Science Foundation conference in Texas in 2016. Instead of technical theorems or complete proofs, the book relies on providing essential points of many arguments through explicit examples, with the hope that they will be useful for applied mathematicians and physicists.
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The Grant Application Writer’s Workbook – NSF FastLane Version covers how applications to the National Science Foundation are prepared and reviewed. In this newly updated version, content has been updated to provide guidance and reflect the changes NSF has made. In addition to the updates, there is discussion about the use of the same review criteria for Intellectual Merit and Broader Impacts. Particular emphasis is placed on what constitutes an appropriate Broader Impacts activity. Principles and fundamentals of good proposal writing are also emphasized, together with essential tips and strategies to produce an end result that is clear, concise, and follows a rational progression. Examples are provided in the Workbook, after which you’re asked to make a comparable response in your area of research interest. As each such response is made, the first draft of your application evolves. You may sample the content of this Workbook by clicking on the links to the right. -- Provided by the publisher.

• Basics: Motivation, first definitions and properties Tensors via linear algebra: Rank and border rank Tensor networks The asymptotic geometry of tensors: Detour into probability and information theory Strassen's laser method and spectral theory Quantum mechanics for quantum information theory Quantum information theory and the asymptotic geometry of tensors Moment maps and moment polytopes Hints and answers to selected exercises Bibliography Index.
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Tensors are used throughout the sciences, especially in solid state physics and quantum information theory. This book brings a geometric perspective to the use of tensors in these areas. It begins with an introduction to the geometry of tensors and provides geometric expositions of the basics of quantum information theory, Strassen's laser method for matrix multiplication, and moment maps in algebraic geometry. It also details several exciting recent developments regarding tensors in general. In particular, it discusses and explains the following material previously only available in the original research papers: (1) Shitov's 2017 refutation of longstanding conjectures of Strassen on rank additivity and Common on symmetric rank; (2) The 2017 Christandl-Vrana-Zuiddam quantum spectral points that bring together quantum information theory, the asymptotic geometry of tensors, matrix multiplication complexity, and moment polytopes in geometric invariant theory; (3) the use of representation theory in quantum information theory, including the solution of the quantum marginal problem; (4) the use of tensor network states in solid state physics, and (5) recent geometric paths towards upper bounds for the complexity of matrix multiplication. Numerous open problems appropriate for graduate students and post-docs are included throughout.
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• Number theoretic zeta and $L$-functions The Selberg zeta function $L$-functions in geometry The Ihara zeta function Spectral graph theory Explicit constructions of Ramanujan graphs Artin $L$-functions and prime distributions for graphs Zeta and $L$-functions of complexes Bibliography Index.
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Zeta and $L$-functions play a central role in number theory. They provide important information of arithmetic nature. This book, which grew out of the author's teaching over several years, explores the interaction between number theory and combinatorics using zeta and $L$-functions as a central theme. It provides a systematic and comprehensive account of these functions in a combinatorial setting and establishes, among other things, the combinatorial counterparts of celebrated results in number theory, such as the prime number theorem and the Chebotarev density theorem. The spectral theory for finite graphs and higher dimensional complexes is studied. Of special interest in theory and applications are the spectrally extremal objects, called Ramanujan graphs and Ramanujan complexes, which can be characterized by their associated zeta functions satisfying the Riemann Hypothesis. Explicit constructions of these extremal combinatorial objects, using number-theoretic and combinatorial means, are presented. Research on zeta and $L$-functions for complexes other than graphs emerged only in recent years. This is the first book for graduate students and researchers offering deep insight into this fascinating and fast developing area.
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• Introduction. Smooth vs the non-smooth categories Spectral pair analysis for IFSs Harmonic analyses on fractals, with an emphasis on iterated function systems (IFS) measures Four kinds of harmonic analysis Harmonic analysis via representations of the Cuntz relations $\textit{ Positive definite functions }$ and kernel analysis Representations of $\textit{Lie groups}$. Non-commutative harmonic analysis Bibliography Index.
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There is a recent and increasing interest in harmonic analysis of non-smooth geometries. Real-world examples where these types of geometry appear include large computer networks, relationships in datasets, and fractal structures such as those found in crystalline substances, light scattering, and other natural phenomena where dynamical systems are present. Notions of harmonic analysis focus on transforms and expansions and involve dual variables. In this book on smooth and non-smooth harmonic analysis, the notion of dual variables will be adapted to fractals. In addition to harmonic analysis via Fourier duality, the author also covers multiresolution wavelet approaches as well as a third tool, namely, $L^2$ spaces derived from appropriate Gaussian processes. The book is based on a series of ten lectures delivered in June 2018 at a CBMS conference held at Iowa State University.
