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Published: 9 March 2023

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1

1938-, Griffiths Phillip, ed. On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Princeton, N.J: Princeton University Press, 2004.

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2

Quantitative arithmetic of projective varieties. Basel: Birkhäuser, 2009.

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3

J, Kovacs Sandor, ed. Classification of higher dimensional algebraic varieties. Basel: Birkhäuser, 2010.

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4

Bjorn, Poonen, and Tschinkel Yuri, eds. Arithmetic of higher-dimensional algebraic varieties. Boston: Birkhäuser, 2004.

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5

Johnsen, Trygve. K3 Projective models in scrolls. Berlin: Springer, 2004.

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6

Johnsen, Trygve. K3 Projective models in scrolls. Berlin: Springer, 2004.

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7

Voisin, Claire. Théorie de Hodge et géométrie algébrique complexe. Paris: Société Mathématique de France, 2002.

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8

Introduction to Algebra with Applications for a Variety of Technologies. 3rd ed. Kendall/Hunt Publishing Company, 2002.

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9

Hart, Jerome B., and Roxane R. Barrows. Introduction to Algebra with Applications for a Variety of Technologies. 2nd ed. Kendall/Hunt Publishing Company, 1999.

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10

Bushko, Andrew. Introduction to Algebra with Applications for a Variety of Technologies. Kendall/Hunt Publishing Company, 1998.

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11

Griffiths, Phillip, and Mark Green. On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety (Annals of Mathematics Studies). Princeton University Press, 2005.

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12

Griffiths, Phillip, and Mark Green. On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (AM-157) (Annals of Mathematics Studies). Princeton University Press, 2004.

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13

Zein, Fouad El, and Lˆe D˜ung Tr ´ang. Mixed Hodge Structures. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0003.

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This chapter discusses mixed Hodge structures (MHS). It first defines the abstract category of Hodge structures and introduces spectral sequences. The decomposition on the cohomology of Kähler manifolds is used to prove the degeneration at rank 1 of the spectral sequence defined by the filtration F on the de Rham complex in the projective nonsingular case. The chapter then introduces an abstract definition of MHS as an object of interest in linear algebra. It then attempts to develop algebraic homology techniques on filtered complexes up to filtered quasi-isomorphisms of complexes. Finally, this chapter provides the construction of the MHS on any algebraic variety.
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14

Earl, Richard, and James Nicholson. The Concise Oxford Dictionary of Mathematics. 6th ed. Oxford University Press, 2021. http://dx.doi.org/10.1093/acref/9780198845355.001.0001.

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Over 4,000 entries This informative A to Z provides clear, jargon-free definitions of a wide variety of mathematical terms. Its articles cover both pure and applied mathematics and statistics, and include key theories, concepts, methods, programmes, people, and terminology. For this sixth edition, around 800 new terms have been defined, expanding on the dictionary’s coverage of algebra, differential geometry, algebraic geometry, representation theory, and statistics. Among this new material are articles such as cardinal arithmetic, first fundamental form, Lagrange’s theorem, Navier-Stokes equations, potential, and splitting field. The existing entries have also been revised and updated to account for developments in the field. Numerous supplementary features complement the text, including detailed appendices on basic algebra, areas and volumes, trigonometric formulae, and Roman numerals. Newly added to these sections is a historical timeline of significant mathematicians’ lives and the emergence of key theorems. There are also illustrations, graphs, and charts throughout the text, as well as useful web links to provide access to further reading.
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15

Lieblich, Max, Will Sawin, Olsson Martin, and János Kollár. What Determines an Algebraic Variety? Princeton University Press, 2023.

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16

Lieblich, Max, Will Sawin, Olsson Martin, and János Kollár. What Determines an Algebraic Variety? Princeton University Press, 2023.

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17

Lieblich, Max, Will Sawin, Olsson Martin, and János Kollár. What Determines an Algebraic Variety? Princeton University Press, 2023.

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18

Monomialization of Morphisms from 3 Folds to Surfaces. Springer, 2002.

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19

Cutkosky, Steven D. Monomialization of Morphisms from 3-Folds to Surfaces. Springer London, Limited, 2004.

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20

Huybrechts, D. Fourier-Mukai Transforms in Algebraic Geometry. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.001.0001.

