Academic literature on the topic 'Ultrafilters (Mathematics) Combinatorial set theory'

The preceding chapters in this volume have documented the substantial recent progress towards understanding the complexity of randomly specified combinatorial problems. This improved understanding has been obtained by combining concepts and ideas from theoretical computer science and discrete mathematics with those developed in statistical mechanics. Techniques such as the cavity method and the replica method, primarily developed by the statistical mechanics community to understand physical phenomena, have yielded important insights into the intrinsic difficulty of solving combinatorial problems when instances are chosen randomly. These insights have ultimately led to the development of efficient algorithms for some of the problems. A potential weakness of these results is their reliance on random instances. Although the typical probability distributions used on the set of instances make the mathematical results tractable, such instances do not, in general, capture the realistic instances that arise in practice. This is because practical applications of graph theory and combinatorial optimization in CAD systems, mechanical engineering, VLSI design, transportation networks, and software engineering involve processing large but regular objects constructed in a systematic manner from smaller and more manageable components. Consequently, the resulting graphs or logical formulas have a regular structure, and are defined systematically in terms of smaller graphs or formulas. It is not unusual for computer scientists and physicists interested in worst-case complexity to study problem instances with regular structure, such as lattice-like or tree-like instances. Motivated by this, we discuss periodic specifications as a method for specifying regular instances. Extensions of the basic formalism that give rise to locally random but globally structured instances are also discussed. These instances provide one method of producing random instances that might capture the structured aspect of practical instances. The specifications also yield methods for constructing hard instances of satisfiability and various graph theoretic problems, important for testing the computational efficiency of algorithms that solve such problems. Periodic specifications are a mechanism for succinctly specifying combinatorial objects with highly regular repetitive substructure. In the past, researchers have also used the term dynamic to refer to such objects specified using periodic specifications (see, for example, Orlin [419], Cohen and Megiddo [103], Kosaraju and Sullivan [347], and Hoppe and Tardos [260]).

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.