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• Basic definitions and examples McMullen's decomposition theorem Valuations on the line McMullen's description of $(n-1)$-homogeneous valuations The Klain-Schneider characterization of simple valuations Digression on the theory of generalized functions on manifolds The Goodey-Weil imbedding Digression on vector bundles The irreducibility theorem Further developments Bibliography.
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Theory of valuations on convex sets is a classical part of convex geometry which goes back at least to the positive solution of the third Hilbert problem by M. Dehn in 1900. Since then the theory has undergone a multifaceted development. The author discusses some of Hadwiger's results on valuations on convex compact sets that are continuous in the Hausdorff metric. The book also discusses the Klain-Schneider theorem as well as the proof of McMullen's conjecture, which led subsequently to many further applications and advances in the theory. The last section gives an overview of more recent developments in the theory of translation-invariant continuous valuations, some of which turn out to be useful in integral geometry. This book grew out of lectures that were given in August 2015 at Kent State University in the framework of the NSF CBMS conference ``Introduction to the Theory of Valuations on Convex Sets''. Only a basic background in general convexity is assumed.
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• Introductory biology Introduction to modeling Models of population dynamics Cancer and the immune system Parameters estimation Mathematical analysis inspired by cancer models Mathematical model of artherosclerosis: Risk of high cholesterol Mathematical analysis inspired by the atherosclerosis model Mathematical models of chronic wounds Mathematical analysis inspired by the chronic wound model Introduction to PDEs Bibliography Index.
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The fast growing field of mathematical biology addresses biological questions using mathematical models from areas such as dynamical systems, probability, statistics, and discrete mathematics. This book considers models that are described by systems of partial differential equations, and it focuses on modeling, rather than on numerical methods and simulations. The models studied are concerned with population dynamics, cancer, risk of plaque growth associated with high cholesterol, and wound healing. A rich variety of open problems demonstrates the exciting challenges and opportunities for research at the interface of mathematics and biology. This book primarily addresses students and researchers in mathematics who do not necessarily have any background in biology and who may have had little exposure to PDEs.
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• Preface
• Chapter 1: Evolutionary game theory
• Chapter 2: All those puzzling voting mysteries!
• Chapter 3: Voting theory applied elsewhere
• Chapter 4: Voting: Symmetry and decompositions
• Chapter 5: Game theory: A decomposition
• Chapter 6: The reductionist approach Bibliography Index.
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The mathematical challenges coming from the social and behavioral sciences differ significantly from typical applied mathematical concerns. ""Change, "" for instance, is ubiquitous, but without knowing the fundamental driving force, standard differential and iterative methods are not appropriate. Although differing forms of aggregation are widely used, a general mathematical assessment of potential pitfalls is missing. These realities provide opportunities to create new mathematical approaches. These themes are described in an introductory, expository, and accessible manner by exploring new ways to handle dynamics and evolutionary game theory, to identify subtleties of decision and voting methods, to recognize unexpected modeling concerns, and to introduce new approaches with which to examine game theory. Applications range from avoiding undesired consequences when designing policy to identifying unanticipated voting (where the ""wrong"" person could win), nonparametric statistical, and economic ""supply and demand"" properties.
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• IntroductionGeometric preliminariesMain resultsModel spaces of constant curvatureLocal structure of eigenfunctionsHadamard parametrics on Riemannian manifoldsLagrangian distributions and Fourier integral operatorsSmall time wave group and Weyl asymptoticsMatrix elements$L^p$ normsQuantum integrable systemsRestriction theoremsNodal sets: Real domainEigenfunctions in the complex domainIndex.
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Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow.The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain.The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.
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• An overview of the Elliott programAn introduction to the Basic Homotopy LemmaMaps to finite dimensional $C^*$-algebrasStable homotopy lemmasThe Basic Homotopy Lemma, finite dimensional cases$C^*$-algebras of generalized tracial rank oneMore Basic Homotopy LemmasAsymptotic unitary equivalenceClassification of simple $C^*$-algebras of finite rankBibliographyIndex.