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This book provides a systematic exposition of the theory of Fourier-Mukai transforms from an algebro-geometric point of view. Assuming a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. The derived category is a subtle invariant of the isomorphism type of a variety, and its group of autoequivalences often shows a rich structure. As it turns out — and this feature is pursued throughout the book — the behaviour of the derived category is determined by the geometric properties of the canonical bundle of the variety. Including notions from other areas, e.g., singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs and exercises are provided. The final chapter summarizes recent research directions, such as connections to orbifolds and the representation theory of finite groups via the McKay correspondence, stability conditions on triangulated categories, and the notion of the derived category of sheaves twisted by a gerbe.
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21

Brown, Justin, and V. Lakshmibai. Grassmannian Variety: Geometric and Representation-Theoretic Aspects. Springer, 2015.

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22

Brown, Justin, and V. Lakshmibai. Grassmannian Variety: Geometric and Representation-Theoretic Aspects. Springer New York, 2015.

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23

Brown, Justin, and V. Lakshmibai. The Grassmannian Variety: Geometric and Representation-Theoretic Aspects. Springer, 2016.

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24

Zuo, Kang. Representations of Fundamental Groups of Algebraic Varieties. Springer London, Limited, 2006.

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25

Zuo, Kang. Representations of Fundamental Groups of Algebraic Varieties. Springer, 2000.

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26

Fenning, Daniel, Adams John, and John Adams Library (Boston Public Librar. Young Algebraist's Companion : Or, a New and Easy Guide to Algebra; Introduced by the Doctrine of Vulgar Fractions: Designed for the Use of Schools ... Illustrated with Variety of Numerical and Literal Examples, and Attempted in Natural and Familar Di. Creative Media Partners, LLC, 2018.

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27

de Cataldo, Mark Andrea, Luca Migliorini Lectures 1–3, and Luca Migliorini. The Hodge Theory of Maps. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0005.

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This chapter summarizes the classical results of Hodge theory concerning algebraic maps. Hodge theory gives nontrivial restrictions on the topology of a nonsingular projective variety, or, more generally, of a compact Kähler manifold: the odd Betti numbers are even, the hard Lefschetz theorem, the formality theorem, stating that the real homotopy type of such a variety is, if simply connected, determined by the cohomology ring. Similarly, Hodge theory gives nontrivial topological constraints on algebraic maps. This chapter focuses on the latter, as it considers how the existence of an algebraic map f : X → Y of complex algebraic varieties is reflected in the topological invariants of X.
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28

Abbes, Ahmed, Michel Gros, and Takeshi Tsuji. The p-adic Simpson Correspondence (AM-193). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691170282.001.0001.

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The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.
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29

André, Yves, Francesco Baldassarri, and Maurizio Cailotto. De Rham Cohomology of Differential Modules on Algebraic Varieties. Springer International Publishing AG, 2021.

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30

André, Yves, and Francesco Baldassarri. De Rham Cohomology of Differential Modules on Algebraic Varieties. Birkhauser Verlag, 2012.

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31

André, Yves, and Francesco Baldassarri. De Rham Cohomology of Differential Modules on Algebraic Varieties (Progress in Mathematics). Birkhäuser Basel, 2001.

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32

André, Yves, Francesco Baldassarri, and Maurizio Cailotto. De Rham Cohomology of Differential Modules on Algebraic Varieties. Birkhäuser, 2020.

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33

Griffiths, Phillip. On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (AM-157). Princeton University Press, 2004.

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34

Griffiths, Phillip, and Mark Green. On the Tangent Space to the Space of Algebraic Cycles on a Smooth Algebraic Variety. (Am-157). Princeton University Press, 2004.

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35

Sherwood, Dennis, and Paul Dalby. Thermodynamics and mathematics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198782957.003.0004.

