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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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• 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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Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, Scholze introduced the concept of "diamonds, " which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. In this book, Peter Scholze and Jared Weinstein show that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. This book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures on p-adic Geometry will be a useful resource for students and scholars working in arithmetic geometry and number theory.
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• Algebraic geometry: a celebration of Emma Previato's 65th birthday Ron Donagi and Tony Shaska--
• 1. Arithmetic analogues of Hamiltonian systems Alexandru Buium--
• 2. Algebraic spectral curves over Q and their tau-functions Boris Dubrovin--
• 3. Frobenius split anticanonical divisors Sandor J. Kovacs--
• 4. Halves of points of an odd degree hyperelliptic curve in its jacobian Yuri G. Zarhin--
• 5. Normal forms for Kummer surfaces Adrian Clingher and Andreas Malmendier--
• 6. -functions: old and new results V. M. Buchstaber, V. Z. Enolski and D. V. Leykin--
• 7. Bergman tau-function: from Einstein equations and Dubrovin-Frobenius manifolds to geometry of moduli spaces Dmitry Korotkin--
• 8. The rigid body dynamics in an ideal fluid: Clebsch top and Kummer surfaces Jean-Pierre Francoise and Daisuke Tarama--
• 9. An extension of Delsarte, Goethals and Mac Williams theorem on minimal weight codewords to a class of Reed-Muller type codes Cicero Carvalho and Victor G. L. Neumann--
• 10. A primer on Lax pairs L. M. Bates and R. C. Churchill--
• 11. Lattice-theoretic characterizations of classes of groups Roland Schmidt--
• 12. Jacobi inversion formulae for a curve in Weierstrass normal form Jiyro Komeda and Shigeki Matsutani--
• 13. Spectral construction of non-holomorphic Eisenstein-type series and their Kronecker limit formula James Cogdell, Jay Jorgenson and Lejla Smajlovic--
• 14. Some topological applications of theta functions Mauro Spera--
• 15. Multiple Dedekind zeta values are periods of mixed Tate motives Ivan Horozov--
• 16. Noncommutative cross-ratio and Schwarz derivative Vladimir Retakh, Vladimir Rubtsov and Georgy Sharygin.
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• Integrable systems: a celebration of Emma Previator's 65th birthday Ron Donagi and Tony Shaska--
• 1. Trace ideal properties of a class of integral operators Fritz Gesztesy and Roger Nichols--
• 2. Explicit symmetries of the Kepler Hamiltonian Horst Knoerrer--
• 3. A note on the commutator of Hamiltonian vector fields Henryk Zoladek--
• 4. Nodal curves and a class of solutions of the Lax equation for shock clustering and Burgers turbulence Luen-Chau Li--
• 5. Solvable dynamical systems in the plane with polynomial interactions Francesco Calogero and Farrin Payandeh--
• 6. The projection method in classical mechanics A. M. Perelomov--
• 7. Pencils of quadrics, billiard double-reflection and confocal incircular nets Vladimir Dragovic, Milena Radnovic and Roger Fidele Ranomenjanahary--
• 8. Bi-flat F-manifolds: a survey Alessandro Arsie and Paolo Lorenzoni--
• 9. The periodic 6-particle Kac-Van Moerbeke system Pol Vanhaecke--
• 10. Integrable mappings from a unified perspective Tova Brown and Nicholas M. Ercolani--
• 11. On an Arnold-Liouville type theorem for the focusing NLS and the focusing mKdV equations T. Kappeler and P. Topalov--
• 12. Commuting Hamiltonian flows of curves in real space forms Albert Chern, Felix Knoeppel, Franz Pedit and Ulrich Pinkall--
• 13. The Kowalewski top revisited F. Magri--
• 14. The Calogero-Francoise integrable system: algebraic geometry, Higgs fields, and the inverse problem Steven Rayan, Thomas Stanley and Jacek Szmigielski--
• 15. Tropical Markov dynamics and Cayley cubic K. Spalding and A. P. Veselov--
• 16. Positive one-point commuting difference operators Gulnara S. Mauleshova and Andrey E. Mironov.
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Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. The articles in this second volume discuss areas related to algebraic geometry, emphasizing the connections of this central subject to integrable systems, arithmetic geometry, Riemann surfaces, coding theory and lattice theory.
