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Research Group Real Algebraic Geometry > Prof. Dr. S. Kuhlmann > Mitarbeiter > Dr. Maria Infusino

Maria Infusino - Teaching

Summer Semester 2020

Topological Algebras
 with Patrick Michalski.

Lecture (2 hours per week): Wednesdays 13:30-15:00
by Dr Maria Infusino.
Tutorial (2 hours every two weeks): Thursdays 13:30-15:00
by Patrick Michalski.
Fragestunde (2 hours every week): Thursdays 11:45-13:15
by Dr Maria Infusino.


Important
Because of the current restrictions due to the Coronavirus, this course will be provided in an online form. All the interested students are kindly asked to please register for this course at ILIAS, where all the organizational details and material is provided.

Contents
The aim of this course is to give an overview on the theory of topological algebras and of the standard tools used in tackling problems involving them. Normed and Banach algebras will be introduced but we will mainly investigate how far one can go beyond this classical framework while still retaining substantial results. Particular attention will be given to locally multiplicatively convex algebras and tensor algebras, but also other special classes of topological algebras will be closely studied e.g. Frechet algebras, locally bounded algebras, projective and inductive limit algebras, topological division algebras, etc. We will also introduce the spectrum of a given topological algebra and study its relation to maximal ideals. Topological algebras play an important role also in some problems appearing in real algebraic geometry, that will be outlined in this course and could be a starting point for a master thesis within the Schwerpunkt Reelle Geometrie und Algebra.

Lecture Notes including references will be provided in ILIAS.

Prerequisites
The course builds on the general theory of topological vector spaces and commutative algebras with connections to real algebraic geometry. However, the presentation will try to be as much self-contained and systematic as possible, giving precise references or briefly recalling notions and results needed from previous courses.

Target group
MA, LA (from 8.semester)

Validation
  • Spezialisierungsmodul im Schwerpunkt Reelle Geometrie und Algebra
  • Wahlmodul MA Mathematik
  • Wahlmodul MA LA
  • Wahlmodul/Spezielles Gebiet GymPO 2009

    • Credits 5 ECTS

    Exam
    Considering the current situation, the form of the final exam cannot be determined at the moment but will be clarified during the lecture period. In any case, in order to be admitted at the exam, students need to achieve at least 50% of the total number of points assigned in the Exercise Sheets as well as present a solution in the tutorial at least once.

    Language
    English

    Exercise and Recap Sheets
    An exercise sheet will be distributed every two weeks on Wednesday (starting from the second lecture week) to both assess the progress of the participants and allow them to explicitly work out more details of some results proposed in the lectures. A recap sheet will be distributed every two weeks on Wednesday (starting from the first week) to help the students in self-assessing their learning process. No solutions are required for the recap sheet. 

    Solutions to exercises and Tutorial
  • Students’ solutions to each exercise sheet will have to be handed in a week after its distribution (deadline will be clearly stated on each exercise sheet).
  • A tutorial is offered by the tutor every two weeks for the discussion of the students' solutions. In the tutorial, model solutions to each exercise sheet will be also provided.

    Fragestunden
  • Lecturer-Fragestunden (2 hours per week): The lecturer is available every week on Thursday 11:45-13:15 for individual meetings to discuss any question, problem and comment related to the course.
  • Tutor-Fragestunden (1 hour every two weeks, in the weeks where no tutorial is held): The tutor is available every two weeks on Thursday 13:30-15:00 for individual meetings to discuss any questions related to the exercises.

    • References
  • V.K. Balachandran, Topological algebras. Reprint of the 1999 original. North-Holland Mathematics Studies, 185. North-Holland Publishing Co., Amsterdam, 2000.
  • E. Beckenstein, L. Narici, S. Suffel, Topological algebras. North-Holland, Mathematics Studies, Vol. 24. Notas de Matemática, No. 60, 1977.
  • A. Mallios, Topological algebras. Selected topics. 109. North-Holland Publishing Co.1986.
  • W. Żelazko, Selected topics in topological algebras. Lecture Notes Series, No. 31. Matematisk Institut, Aarhus Universitet, Aarhus,1971.
    • Further references are provided in the Lecture Notes.





    Last update: 13.05.2020