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Most school mathematics is about how one variable, y, varies with respect to one other variable, x, according to an equation such as y = 3x2. Equations like this underpin the student’s knowledge of algebra, and differential and integral calculus. Thermodynamics, however, is necessarily about how a variable, such as the pressure P, varies with respect not to one but to three variables simultaneously – for example, the mole number n, the volume V, and the temperature T. This makes the algebra of thermodynamics more complex, and also implies that mutual changes between pairs of variables is described not in terms of total derivatives of the form dy/dx, but rather by partial derivatives of the form (∂P/∂T)V. Many students find the leap from dy/dx to (∂P/∂T)V very difficult - the purpose of this chapter is therefore to build the reader’s confidence in understanding, and manipulating, functions of two and three variables.
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36

Zein, Fouad El, and Loring W. Tu. From Sheaf Cohomology to the Algebraic de Rham Theorem. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0002.

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This chapter proves Grothendieck's algebraic de Rham theorem. It first proves Grothendieck's algebraic de Rham theorem more or less from scratch for a smooth complex projective variety X, namely, that there is an isomorphism H*(Xₐₙ,ℂ) ≃ H*X,Ω‎subscript alg superscript bullet) between the complex singular cohomology of Xan and the hypercohomology of the complex Ω‎subscript alg superscript bullet of sheaves of algebraic differential forms on X. The proof necessitates a discussion of sheaf cohomology, coherent sheaves, and hypercohomology. The chapter then develops more machinery, mainly the Čech cohomology of a sheaf and the Čech cohomology of a complex of sheaves, as tools for computing hypercohomology. The chapter thus proves that the general case of Grothendieck's theorem is equivalent to the affine case.
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37

Bouamra, Faiza, ed. Abstracts of 1st International Conference on Computational & Applied Physics. AIJR Publisher, 2022. http://dx.doi.org/10.21467/abstracts.122.

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This book contains the abstracts of the papers presented at the International Conference on Computational & Applied Physics (ICCAP’2021) Organized by the Surfaces, Interfaces and Thin Films Laboratory (LASICOM), Department of Physics, Faculty of Science, University Saad Dahleb Blida 1, Algeria, held on 26–28 September 2021. The Conference had a variety of Plenary Lectures, Oral sessions, and E-Poster Presentations.
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38

Geometrie D'Arakelov Des Varietes Toriques Et Fibres En Droites Integrables. Societe Mathematique De France, 2000.

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39

Hrushovski, Ehud, and François Loeser. Non-Archimedean Tame Topology and Stably Dominated Types (AM-192). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161686.001.0001.

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Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.
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40

Abban, Hamid, Gavin Brown, Alexander Kasprzyk, and Shigefumi Mori, eds. Recent Developments in Algebraic Geometry. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781009180849.

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Written in celebration of Miles Reid's 70th birthday, this illuminating volume contains 11 papers by leading mathematicians in and around algebraic geometry, broadly related to the themes and interests of Reid's varied career. Just as in Reid's own scientific output, some of the papers give comprehensive accounts of the state of the art of foundational matters, while others give expositions of subject areas or techniques in concrete terms. Reid has been one of the major expositors of algebraic geometry and a great influence on many in this field – this book hopes to inspire a new generation of graduate students and researchers in his tradition.
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41

Cataldo, Mark Andrea de, Luca Migliorini Lectures 4–5, and Mark Andrea de Cataldo. The Hodge Theory of Maps. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0006.

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This chapter showcases two further lectures on the Hodge theory of maps, and they are mostly composed of exercises. The first lecture details a minimalist approach to sheaf cohomology, and then turns to the intersection cohomology complex, which is limited to the definition and calculation of the intersection complex Isubscript X of a variety of dimension d with one isolated singularity. Finally, this lecture discusses the Verdier duality. The second lecture sets out the Decomposition theorem, which is the deepest known fact concerning the homology of algebraic varieties. It then considers the relative hard Lefschetz and the hard Lefschetz for intersection cohomology groups.
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42

Voisin, Claire. Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691160504.001.0001.

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This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.
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43

Sobczyk, Garret, and D. Hestenes. Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer, 2012.

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44

Huybrechts, D. Where to Go from Here. Oxford University Press, 2007. http://dx.doi.org/10.1093/acprof:oso/9780199296866.003.0013.

