Department
of Mathematics and Statistics 		University
of Konstanz 		  Logo der Universität Konstanz
Research Group Real Algebraic Geometry > Prof. Dr. S. Kuhlmann > Mitarbeiter > Dr. Maria Infusino

Maria Infusino - Teaching

Summer semester 2018

Topological Algebras
with Patrick Michalski.

Lecture (2 hours per week): Wednesday 13:30-15:00, Room D406
Tutorial (2 hours every two weeks): Friday 10:00 – 11:30, Room F420

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. locally bounded algebras, Frechet algebras and projective limit algebras. Topological algebras also play an important role in some problems appearing in real algebraic geometry, that will be outlined in this course and could be a starting point for a bachelor/master thesis within the Schwerpunkt Reelle Geometrie und Algebra.

Prerequisites
The presentation will try to be as much self-contained and systematic as possible, so the only prerequisite is a basic knowledge of algebra and analysis. A prior exposure to functional analysis and some familiarity with the general theory of topological vector spaces (tvs) would be helpful, but not required. Indeed, notions and results from tvs theory will be recalled where needed or precise references to them will be provided.

Target group
BA, MA, LA, Diplom (from 4.semester).

Validation
  • Ergänzungsmodul BA Mathematik
  • Wahlmodul MA Mathematik
  • Wahlmodul MA LA
  • Wahlmodul/Spezielles Gebiet GymPO 2009

    • Exam
    The final exam will be oral and to be scheduled individually. To enroll for the exam please contact Frau Gisela Cassola (Room F439).

    Language
    English

    Tutorial and Weekly Office hours
    An exercise sheet will be distributed every two weeks 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 tutorial is offered every two weeks (Friday 10.00 – 11.30, Room F420) to all the participants in order to discuss their solutions to the exercise sheets. During the weeks when there is no tutorial, a set of recap questions will be distributed to help the participants in self-assessing their learning process in preparation for the oral exam. Moreover, the lecturer is available every week (Thursday 14:00-15:00, Room F408) for individual meetings to discuss any problem, comment and/or questions related to the course.

    Tutorial Calendar
    27.04 Recap about topological preliminaries
    04.05 Discussion of solutions to Sheet 1
    18.05 Discussion of solutions to Sheet 2
    08.06 Discussion of solutions to Sheet 3
    15.06 Discussion of solutions to Sheet 4
    29.06 Discussion of solutions to Sheet 5
    13.07 Discussion of solutions to Sheet 6


    Lecture Notes
    Lecture 1: Introduction to the course and basics about topological algebras (last update on 23.04.18)
    Lecture 2: More examples and further properties of TAs: Hausdorfness, unitizations, subalgebras. (last update on 27.04.18)
    Lecture 3: Weak and strong operator topology. Subalgebras and quotients of a TA. (last update on 02.05.18)
    Lecture 4: Complements on quotients of TAs and an introduction to lmc algebras. (last update on 09.05.18)
    Lecture 5: Seminorm characterization of lmc algebras and examples. (last update on 08.06.18)
    Lecture 6: More on seminorm characterization of lmc algebras: Hausdorff lmc algebras and finest lmc topology. (last update on 08.06.18)
    Lecture 7: Remarks on the finest lmc topology. Topological algebras admitting lmc topologies. (last update on 21.06.18)
    Lecture 8: Metrizable and Frechet algebras. (last update on 21.06.18)
    Lecture 9: Examples of Frechet algebras and introduction to locally bounded algebras. (last update on 21.06.18)
    Lecture 10: More on locally bounded algebras and introduction to projective topologies. (last update on 04.08.18)
    Lecture 11: Projective systems and projective limits of TAs. (last update on 04.08.18)
    Lecture 12: Arens-Michael decomposition. (last update on 04.08.18)
    Lecture 13: Symmetric tensor algebras. (last update on 04.08.18)
    Lecture 14: More on symmetric tensor algebras and short overview on the moment problem (last update on 04.08.18)

    Bibliography (last update on 04.08.18)


    Lecture Notes (unique pdf file) (last update on 04.08.18)


    Problem Sheets
    Exercise Sheet 1 (to be handed in by 02.05.18)
    Exercise Sheet 2 (to be handed in by 16.05.18)
    Exercise Sheet 3 (to be handed in by 30.05.18)
    Exercise Sheet 4 (to be handed in by 13.06.18)
    Exercise Sheet 5 (to be handed in by 27.06.18)
    Exercise Sheet 6 (to be handed in by 11.07.18)


    Recap Sheets
    Recap Sheet 1
    Recap Sheet 2
    Recap Sheet 3
    Recap Sheet 4
    Recap Sheet 5
    Recap Sheet 6


    Announcements




    Last update: 04.08.2018