(source: Nielsen Book Data)
Created as a celebration of mathematical pioneer Emma Previato, this comprehensive book highlights the connections between algebraic geometry and integrable systems, differential equations, mathematical physics, and many other areas. The authors, many of whom have been at the forefront of research into these topics for the last decades, have all been influenced by Previato's research, as her collaborators, students, or colleagues. The diverse articles in the book demonstrate the wide scope of Previato's work and the inclusion of several survey and introductory articles makes the text accessible to graduate students and non-experts, as well as researchers. This first volume covers a wide range of areas related to integrable systems, often emphasizing the deep connections with algebraic geometry. Common themes include theta functions and Abelian varieties, Lax equations, integrable hierarchies, Hamiltonian flows and difference operators. These powerful tools are applied to spinning top, Hitchin, Painleve and many other notable special equations.
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• Introduction
• Notation and preliminary material
• Twisted (co)homology and integrals of hypergeometric type
• An explicit expression for Veech's map and some consequences
• Flat tori with two cone points
• Some explicit computations and a proof of Veech's volume conjecture when g = 1 and n = 2
• Appendix A. 1-dimensional complex hyperbolic conifolds
• Appendix B. Manin connection associated to Veech's map

"... Our starting point is an explicit formula for flat metrics with cone singularities on elliptic curves, in terms of theta functions. From this, we deduce an explicit description of Veech's foliation: at the level of the Torelli space of n-marked elliptic curves, it is given by an explicit affine first integral. From the preceding result, one determines exactly which leaves of Veech's foliation are closed subvarieties of the moduli space M1,n of n-marked elliptic curves. We also give a local explicit expression, in terms of hypergeometric elliptic integrals, for the Veech map by means of which is defined the complex hyperbolic structure of a leaf."--Page iii

• Preface /
• Pablo A. Parrilo, Rekha R. Thomas
• A brief introduction to sums of squares / Grigoriy Blekherman
• The geometry of spectrahedra / Cynthia Vinzant
• Lifts of convex sets / Hamza Fawzi
• Algebraic geometry and sums of squares / Mauricio Velasco
• Sums of squares in theoretical computer science / Ankur Moitra
• Applications of sums of squares / Georgina Hall

This volume is based on lectures delivered at the 2019 AMS Short Course ""Sum of Squares: Theory and Applications'', held January 14-15, 2019, in Baltimore, Maryland. This book provides a concise state-of-the-art overview of the theory and applications of polynomials that are sums of squares. This is an exciting and timely topic, with rich connections to many areas of mathematics, including polynomial and semidefinite optimization, real and convex algebraic geometry, and theoretical computer science. The six chapters introduce and survey recent developments in this area; specific topics include the algebraic and geometric aspects of sums of squares and spectrahedra, lifted representations of convex sets, and the algorithmic and computational implications of viewing sums of squares as a meta algorithm. The book also showcases practical applications of the techniques across a variety of areas, including control theory, statistics, finance and machine learning.
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• Introduction.- Hermitian vector bundles over arithmetic curves.- -Invariants of Hermitian vector bundles over arithmetic curves.- Geometry of numbers and -invariants.- Countably generated projective modules and linearly compact Tate spaces over Dedekind rings.- Ind- and pro-Hermitian vector bundles over arithmetic curves.- -Invariants of infinite dimensional Hermitian vector bundles: denitions and first properties.- Summable projective systems of Hermitian vector bundles and niteness of -invariants.- Exact sequences of infinite dimensional Hermitian vector bundles and subadditivity of their -invariants.- Infinite dimensional vector bundles over smooth projective curves.- Epilogue: formal-analytic arithmetic surfaces and algebraization.- Appendix A. Large deviations and Cramer's theorem.- Appendix B. Non-complete discrete valuation rings and continuity of linear forms on prodiscrete modules.- Appendix C. Measures on countable sets and their projective limits.- Appendix D. Exact categories.- Appendix E. Upper bounds on the dimension of spaces of holomorphic sections of line bundles over compact complex manifolds.- Appendix F. John ellipsoids and finite dimensional normed spaces.
