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This thesis consists of two parts. The first part is about the semigroup of values SV and the Poincaré series PV(t) associated to a finite set of divisorial valuations coming from a modification of Kd, where K is any field. When K is infinite, we can prove that SV is finitely generated whenever there exists some finite generating sequence L for V . The existence of such a finite L also implies that PV(t) is a rational function whose denominator can be expressed in terms of the valuation vectors of the elements of L. Here K can even be a finite field. However, a finite generating sequence does not always exist. This is the case for the modification of C3 where we blow up in nine very general points on the first exceptional divisor. In that specific example, the semigroup of values is not finitely generated.The second part is about lattice polytopes. It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of integrally closed polytopes of small dimension and for lattice parallelepipeds. This is the first non-trivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulation.

academic collection --- 512.7 <043.3> --- 519.1 <043> --- Algebraic geometry. Commutative rings and algebras--dissertaties --- Combinatorics. Graph theory--Dissertaties --- Theses --- 519.1 <043> Combinatorics. Graph theory--Dissertaties

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Originally published in 1985, this classic textbook is an English translation of Einführung in die kommutative Algebra und algebraische Geometrie. As part of the Modern Birkhäuser Classics series, the publisher is proud to make Introduction to Commutative Algebra and Algebraic Geometry available to a wider audience.   Aimed at students who have taken a basic course in algebra, the goal of the text is to present important results concerning the representation of algebraic varieties as intersections of the least possible number of hypersurfaces and—a closely related problem—with the most economical generation of ideals in Noetherian rings. Along the way, one encounters many basic concepts of commutative algebra and algebraic geometry and proves many facts which can then serve as a basic stock for a deeper study of these subjects.

Ordered algebraic structures --- Algebraic geometry --- algebra --- landmeetkunde --- wiskunde --- geometrie --- Geometry, algebraic. --- Algebra. --- Algebraic Geometry. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Geometry --- Commutative algebra. --- Geometry, Algebraic. --- Algebraic geometry. --- Commutative rings. --- Algebra --- Rings (Algebra)

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Lectures on Finitely Generated Solvable Groups are based on the “Topics in Group Theory" course focused on finitely generated solvable groups that was given by Gilbert G. Baumslag at the Graduate School and University Center of the City University of New York.  While knowledge about finitely generated nilpotent groups is extensive, much less is known about the more general class of solvable groups containing them.  The study of finitely generated solvable groups involves many different threads; therefore these notes contain discussions on HNN extensions; amalgamated and wreath products; and other concepts from combinatorial group theory as well as commutative algebra.  Along with Baumslag’s Embedding Theorem for Finitely Generated Metabelian Groups, two theorems of Bieri and Strebel are presented to provide a solid foundation for understanding the fascinating class of finitely generated solvable groups.  Examples are also supplied, which help illuminate many of the key concepts contained in the notes.  Requiring only a modest initial group theory background from graduate and post-graduate students, these notes provide a field guide to the class of finitely generated solvable groups from a combinatorial group theory perspective.

Functions of several complex variables. --- Mathematics. --- Solvable groups. --- Combinatorial analysis --- Mathematics --- Algebra --- Group theory --- Physical Sciences & Mathematics --- Combinatorial group theory. --- Soluble groups --- Combinatorial groups --- Groups, Combinatorial --- Algebra. --- Commutative algebra. --- Commutative rings. --- Group theory. --- Combinatorics. --- Group Theory and Generalizations. --- General Algebraic Systems. --- Commutative Rings and Algebras. --- Mathematical analysis --- Combinatorics --- Groups, Theory of --- Substitutions (Mathematics) --- Rings (Algebra)

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“Rational homotopy theory is today one of the major trends in algebraic topology. Despite the great progress made in only a few years, a textbook properly devoted to this subject still was lacking until now… The appearance of the text in book form is highly welcome, since it will satisfy the need of many interested people. Moreover, it contains an approach and point of view that do not appear explicitly in the current literature.” —Zentralblatt MATH (Review of First Edition)   “The monograph is intended as an introduction to the theory of minimal models. Anyone who wishes to learn about the theory will find this book a very helpful and enlightening one. There are plenty of examples, illustrations, diagrams and exercises. The material is developed with patience and clarity. Efforts are made to avoid generalities and technicalities that may distract the reader or obscure the main theme. The theory and its power are elegantly presented. This is an excellent monograph.” —Bulletin of the American Mathematical Society (Review of First Edition)   This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then followed by an in-depth look at differential forms and de Tham’s theorem on simplical complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented.   New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory   With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.

Differential forms. --- Homology. --- Homotopy theory. --- Homotopy theory --- Differential forms --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Deformations, Continuous --- Forms, Differential --- Mathematics. --- Algebra. --- Category theory (Mathematics). --- Homological algebra. --- Commutative algebra. --- Commutative rings. --- Topology. --- Algebraic topology. --- Algebraic Topology. --- Category Theory, Homological Algebra. --- Commutative Rings and Algebras. --- Topology --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Rings (Algebra) --- Algebra --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory --- Mathematical analysis --- Math --- Science --- Continuous groups --- Geometry, Differential --- Algebraic topology