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This chapter gives pointers for more advanced topics, which require prerequisites that are beyond standard introductions to algebraic geometry. The Mckay correspondence relates the equivariant-derived category of a variety endowed with the action of a finite group and the derived category of a crepant resolution of the quotient. This chapter gives the results from Bridgeland, King, and Reid for a special crepant resolution provided by Hilbert schemes and of Bezrukavnikov and Kaledin for symplectic vector spaces. A brief discussion of Kontsevich's homological mirror symmetry is included, as well as a discussion of stability conditions on triangulated categories. Twisted sheaves and their derived categories can be dealt with in a similar way, and some of the results in particular for K3 surfaces are presented.
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45

Hrushovski, Ehud, and François Loeser. Γ‎-internal spaces. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161686.003.0006.

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This chapter describes the topological structure of Γ‎-internal spaces. Let V be an algebraic variety over a valued field. An iso-definable subset X of unit vector V is said to be Γ‎-internal if it is in pro-definable bijection with a definable set which is Γ‎-internal. A number of delicate issues arise here. A pro-definable subset X of unit vector V is Γ‎-parameterized if there exists a definable subset Y of Γ‎ⁿ, for some n, and a pro-definable map g : Y → unit vector V with image X. The chapter presents an example showing that there exists Γ‎-parameterized subsets of unit vector V which are not iso-definable, whence not Γ‎-internal. It also presents the main results about the topological structure of Γ‎-internal spaces.
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46

Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics). Springer, 1987.

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47

Brosnan, Patrick, and Fouad El Zein. Variations of Mixed Hodge Structure. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0008.

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This chapter discusses the definition of admissible variations of mixed Hodge structure (VMHS), the results of M. Kashiwara in A study of variation of mixed Hodge structure (1986), and applications to the proof of algebraicity of the locus of certain Hodge cycles. It begins by recalling the relations between local systems and linear differential equations as well as the Thom–Whitney results on the topological properties of morphisms of algebraic varieties. The definition of a VMHS on a smooth variety is given, and the singularities of local systems are discussed. The chapter then studies the properties of degenerating geometric VMHS. Next it gives the definition and properties of admissible VMHS and reviews important local results of Kashiwara. Finally, the chapter recalls the definition of normal functions and explains recent results on the algebraicity of the zero set of normal functions.
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48

Geometrie differentielle: Geometrie differentielle, varietes complexes, feuilletages riemanniens : Colloque Geometrie et physique de 1986 en l'honneur d'Andre Lichnerowicz (Travaux en cours). Hermann, 1988.

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49

Hrushovski, Ehud, and François Loeser. Introduction. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161686.003.0001.

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This book deals with non-archimedean tame topology and stably dominated types. It considers o-minimality as an analogy and reduces questions over valued fields to the o-minimal setting. A fundamental tool, imported from stability theory, is the notion of a definable type, which plays a number of roles, starting from the definition of a point of the fundamental spaces. One of the roles of definable types is to be a substitute for the classical notion of a sequence, especially in situations where one is willing to refine to a subsequence. To each algebraic variety V over a valued field K, the book associates in a canonical way a projective limit unit vector V of spaces, which is the stable completion of V. In case the value group is ℝ, the results presented in this book relate to similar tameness theorems for Berkovich spaces.
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50

Medvetz, Thomas, and Jeffrey J. Sallaz, eds. The Oxford Handbook of Pierre Bourdieu. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780199357192.001.0001.

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Pierre Bourdieu was arguably the most important social theorist of the twentieth century. A French sociologist, he produced during his lifetime scores of empirical studies that laid the foundation for a rich theoretical program. These included studies of French colonialism in Algeria, the education system in France, new forms of state power, and the rise of autonomous artistic and scientific fields. Bourdieu’s research program was grounded in concepts such as habitus, field, forms of capital, and symbolic domination. Although most of these concepts have long historical legacies, Bourdieu elaborated conjoined them in an entirely originzal way, This Handbook assesses Pierred Bourdieu’s legacy from the standpoint of the early twenty-first century. It brings together a diverse array of contributors who consider how Bourdieu has advanced research and thinking in a variety of fields and areas. In particular, it considers how Bourdieu’s work has been appropriated for study in various regions of the world; how scholars have used Bourdieu to understand emergent transnational phenomena; how Bourdieu’s ideas have reshaped various disciplines and subfields; the ways in which Bourdieu’s concepts are embedded in long-standing theoretical traditions and debates; and the many ways in which Bourdieu’s research has generated entirely new fields and objects of study.
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