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This book presents the most up-to-date and sophisticated account of the theory of Euclidean lattices and sequences of Euclidean lattices, in the framework of Arakelov geometry, where Euclidean lattices are considered as vector bundles over arithmetic curves. It contains a complete description of the theta invariants which give rise to a closer parallel with the geometric case. The author then unfolds his theory of infinite Hermitian vector bundles over arithmetic curves and their theta invariants, which provides a conceptual framework to deal with the sequences of lattices occurring in many diophantine constructions. The book contains many interesting original insights and ties to other theories. It is written with extreme care, with a clear and pleasant style, and never sacrifices accessibility to sophistication. .
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• Curves with many points. I: Modular curves Class field theory Curves with many points. II Infinite global fields Decoding: Some examples Sphere packings Codes from multidimensional varieties Applications Appendix: Some basic facts from Volume 1 Bibliography List of names Index.
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Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to several domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.
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• Preliminary Material
• Holomorphic Mappings
• Definitions
• The Case Where E is Finite-Dimensional
• Holomorphic = Analytic for Arbitrary E
• Complex Manifolds
• Manifolds
• Vector Bundles
• Projective Space P(E)
• Grassmannians
• Further Examples of Complex Manifolds
• Integration on Oriented Manifolds and Stokes' Formula
• Differential Forms on a Complex Manifold
• Symmetric Products of C
• Continuity of Roots
• Weierstrass Preparation Theorem
• The Symmetric Product of C
• Symmetric Products of Topological Spaces
• Vector Symmetric Functions
• Vertical Localization
• Canonical Equations
• Complex Structure on Symk(Cp)
• Stratification
• Multigraphs and Reduced Complex Spaces
• Reduced Multigraph
• Proper Mappings
• Analytic Subsets
• Analytic Continuation
• Analytic Étale Coverings
• Reduced Multigraphs
• Local Study of Reduced Multigraphs
• Irreducibility of Reduced Multigraphs
• Multigraphs
• Basic Definitions
• Classification Map and the Canonical Equation
• Analytic Subsets
• Local Parameterization Theorem : First Version
• Irreducible Components and Singular Locus
• Maximum Principle
• Ramified Covers
• Local Parameterization Theorem : Final Version
• Analyticity of the Singular Locus
• Reduced Complex Spaces
• Definitions and Elementary Properties
• Singular Locus and Irreducible Components
• Dimension and Local Irreducibility
• Minimal Embeddings and the Zariski Tangent Space
• Fiber Dimension and Generic Rank
• Symmetric Product of a Reduced Complex Space
• Proper Finite Mappings
• Remmert-Stein Theorem
• Notes on Chapter 1 and this chapter
• The Preparation and Division Theorems
• The Local Parameterization Theorem
• The Three Definitions of Ramified Covers
• Complex Spaces
• The Theorem of Remmert-Stein
• Analysis and Geometry on a Reduced Complex Space
• Transversality and the Zariski Tangent Cone
• Transverse Planes to an Analytic Subset
• Algebraic Cones
• Transversality and the Tangent Cone
• Zariski Tangent Cycle
• The Theorem of P. Lelong
• Introduction
• Preliminaries
• The Case of a Reduced Multigraph
• Differential Forms on a Reduced Complex Space
• Lelong's Theorem : General Case
• Volume
• Coherent Sheaves
• Coherent Sheaves on a Reduced Complex Space
• Canonical Topology
• Modifications and Blowups
• Modifications
• Blowups
• Meromorphic Mappings
• Normalization
• Normal Spaces
• Meromorphic Functions
• Locally Bounded Meromorphic Functions
• Universal Denominators
• Additional Material on Normality
• Normalization
• The Weak Normalization
• Complementary Material on Meromorphic Functions
• Local Bound of Volume of General Fibers
• Local Blowing Up
• The Theorem
• Direct Image and Enclosure
• Holomorphic Mappings with Values in a TVS
• Reduced Multigraphs in a Sequentially Complete TVS
• The Direct Image Theorem
• Theorem on Encloseability
• Holomorphic Convexity : The Quotient Theorem
• Dirac Mapping
• Holomorphically Convex Spaces
• Stein Spaces and the Remmert Reduction
• The Quotient Theorem of H. Cartan
• Notes on This Chapter
• Transversality and the Zariski Tangent Cone
• Algebraic Cones
• The Theorem of P. Lelong, Canonical Topology, Modifications and Blowups
• Normalization
• Weak Normalization
• Bound of Volume of General Fibers
• Direct Image and Encloseability : Holomorphic Convexity
• Quotient Theorem
• Families of Cycles in Complex Geometry
• Families of Cycles
• Cycles
• Elementary Operations on Cycles
• Functorial Properties
• Continuous Families of Cycles
• Scales
• Topology of C...(M) and of Cn(M)
• Functions Denned by Integration
• C...(M) and Cn(M) are Second Countable
• Continuity of Direct Image Maps
• Integration of Cohomology Classes : Topological Case
• Compactness and the Theorem of E. Bishop
• Cycles as Currents
• Analytic Families of Cycles
• Basic Definitions
• Multiplicity of a Point in a Cycle
• Graph of an Analytic Family of Cycles
• The Case of a Normal Parameter Space
• Stability of Analytic Families by Direct Images
• Fundamental Counterexample
• What Does Not Work!
