| CODE | MAT5210 | ||||||||
| TITLE | Combinatorial Set Theory | ||||||||
| UM LEVEL | 05 - Postgraduate Modular Diploma or Degree Course | ||||||||
| MQF LEVEL | 7 | ||||||||
| ECTS CREDITS | 15 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | Combinatorial Set Theory, also known as Infinite Combinatorics studies the size of infinite collections of sets satisfying certain conditions. It generalizes a number of existing finite combinatorial principles such as Ramsey’s Theorem. The principles developed in Infinitary Combinatorics have interesting applications in Analysis, Topology and Measure Theory. For example, the famous Suslin’s problem asks whether any dense, complete linear order, without end points and satisfying the countable chain condition is isomorphic to the real line. This topological problem was solved by first reducing it to a problem in Infinitary Combinatorics. This study-unit will cover: 1) Basic Concepts: Boolean Algebras, Partial Orders, Trees, Ultrafilters and Ideals; 2) Cardinal Characteristics of the Continuum: (a) Basic Definitions and inequalities. (b) applications. (c) Cardinal Invariants for Ideals; 3) Martin's Axiom: (a) Equivalents of MA. (b) Consequences of MA in Measure and Category. (c) Suslin Hypothesis and MA; 4) Infinite Ramsey Theory - Applications; 5) Closed and Unbounded sets, Stationary Sets. Diamond Principle. Study-Unit Aims: - Present a survey of the crucial results in Combinatorial SetTheory; - Acquaint students with common infinite mathematical structures; - Provide students basic tools in Combinatorial Set Theory that can be applied to research in areas such as Analysis, Topology, Measure Theory or Set Theory itself. Learning Outcomes: 1. Knowledge & Understanding By the end of the study-unit the student will be able to: - Develop their ability to prove results involving infinite objects; - Analyse how certain combinatorial principles for finite sets extend to the infinite realm; - Gain familiarity with techniques and approaches used in Infinitary Combinatorics. 2. Skills By the end of the study-unit the student will be able to: - Compare sizes of different infinite collections of sets; - Apply combinatorial principles in the construction of examples and counterexamples in Analysis and Topology; - Analyse the use of cardinal invariants in the investigation of structures in Mathematics; - Determine where Infinitary Combinatorics may be applied; - Appreciate the interplay between Infinitary Combinatorics and results of independence in Mathematics. Main Text/s and any supplementary readings: Main Texts: - Kunen K. (1983) Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics. Elsevier. - Just W. & Weese M. (1997) Discovering Modern Set Theory II : Set Theoretic tools for every Mathematician. American Mathematical Soc. - Jech T. (2002) Set Theory: The Third Millennium Edition. Springer Monographs in Mathematics. - Ciesielski K. (1997) Set Theory for the Working Mathematician. Cambridge University Press. |
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| ADDITIONAL NOTES | Pre-requisite Qualifications: B.Sc with Mathematics as a main area Follows from: MAT3000 |
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| STUDY-UNIT TYPE | Lecture | ||||||||
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The University makes every effort to ensure that the published Courses Plans, Programmes of Study and Study-Unit information are complete and up-to-date at the time of publication. The University reserves the right to make changes in case errors are detected after publication.
The availability of optional units may be subject to timetabling constraints. Units not attracting a sufficient number of registrations may be withdrawn without notice. It should be noted that all the information in the description above applies to study-units available during the academic year 2024/5. It may be subject to change in subsequent years. |
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