Series: Publications of the Scuola Normale Superiore, Vol. 18
Subseries: CRM Series
2015, Approx. 160 p.
Softcover
Easily accessible surveys directed to a broad community in mathematics and computer science
This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include: graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.
Imre Barany: Tensors, colours, octahedral.- Maria Chudnovsky: Cliques and stable sets in undirected graphs.- Mauro Di Nasso: A taste of nonstandard methods in combinatorics of numbers.- Bela Bollobas, Zoltan Furedi, Ida Kantor, G. O. H. Katona and Imre Leader: A coding problem for pairs of subsets.- Jiri Matousek: String graphs and separators.- Jaroslav Nesetril and Patrice Ossona de Mendez: On first-order definable colorings.- Ryan Schwartz and Jozsef Solymosi: Combinatorial applications of the subspace theorem.- Peter Hegarty and Dmitry Zhelezov: Can connected commuting graphs of finite groups have arbitrarily large diameter?
Series: Springer Monographs in Mathematics
2015, X, 200 p. 10 illus.
Hardcover
ISBN 978-3-319-15113-7
Due: April 14, 2015
Contains open problems and new ideas for research
Develops a geometric theory in an extended context
Uses the lattice structure for all representations of boolean representations of simplicial complexes
This self-contained monograph explores a new theory centered around boolean representations of simplicial complexes leading to a new class of complexes featuring matroids as central to the theory. The book illustrates these new tools to study the classical theory of matroids as well as their important geometric connections. Moreover, many geometric and topological features of the theory of matroids find their counterparts in this extended context.
Graduate students and researchers working in the areas of combinatorics, geometry, topology, algebra and lattice theory will find this monograph appealing due to the wide range of new problems raised by the theory. Combinatorialists will find this extension of the theory of matroids useful as it opens new lines of research within and beyond matroids. The geometric features and geometric/topological applications will appeal to geometers. Topologists who desire to perform algebraic topology computations will appreciate the algorithmic potential of boolean representable complexes.
1. Introduction.- 2. Boolean and superboolean matrices.- 3. Posets and lattices.- 4. Simplicial complexes.- 5. Boolean representations.- 6. Paving simplicial complexes.- 7. Shellability and homotopy type .- 8. Operations on simplicial complexes.- 9. Open questions.
Series: Progress in Mathematics, Vol. 309
2015, Approx. 450 p.
Hardcover
ISBN 978-3-0348-0902-3
Due: August 6, 2015
Comprehensive book on aperiodic order describing the state of the art in the field for researchers and students
Evolving research area where many different mathematical theories meet
Yields a pool of interesting examples for various abstract mathematical theories
Enhanced by the 2011 Nobel Prize in chemistry, awarded to D. Shechtman for the discovery of quasicrystals, the mathematical study of periodically ordered tilings meets with renewed vivid interest
What is order which is not based on simple repetition, that is, periodicity? How must atoms be arranged in a material so that it diffracts like a quasicrystal? How can we describe aperiodically ordered systems mathematically? Originally triggered by the ? later nobel prize winning ? discovery of quasicrystals, the investigation of aperiodic order has by now become a well-established and strongly evolving field of mathematical research. It is closely tied to a surprising variety of branches of mathematics and physics. The book offers an overview over the state of the art in the field of aperiodic order. It comprises carefully selected surveys which are written by leading researchers. Since the book is written for a readership of non-experts which have a general background in mathematics, theoretical physics or computer science, it will serve as a highly accessible first hand source of information to anybody interested in this rich and exciting field. Topics covered in the book include the mathematical theory of diffraction, the dynamical systems of tilings or Delone sets, their cohomology and non-commutative geometry, the Pisot substitution conjecture, aperiodic Schrodinger operators, and connections to arithmetic number theory.
0. Introductory chapter.- 1. Mathematical diffraction theory.- 2. Pisot substitution conjecture.- 3. Topology of tiling spaces.- 4. Proximality in tiling spaces.- 5. Linear repetitive Delone sets of finite local complexity.- 6. Tilings with infinite local complexity.- 7. Stability of non-periodic solids.- 8. Aperiodic Schrodinger operators.- 9. Non commutative geometry of tilings.- 10. Arithmetic properties of sub shifts.