• Example
• Characterization of Isotropic Morphisms : Applications
• Isotropic Morphisms
• Integration of Cohomology Classes
• Finiteness of the Space of Cycles : Applications
• Finiteness Theorem
• Some Consequences
• Theorem on Connectedness
• Number of Irreducible Components
• Connected Cycles
• Relative Cycles
• Preliminaries
• The Space of Cycles Relative to a Morphism
• Fibers of a Proper Meromorphic Mapping
• The Case of a Proper Holomorphic Map
• The Case of a Proper Meromorphic Map
• Almost Holomorphic Mappings
• Analytic Families of Holomorphic Mappings
• Appendix I : Complexification
• Conjugation on a Complex Vector Space
• Complexification of a Real Vector Space
• The Complex Case
• Orientation of a Complex Vector Space
• The Space ...(E)
• Positivity in the Sense of P. Lelong
• Appendix II : Locally Convex Topological Vector Spaces
• Bibliography
• Index

• Foreword Matthew Ando-- Preface-- Introduction--
• 1. Unoriented bordism--
• 2. Complex bordism--
• 3. Finite spectra--
• 4. Unstable cooperations--
• 5. The -orientation-- Appendix A. Power operations-- Appendix B. Loose ends-- References-- Index.
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This text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject 'elliptic cohomology' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten's genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.
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• Introduction Preliminaries Saito's Hodge filtration and Hodge modules Birational definition of Hodge ideals Basic properties of Hodge ideals Local study of Hodge ideals Vanishing theorems Vanishing on $\mathbf{P} ^n$ and abelian varieties, with applications Appendix: Higher direct images of forms with log poles References.
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The authors use methods from birational geometry to study the Hodge filtration on the localization along a hypersurface. This filtration leads to a sequence of ideal sheaves, called Hodge ideals, the first of which is a multiplier ideal. They analyze their local and global properties, and use them for applications related to the singularities and Hodge theory of hypersurfaces and their complements.
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• Introduction and main result Preliminarities Quasi-characters Strongly cuspidal functions Statement of the Trace formula Proof of Theorem 1.3 Localization Integral transfer Calculation of the limit $\lim _N\rightarrow \infty I_x, \omega , N(f)$ Proof of Theorem 5.4 and Theorem 5.7 Appendix A. The proof of Lemma 9.1 and Lemma 9.11 Appendix B. The reduced model Appendix B. The reduced model Bibliography.
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Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, the author is able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, the author proves a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, the author proves the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.
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• Introduction Background Schubert systems First applications Schubert decompositions for type $\widetilde{D}_n$ Proof of Theorem 4.1 Appendix A. Representations for quivers of type $\widetilde{D}_n$ Appendix B. Bases for representations of type $\widetilde{D}_n$ Bibliography.
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Let $Q$ be a quiver of extended Dynkin type $\widetilde{D}_n$. In this first of two papers, the authors show that the quiver Grassmannian $\mathrm{Gr}_{\underline{e}}(M)$ has a decomposition into affine spaces for every dimension vector $\underline{e}$ and every indecomposable representation $M$ of defect $-1$ and defect $0$, with the exception of the non-Schurian representations in homogeneous tubes. The authors characterize the affine spaces in terms of the combinatorics of a fixed coefficient quiver for $M$. The method of proof is to exhibit explicit equations for the Schubert cells of $\mathrm{Gr}_{\underline{e}}(M)$ and to solve this system of equations successively in linear terms. This leads to an intricate combinatorial problem, for whose solution the authors develop the theory of Schubert systems. In Part 2 of this pair of papers, they extend the result of this paper to all indecomposable representations $M$ of $Q$ and determine explicit formulae for the $F$-polynomial of $M$.
