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Combinatorial Convexity and Algebraic Geometry
The book is an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry, and to the connections between these fields, known as the theory of toric varieties. The first part of the book covers the theory of polytopes and provides large parts of the mathematical background of linear optimization and of the geometrical aspects in computer science. The second part introduces toric varieties in an elementary way.
Contributions to Functional Analysis
The volume in hand contains a selection from the numerous contributions dedicated to Professor Dr. Gottfried Köthe on the occasion of his 60th birthday. This selection only takes into consideration the papers on Functional Analysis as far as they have reached us in time to be included in the volume. All of these papers have been published in [the journal] "Mathematische Annalen", volume 162.
Lectures on Polytopes
Based on a graduate course at the Technische Universität, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The straightforward exposition features many illustrations, and complete proofs for most theorems. With only linear algebra as a prerequisite, it takes the reader quickly from the basics to topics of recent research. The lectures introduce basic facts about polytopes, with an emphasis on methods that yield the results, discuss important examples and elegant constructions, and show the excitement of current work in the field. They will provide interesting and enjoyable reading for researchers as well as students.
Pillars of a Mature Worldview
Our worldview determines our perception of reality and thus shapes our attitude to life and the way we live. If our worldview adequately reflects reality, we can lead an authentic and meaningful life that we will later look back on with satisfaction. A mature worldview is much more than the accumulation of facts and experiences. Whereas in the past there was primarily a lack of scientific insight, today it is above all the immense flood of information and opinions that obscures our view of essential knowledge. It is therefore not surprising that people's views sometimes differ considerably with regard to fundamental questions of life. The thesis of this book is that in this age it is possibl...
Valuation Theory and Its Applications
This book is the second of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). It contains the most recent applications of valuation theory to a broad range of mathematical ideas. Valuation theory arose in the early part of the twentieth century in connection with number theory and continues to have many important applications to algebra, geometry, and analysis. The research and survey papers in this volume cover a variety of topics, including Galois theory, the Grunwald-Wang Theorem, algebraic geometry, resolution of singularities, curves over Prufer domains, model theory of valued fields and the Frobenius, Hardy fields, Hensel's Lemma, fixed point theorems, and computations in valued fields. It is suitable for graduate students and research mathematicians interested in algebra, algebraic geometry, number theory, and mathematical logic.
Toward a Theology of Nature
Pannenberg poses theological questions to natural scientists that illuminate his personal position on issues dealing with theology and the natural sciences, especially physics, reviewing the relationship between natural law and contingency, the importance of the spirit in the phenomenon of life, field theory, language, and the theological account for the nature of God and God's creative activity.
Hamiltonian Submanifolds of Regular Polytopes
This work is set in the field of combinatorial topology, sometimes also referred to as discrete geometric topology, a field of research in the intersection of topology, geometry, polytope theory and combinatorics. The main objects of interest in the field are simplicial complexes that carry some additional structure, forming combinatorial triangulations of the underlying PL manifolds. In particular, polyhedral manifolds as subcomplexes of the boundary complex of a convex regular polytope are investigated. Such a subcomplex is called k-Hamiltonian if it contains the full k-skeleton of the polytope. The notion of tightness of a PL-embedding of a triangulated manifold is closely related to its ...
Essentials of Tropical Combinatorics
The goal of this book is to explain, at the graduate student level, connections between tropical geometry and optimization. Building bridges between these two subject areas is fruitful in two ways. Through tropical geometry optimization algorithms become applicable to questions in algebraic geometry. Conversely, looking at topics in optimization through the tropical geometry lens adds an additional layer of structure. The author covers contemporary research topics that are relevant for applications such as phylogenetics, neural networks, combinatorial auctions, game theory, and computational complexity. This self-contained book grew out of several courses given at Technische Universität Berlin and elsewhere, and the main prerequisite for the reader is a basic knowledge in polytope theory. It contains a good number of exercises, many examples, beautiful figures, as well as explicit tools for computations using $texttt{polymake}$.
