2013/10/03 by Jakub Konieczny, Konieczny, Jakub
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Computability, Logic, AI Algorithms #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.1310.1056
Ultrafilters are very useful and versatile objects with applications\nthroughout mathematics: in topology, analysis, combinarotics, model theory, and\neven theory of social choice. Proofs based on ultrafilters tend to be shorter\nand more elegant than their classical counterparts. In this thesis, we survey\nsome of the most striking ways in which ultrafilters can be exploited in\ncombinatorics and ergodic theory, with a brief mention of model theory.\n In the initial sections, we establish the basics of the theory of\nultrafilters in the hope of keeping our exposition possibly self-contained, and\nthen proceed to specific applications. Important combinatorial results we\ndiscuss are the theorems of Hindman, van der Waerden and Hales-Jewett. Each of\nthem asserts essentially that in a finite partition of, respectively, the\nnatural numbers or words over a finite alphabet, one cell much of the\ncombinatorial structure. We next turn to results in ergodic theory, which rely\nstrongly on combinatorial preliminaries. They assert essentially that certain\nsets of return times are combinatorially rich. We finish by presenting the\nultrafilter proof of the famous Arrow's Impossibility Theorem and the\nconstruction of the ultraproduct in model theory.\n