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• Introduction--
• 1. Topological rings and modules and their completions--
• 2. Inverse limits--
• 3. The idea of slenderness--
• 4. Objects of type / \coprod--
• 5. Concrete examples. Slender rings--
• 6. More examples of slender objects-- Appendix. Ordered sets and measurable cardinals-- References-- Notation index-- Name Index-- Subject index.
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Slenderness is a concept relevant to the fields of algebra, set theory, and topology. This first book on the subject is systematically presented and largely self-contained, making it ideal for researchers and graduate students. The appendix gives an introduction to the necessary set theory, in particular to the (non-)measurable cardinals, to help the reader make smooth progress through the text. A detailed index shows the numerous connections among the topics treated. Every chapter has a historical section to show the original sources for results and the subsequent development of ideas, and is rounded off with numerous exercises. More than 100 open problems and projects are presented, ready to inspire the keen graduate student or researcher. Many of the results are appearing in print for the first time, and many of the older results are presented in a new light.
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• Preface--
• 1. Preliminaries--
• 2. Stability of linear stochastic differential equations--
• 3. Stability of non linear stochastic differential equations--
• 4. Stability of stochastic functional differential equations--
• 5. Some applications related to stochastic stability-- Appendix-- References-- Index.
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The stability of stochastic differential equations in abstract, mainly Hilbert, spaces receives a unified treatment in this self-contained book. It covers basic theory as well as computational techniques for handling the stochastic stability of systems from mathematical, physical and biological problems. Its core material is divided into three parts devoted respectively to the stochastic stability of linear systems, non-linear systems, and time-delay systems. The focus is on stability of stochastic dynamical processes affected by white noise, which are described by partial differential equations such as the Navier-Stokes equations. A range of mathematicians and scientists, including those involved in numerical computation, will find this book useful. It is also ideal for engineers working on stochastic systems and their control, and researchers in mathematical physics or biology.
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• Part 1: A. Bayer, Wall-crossing implies Brill-Noether applications of stability conditions on surfaces R. J. Berman, Kahler-Einstein metrics, canonical random point processes and birational geometry T. Bridgeland, Hall algebras and Doanldson-Thomas invariants S. Cantat, The Cremona group A.-M. Castravet, Mori dream spaces and blow-ups T. de Fernex, The space of arcs of an algebraic variety S. Donaldson, Stability of algebraic varieties and Kahler geometry L. Ein and R. Lazarsfeld, Syzygies of projective varieties of large degree: Recent progress and open problems E. Gonzalez, P. Solis, and C. T. Woodward, Stable gauged maps D. Greb, S. Kebekus, and B. Taji, Uniformisation of higher-dimensional minimal varieties H. D. Hacon, J. McKernan, and C. Xu, Boundedness of varieties of log general type D. Halpern-Leistner, $\Theta$-stratifications, $\Theta$-reductive stacks, and applications A. Horing and T. Peternell, Bimeromorphic geometry of Kahler threefolds S. J. Kovacs, Moduli of stable log-varieties-An update A. Okounkov, Enumerative geometry and geometric representation theory R. Pandharipande, A calculus for the moduli space of curves Z. Patakfalvi, Frobenius techniques in birational geometry M. Paun, Singualar Hermitian metrics and positivity of direct images of pluricanonical bundles M. Popa, Positivity for Hodge modules and geometric applications R. P. Thomas, Notes on homological projective duality Y. Toda, Non-commutative deformations and Donaldson-Thomas invariants V. Tosatti, Nakamaye's theorem on complex manifolds.