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This book examines some recent developments in the theory of $C^*$-algebras, which are algebras of operators on Hilbert spaces. An elementary introduction to the technical part of the theory is given via a basic homotopy lemma concerning a pair of almost commuting unitaries. The book presents an outline of the background as well as some recent results of the classification of simple amenable $C^*$-algebra, otherwise known as the Elliott program. This includes some stable uniqueness theorems and a revisiting of Bott maps via stable homotopy. Furthermore, $KK$-theory related rotation maps are introduced. The book is based on lecture notes from the CBMS lecture sequence at the University of Wyoming in the summer of 2015.
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• Before you begin to write. What you need to know about NSF and its funding of research ; How to develop and optimize the idea for your grant application ; How to choose the appropriate funding opportunity and grant mechanism for your idea ; Preliminary proposals ; FastLane or Grants.gov for preparation and submission of your proposal? ; Merit review principles and criteria : tips on how to write for reviewers
• Template/Master plan for your proposal. Overview and objectives section ; Writing the overview and objectives section ; Expected significance section ; How to maximize your application's programmatic relevance to NSF
• Developing the rest of your application. Relation to the principal investigator's longer-term goals ; Research plan : narrative description of what will be done ; Background : review of relevant literature, results from prior NSF support, preliminary studies - and references cited sections ; Relation to other work in progress ; Broader impacts of the proposed work ; Budget and budget justification ; Current and pending support section ; Biographical sketch : Facilities, equipment and other resources section : Data management plan, posdoctoral researcher mentoring plan, letters of collaboration, and appendices
• Maximizing the competitiveness of your proposal. Pre-submission review ; How to create an eye-catching title for your proposal ; How to write the project summary : proprietary/privileged information
• Index.

• The local theory of the Cauchy problem The "road map": The concentration compactness/rigidity theorem method for critical problems I The "road map": The concentration compactness/rigidity theorem method for critical problems II Properties of compact solutions and some more rigidity theorems, with applications to an extension of Theorem 2.6 Proof of the rigidity theorems Type II blow-up solutions Channels of energy and outer energy lower bounds Universal type II blow-up profiles Soliton resolution for radial solutions to (NLW), I Soliton resolution for radial solutions to (NLW), II Soliton resolution for radial solutions to (NLW), III Bibliography.
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This monograph deals with recent advances in the study of the long-time asymptotics of large solutions to critical nonlinear dispersive equations. The first part of the monograph describes, in the context of the energy critical wave equation, the "concentration-compactness/rigidity theorem method" introduced by C. Kenig and F. Merle. This approach has become the canonical method for the study of the "global regularity and well-posedness" conjecture (defocusing case) and the "ground-state" conjecture (focusing case) in critical dispersive problems. The second part of the monograph describes the "channel of energy" method, introduced by T. Duyckaerts, C. Kenig, and F. Merle, to study soliton resolution for nonlinear wave equations. This culminates in a presentation of the proof of the soliton resolution conjecture, for the three-dimensional radial focusing energy critical wave equation. It is the intent that the results described in this book will be a model for what to strive for in the study of other nonlinear dispersive equations.
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• * Introduction* Three views of Ramsey theory* Ramsey's theorem van der Waerden's theorem* The Hales-Jewett theorem* Szemeredi's theorem* Graph Ramsey theory* Euclidean Ramsey theory* A general Ramsey product theorem* The theorems of Schur, Folkman, and Hindman* Rado's theorem* Current trends* Bibliography.
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In every sufficiently large structure which has been partitioned there will always be some well-behaved structure in one of the parts. This takes many forms. For example, colorings of the integers by finitely many colors must have long monochromatic arithmetic progressions (van der Waerden's theorem); and colorings of the edges of large graphs must have monochromatic subgraphs of a specified type (Ramsey's theorem). This book explores many of the basic results and variations of this theory. Since the first edition of this book there have been many advances in this field. In the second edition the authors update the exposition to reflect the current state of the art. They also include many pointers to modern results.
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• Mathematical shapes of uncertainty Gap theorems A problem by Polya and Levinson Determinacy of measures and oscillations of high-pass signals Beurling-Malliavin and Bernstein's problems The type problem Toeplitz approach to UP Toeplitz version of the Beurling-Malliavin theory Bibliography.
