| CODE | MAT3226 | ||||||||
| TITLE | Geometry of Euclidean Spaces | ||||||||
| UM LEVEL | 03 - Years 2, 3, 4 in Modular Undergraduate Course | ||||||||
| MQF LEVEL | 6 | ||||||||
| ECTS CREDITS | 5 | ||||||||
| DEPARTMENT | Mathematics | ||||||||
| DESCRIPTION | (I) Elementary Combinatorial Techniques: - Affine Notions (linear, affine and convex combinations, linear and affine independence); - Simplexes (n-dimensional simplex, affine coordinates, vertices and faces, geometric simplex, boundary and barycenter); - Triangulation (simplicial complex, geometric realization, triangulation, barycentric triangulation); - Simplexes in Euclidean Spaces (faces of n-dimensional simplexes, Sperner's lemma). (II) Fundamental Properties of Euclidean Spaces: - Knaster–Kuratowski–Mazurkiewicz Theorem; - Brouwer Fixed-Point Theorem; - Borsuk Non-Retraction Theorem; - Non-Homogeneity Theorem; - Brouwer Dimension Invariance Theorem. Study-unit Aims: The aim of this study-unit is to give a connected and simple account of the most essential topological properties of Euclidean spaces. Students will be exposed to several classical results, for instance Sperner's lemma and Brouwer Fixed-Point theorem, which have many applications also outside topology. Another notable result the students will see is the Brouwer Dimension Invariance Theorem that the Euclidean n-space and the Euclidean m-space are not topologically equivalent unless n equals m. This study-unit will give also a brief account of a topological invariant of Euclidean spaces - the so called Lebesgue's dimension. Learning Outcomes: 1. Knowledge & Understanding: By the end of the study-unit the student will be able to: - Use basic combinatorial techniques and work with such fundamental concepts as simplex, simplicial complex, triangulation and geometric realization; - Know more about topological properties of Euclidean spaces, and their special subsets (such as spheres, balls, cubes, simplexes); - See applications of such fundamental topological concepts as compactness and connectedness. 2. Skills: By the end of the study-unit the student will: - Have a rough idea of the proofs of several classical results (such as Sperner's lemma, Brouwer Fixed-Point theorem, Non-Retraction theorem, Non-Homogeneity theorem, Knaster–Kuratowski–Mazurkiewicz theorem) and the methods used in these proofs. - Be able to improve the handling of several topological techniques by seeing how abstract concepts are deployed in particular situations. - Have a foundation to follow related more advanced study-units. Main Text/s and any supplementary readings: Lecture Notes covering all topics. Textbooks: - Jan van Mill, Infinite-Dimensional Topology, Prerequisites and Introduction, Volume 43 (North-Holland Mathematical Library), Elsevier Science Publisher B.V., 1989. - Jan van Mill, The Infinite-Dimensional Topology of Function Spaces, Volume 64 (North-Holland Mathematical Library), Elsevier Science Publisher B.V., 2001. |
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| ADDITIONAL NOTES | Follows from: MAT3215, MAT3225 | ||||||||
| STUDY-UNIT TYPE | Lecture and Independent Study | ||||||||
| METHOD OF ASSESSMENT |
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| LECTURER/S | Valentin Gutev |
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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 2023/4. It may be subject to change in subsequent years. |
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