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• Part 2: D. Ben-Zvi and D. Nadler, Betti geometric Langlands B. Bhatt, Specializing varieties and their cohomology from characteristic 0 to characteristic $p$ T. D. Browning, How often does the Hasse principle hold? L. Caporaso, Tropical methods in the moduli theory of algebraic curves R. Cavalieri, P. Johnson, H. Markwig, and D. Ranganathan, A graphical interface for the Gromov-witten theory of curves H. Esnault, Some fundamental groups in arithmetic geometry L. Fargues, From local class field to the curve and vice versa M. Gross and B. Siebert, Intrinsic mirror symmetry and punctured Gromov-Witten invariants E. Katz, J. Rabinoff, and D. Zureick-Brown, Diophantine and tropical geometry, and uniformity of rational points on curves K. S. Kedlaya and J. Pottharst, On categories of $(\varphi, \Gamma)$-modules M. Kim, Principal bundles and reciprocity laws in number theory B. Klingler, E. Ullmo, and A. Yafaev, Bi-algebraic geometry and the Andre-Ooert conjecture M. Lieblich, Moduli of sheaves: A modern primer J. Nicaise, Geometric invariants for non-archimedean semialgebraic sets T. Pantev and G. Vezzosi, Symplectic and Poisson derived geometry and deformation quantization A. Pirutka, Varieties that are not stably rational, zero-cycles and unramified cohomology T. Saito, On the proper push-forward of the characteristic cycle of a constructible sheaf T. Szamuely and G. Zabradi, The $p$-adic Hodge decomposition according to Beilinson A. Tamagawa, Specialization of $\ell$-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varieties O. Wittenberg, Rational points and zero-cycles on rationally connected varieties over number fields.
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This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic $p$ and $p$-adic tools, etc. The resulting articles will be important references in these areas for years to come.
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• Introduction Probability Primer Algebra Primer Conditional Independence Statistics Primer Exponential Families Likelihood Inference The Cone of Sufficient Statistics Fisher's Exact Test Bounds on Cell Entries Exponential Random Graph Models Design of Experiments Graphical Models Hidden Variables Phylogenetic Models Identifiability Model Selection and Bayesian Integrals MAP Estimation and Parametric Inference Finite Metric Spaces Bibliography Index.
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Algebraic statistics uses tools from algebraic geometry, commutative algebra, combinatorics, and their computational sides to address problems in statistics and its applications. The starting point for this connection is the observation that many statistical models are semialgebraic sets. The algebra/statistics connection is now over twenty years old, and this book presents the first broad introductory treatment of the subject. Along with background material in probability, algebra, and statistics, this book covers a range of topics in algebraic statistics including algebraic exponential families, likelihood inference, Fisher's exact test, bounds on entries of contingency tables, design of experiments, identifiability of hidden variable models, phylogenetic models, and model selection. With numerous examples, references, and over 150 exercises, this book is suitable for both classroom use and independent study.
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• 1. Notes on Weyl algebras and D-modules / Markus Brodmann
• 2. Inverse systems of local rings / Juan Elias
• 3. Lectures on the representation type of a projective variety / Rosa M. Miró-Roig
• 4. Simplicial toric varieties which are set-theoretic complete intersections / Marcel Morales.

This book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.
(source: Nielsen Book Data)

• A crash course in commutative algebra Affine varieties Projective varieties Regular and rational maps of quasi-projective varieties Products The blow-up of an ideal Finite maps of quasi-projective varieties Dimension of quasi-projective algebraic sets Zariski's main theorem Nonsingularity Sheaves Applications to regular and rational maps Divisors Differential forms and the canonical divisor Schemes The degree of a projective variety Cohomology Curves An introduction to intersection theory Surfaces Ramification and etale maps Bertini's theorem and general fibers of maps Bibliography Index.
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This book presents a readable and accessible introductory course in algebraic geometry, with most of the fundamental classical results presented with complete proofs. An emphasis is placed on developing connections between geometric and algebraic aspects of the theory. Differences between the theory in characteristic $0$ and positive characteristic are emphasized. The basic tools of classical and modern algebraic geometry are introduced, including varieties, schemes, singularities, sheaves, sheaf cohomology, and intersection theory. Basic classical results on curves and surfaces are proved. More advanced topics such as ramification theory, Zariski's main theorem, and Bertini's theorems for general linear systems are presented, with proofs, in the final chapters. With more than 200 exercises, the book is an excellent resource for teaching and learning introductory algebraic geometry.
(source: Nielsen Book Data)