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The Uncertainty Principle in Harmonic Analysis (UP) is a classical, yet rapidly developing, area of modern mathematics. Its first significant results and open problems date back to the work of Norbert Wiener, Andrei Kolmogorov, Mark Krein and Arne Beurling. At present, it encompasses a large part of mathematics, from Fourier analysis, frames and completeness problems for various systems of functions to spectral problems for differential operators and canonical systems. These notes are devoted to the so-called Toeplitz approach to UP which recently brought solutions to some of the long-standing problems posed by the classics. After a short overview of the general area of UP the discussion turns to the outline of the new approach and its results. Among those are solutions to Beurling's Gap Problem in Fourier analysis, the Type Problem on completeness of exponential systems, a problem by Polya and Levinson on sampling sets for entire functions, Bernstein's problem on uniform polynomial approximation, problems on asymptotics of Fourier integrals and a Toeplitz version of the Beurling-Malliavin theory. One of the main goals of the book is to present new directions for future research opened by the new approach to the experts and young analysts.
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• Prelude
• Wiener integrals
• A linear heat equation
• Walsh-Damang integrals
• A non-linear heat equation
• Intermezzo: A parabolic Anderson model
• Intermittency
• Intermittency fronts
• Intermittency islands
• Correlation length
• Appendix A: Some special integrals
• Appendix B: A Burkholder-Davis-Gundy inequality
• Appendix C: Reguarity theory.

The general area of stochastic PDEs is interesting to mathematicians because it contains an enormous number of challenging open problems. There is also a great deal of interest in this topic because it has deep applications in disciplines that range from applied mathematics, statistical mechanics, and theoretical physics, to theoretical neuroscience, theory of complex chemical reactions [including polymer science], fluid dynamics, and mathematical finance. The stochastic PDEs that are studied in this book are similar to the familiar PDE for heat in a thin rod, but with the additional restriction that the external forcing density is a two-parameter stochastic process, or what is more commonly the case, the forcing is a "random noise, " also known as a "generalized random field." At several points in the lectures, there are examples that highlight the phenomenon that stochastic PDEs are not a subset of PDEs. In fact, the introduction of noise in some partial differential equations can bring about not a small perturbation, but truly fundamental changes to the system that the underlying PDE is attempting to describe. The topics covered include a brief introduction to the stochastic heat equation, structure theory for the linear stochastic heat equation, and an in-depth look at intermittency properties of the solution to semilinear stochastic heat equations. Specific topics include stochastic integrals a la Norbert Wiener, an infinite-dimensional Ito-type stochastic integral, an example of a parabolic Anderson model, and intermittency fronts. There are many possible approaches to stochastic PDEs. The selection of topics and techniques presented here are informed by the guiding example of the stochastic heat equation.
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• Introduction to fractals Dimension Trees and fractals Invariant sets Probability trees Galleries Probability trees revisited Elements of ergodic theory Galleries of trees General remarks on Markov systems Markov operator $\mathcal{T}$ and measure preserving transformation $T$ Probability trees and galleries Ergodic theorem and the proof of the main theorem An application: The $k$-lane property Dimension and energy Dimension conservation Ergodic theorem for sequences of functions Dimension conservation for homogeneous fractals: The main steps in the proof Verifying the conditions of the ergodic theorem for sequences of functions Bibliography Index.
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Fractal geometry represents a radical departure from classical geometry, which focuses on smooth objects that "straighten out" under magnification. Fractals, which take their name from the shape of fractured objects, can be characterized as retaining their lack of smoothness under magnification. The properties of fractals come to light under repeated magnification, which we refer to informally as "zooming in". This zooming-in process has its parallels in dynamics, and the varying "scenery" corresponds to the evolution of dynamical variables. The present monograph focuses on applications of one branch of dynamics--ergodic theory--to the geometry of fractals. Much attention is given to the all-important notion of fractal dimension, which is shown to be intimately related to the study of ergodic averages. It has been long known that dynamical systems serve as a rich source of fractal examples. The primary goal in this monograph is to demonstrate how the minute structure of fractals is unfolded when seen in the light of related dynamics.
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• Preface--
• 1. Inverse scattering in two dimensions--
• 2. Maxwell's equations--
• 3. The inverse problem for obstacles--
• 4. The inverse scattering problem for anisotropic media--
• 5. The inverse scattering problem for thin objects--
• 6. The inverse scattering problem for buried objects-- Bibliography-- Index.
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The linear sampling method is the oldest and most developed of the qualitative methods in inverse scattering theory. It is based on solving a linear integral equation and then using the equation's solution as an indicator function for the determination of the support of the scattering object. This book describes the linear sampling method for a variety of electromagnetic scattering problems. It presents uniqueness theorems and the derivation of various inequalities on the material properties of the scattering object from a knowledge of the far field pattern of the scattered wave. Also covered are: the approximation properties of Herglotz wave functions; the behavior of solutions to the interior transmission problem, a novel interior boundary value problem; and numerical examples of the inversion scheme.
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