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| Semester | fall semester 2023 |
| Course frequency | Irregular |
| Lecturers | Luigi De Rosa (luigi.derosa@unibas.ch, Assessor) |
| Content | Differentiation of measures, the Lebesgue-Radon-Nicodym theorem, the Riesz representation theorem for linear functionals on the space of continuous functions, weak compactness for measures, Hausdorff measure and fractal dimensions, main function spaces, theory of distributions, smooth approximations, Onsager's conjecture on incompressible Euler equations, Constantin-E-Titi proof of energy conservation, Duchon-Robert approach and anomalous dissipation, lower dimensional accumulation of energy. |
| Learning objectives | The aim of the course is to provide a detailed account on more advanced topics of measure theory and functional analysis. Such tools are quite general and they find thousands of applications in the vast world of Partial Differential Equations. In order not to keep this course only abstract, we are going to study basic properties of rough incompressible fluids which naturally arise in the theory of Turbulence. |
| Bibliography | [1] Lecture notes of the course [2] Evans, L.C., & Gariepy, R.F. (2015). Measure Theory and Fine Properties of Functions, Revised Edition (1st ed.) [3] Friedlander, G. and Joshi, M. (1998) Introduction to the Theory of Distributions. 2nd Edition, Cambridge University Press, UK. |
| Admission requirements | Having successfully passed the Analysis 1 and Analysis 2 courses. Familiarity with basic tools from measure theory and functional analysis. |
| Language of instruction | English |
| Use of digital media | No specific media used |
| Course auditors welcome |
| Interval | Weekday | Time | Room |
|---|---|---|---|
| wöchentlich | Wednesday | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| wöchentlich | Thursday | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Date | Time | Room |
|---|---|---|
| Wednesday 20.09.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 21.09.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 27.09.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 28.09.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 04.10.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 05.10.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 11.10.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 12.10.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 18.10.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 19.10.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 25.10.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 26.10.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 01.11.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 02.11.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 08.11.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 09.11.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 15.11.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 16.11.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 22.11.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 23.11.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 29.11.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 30.11.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 06.12.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 07.12.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 13.12.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 14.12.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Wednesday 20.12.2023 | 10.15-12.00 | Kollegienhaus, Seminarraum 104 |
| Thursday 21.12.2023 | 08.15-10.00 | Spiegelgasse 5, Seminarraum 05.002 |
| Modules |
Module: Analysis and Geometry (Bachelor's Studies: Mathematics) |
| Assessment format | continuous assessment |
| Assessment details | The final evaluation consist in an oral exam of 30 minutes, which gives the final grade. To access the final oral exam, and therefore to pass the course, it is necessary to pass 2/3 of the weekly exercise sheets. An exercise sheet counts as "passed" if there is a substantial amount of correct solutions. |
| Assessment registration/deregistration | Reg.: course registration, dereg: cancel course registration |
| Repeat examination | no repeat examination |
| Scale | 1-6 0,5 |
| Repeated registration | as often as necessary |
| Responsible faculty | Faculty of Science, studiendekanat-philnat@unibas.ch |
| Offered by | Fachbereich Mathematik |