Maria Infusino - Teaching
Summer Semester 2019
Positive polynomials and Moment Problems with
Patrick Michalski.
Lecture (2 hours per week): Tuesdays 15:15-16:45, Room G306 by Dr Maria Infusino.
Tutorial (2 hours every two weeks): Wednesdays 11:45-13:15, Room M628 by Patrick Michalski.
Contents
This course is meant to be a natural follow-up of the course “Real Algebraic Geometry I" from the WS 2018/2019.
The main purpose is to deepen the study of the fundamental problem in real algebraic geometry
of characterizing the cone of nonnegative polynomials on a given subset K of
$\mathbb{R}^d$.
This is the dual facet of the so-called K-moment problem, which addresses the question of
deciding whether a given sequence of real numbers is the moment sequence of a nonnegative
Radon measure supported on a fixed subset K of
$\mathbb{R}^d$. We will explore this beautiful
duality
established by the Riesz-Haviland theorem, by studying the famous Positivstellensätze
(e.g. Schmüdgen, Putinar, Jacobi) and how they illuminated the theory of the moment
problem for basic closed semialgebraic sets of
$\mathbb{R}^d$.
Particular attention will be given to then non-compact case which is still open
in many of its aspects. Furthermore, we will introduce some infinite dimensional versions
of the moment problem, i.e. for measures supported on infinite dimensional spaces,
and present some recent results and questions connected to them.
Prerequisites
This course will build up on the contents of the courses “Real Algebraic Geometry I'' and “Topological Vector Spaces” from the WS 2018/2019.
Target group
BA, MA, LA (from 6.semester)
Validation
Spezialisierungsmodul im Schwerpunkt Reelle Geometrie und Algebra
Wahlmodul MA Mathematik
Wahlmodul MA LA
Wahlmodul/Spezielles Gebiet GymPO 2009
Exam
The final exam will be oral and to be scheduled individually.
Language
English
Exercise and Recap Sheets
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 to all the participants
for the discussion of their solutions to the exercise sheets. In order to be admitted at the exam, the participants 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.
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.
Tutorial Calendar
| 08.05.2019 | Discussion of solutions to Exercise Sheet 1 |
15.05.2019 | Discussion of solutions to Exercise Sheet 2 |
29.05.2019 | Discussion of solutions to Exercise Sheet 3 |
12.06.2019 | Discussion of solutions to Exercise Sheet 4 |
26.06.2019 | Discussion of solutions to Exercise Sheet 5 |
10.07.2019 | Discussion of solutions to Exercise Sheet 6 |
17.07.2019 | Discussion of solutions to Bonus Sheet |
Lecture Notes
Lecture 1: Introduction to the course and preliminaries on positive polynomials.
(last update on 25.04.19)
Lecture 2: Relation between Psd(K
S) and T
S.
(last update on 13.05.19)
Lecture 3: Representation Theorem and its applications: Positivstellensätze.
(last update on 14.06.19)
Lecture 4: Nichtnegativstellensätze and closure of even power modules.
(last update on 28.09.19)
Lecture 5: Closure of even power modules.
(last update on 20.05.19)
Lecture 6: Formulation of the K-moment problem and Riesz-Haviland Theorem.
(last update on 5.06.19)
Lecture 7: Proof of generalized Riesz-Haviland Theorem.
(last update on 16.06.19)
Lecture 8: Solving the classical KMP by characterizing Psd(K).
(last update on 14.06.19)
Lecture 9: Solving the general KMP by characterizing Psd(K).
(last update on 28.09.19)
Lecture 10: An operator theoretical approach to the KMP for K compact bcsas.
(last update on 24.06.19)
Lecture 11: An operator theoretical approach to the KMP for K=
$\mathbb{R}^n$(last update on 5.07.19)
Lecture 12: More on an operator theoretical approach to the KMP for K=
$\mathbb{R}^n$(last update on 9.07.19)
Lecture 13: Solving the KMP for K not necessarily compact bcsas.
(last update on 15.07.19)
Lecture 14: Determinacy of the KMP.
(last update on 18.07.19)
Lecture Notes (unique pdf file) (last update on 28.09.19)
Bibliography (last update on 18.07.19)
Exercise Sheets
Exercise Sheet 1 (due by 30.04.19)
Exercise Sheet 2 (due by 14.05.19)
Exercise Sheet 3 (due by 21.05.19)
Exercise Sheet 4 (due by 4.06.19)
Exercise Sheet 5 (due by 26.06.19)
Exercise Sheet 6 (due by 10.07.19)
Bonus Sheet (due by 16.07.19)
Recap Sheets
Recap Sheet 1 (handed out on 30.04.19)
Recap Sheet 2 (handed out on 21.05.19)
Recap Sheet 3 (handed out on 4.06.19)
Recap Sheet 4 (handed out on 18.06.19)
Recap Sheet 5 (handed out on 2.07.19)
Recap Sheet 6 (handed out on 17.07.19)
Announcements
The lecture on Tuesday 28.05.2019 is cancelled and will be recovered on Wednesday 22.05.2019, 11:45-13:15, Room M628
The final oral exams (validation as Wahlmodul) will take place on 17.10.2019
NEW: To officially enroll for the final exam, please contact Frau Barjasic (Room F439) before 2.08.2019 .
Last update: 28.09.2019
